Search: keyword:new
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A373482
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Numbers k for which A003415(k) is a multiple of A001414(k), where A003415 is the arithmetic derivative, and A001414 is the sum of prime factors with multiplicity.
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0
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1, 4, 6, 8, 9, 10, 14, 15, 16, 21, 22, 25, 26, 27, 32, 33, 34, 35, 36, 38, 39, 46, 49, 51, 55, 57, 58, 62, 64, 65, 69, 72, 74, 77, 81, 82, 85, 86, 87, 91, 93, 94, 95, 100, 106, 111, 112, 115, 118, 119, 121, 122, 123, 125, 126, 128, 129, 133, 134, 141, 142, 143, 145, 146, 155, 156, 158, 159, 161, 166, 169, 177, 178
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OFFSET
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1,2
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LINKS
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MATHEMATICA
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Select[Range[180],
Divisible[If[#1 < 2, 0, #1 Total[#2/#1 & @@@ #2]],
Total[Times @@@ #2]] & @@
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PROG
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CROSSREFS
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After the initial 1, positions of 0's in A373480.
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KEYWORD
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nonn,new
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AUTHOR
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STATUS
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approved
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A373481
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a(n) = 1 if A003415(n) is a multiple of A001414(n), otherwise 0, where A003415 is the arithmetic derivative, and A001414 is the sum of prime factors with multiplicity.
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0
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1, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 1, 0, 1
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OFFSET
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1
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LINKS
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FORMULA
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a(n) = [n==1 or A373480(n)==0], where [ ] is the Iverson bracket.
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MATHEMATICA
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Array[Boole@ Divisible[If[#1 < 2, 0, #1 Total[#2/#1 & @@@ #2]], Total[Times @@@ #2]] & @@ {#, FactorInteger[#]} &, 120] (* Michael De Vlieger, Jun 08 2024 *)
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PROG
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(PARI)
A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
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CROSSREFS
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Characteristic function of A373482.
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KEYWORD
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nonn,new
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AUTHOR
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STATUS
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approved
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1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 2, 1, 0, 0, 0, 1, 5, 1, 6, 0, 0, 1, 8, 0, 0, 0, 10, 1, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 2, 1, 5, 1, 3, 6, 0, 1, 2, 0, 9, 0, 5, 1, 4, 0, 1, 0, 0, 1, 8, 1, 0, 12, 0, 0, 13, 1, 9, 0, 3, 1, 0, 1, 0, 3, 11, 0, 17, 1, 7, 0, 0, 1, 12, 0, 0, 0, 4, 1, 6, 0, 15, 0, 0, 0, 12, 1, 13, 7, 0, 1, 3, 1, 12, 11
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OFFSET
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2,11
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LINKS
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MATHEMATICA
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Array[Mod[If[#1 < 2, 0, #1 Total[#2/#1 & @@@ #2]], Total[Times @@@ #2]] & @@ {#, FactorInteger[#]} &, 120, 2] (* Michael De Vlieger, Jun 08 2024 *)
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PROG
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(PARI)
A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
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CROSSREFS
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KEYWORD
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nonn,new
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AUTHOR
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STATUS
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approved
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A373466
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Palindromes with exactly 6 distinct prime divisors.
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0
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222222, 282282, 414414, 444444, 474474, 555555, 606606, 636636, 646646, 666666, 696696, 828828, 888888, 969969, 2040402, 2065602, 2141412, 2206022, 2343432, 2417142, 2444442, 2572752, 2646462, 2673762, 2747472, 2848482, 2875782, 2949492, 2976792
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OFFSET
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1,1
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COMMENTS
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The term "exactly" clarifies that we don't mean "at least". But the prime divisors may occur to higher powers in the factorization, cf. Examples.
This is different from A046396 which excludes nonsquarefree terms, i.e., terms where one or more of the distinct prime factors occur to a power greater than 1, as it is possible here, cf. Examples.
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LINKS
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FORMULA
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EXAMPLE
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a(1) = 222222 = 2 * 3 * 7 * 11 * 13 * 37 has exactly 6 distinct prime divisors.
a(3) = 414414 = 2 * 3^2 * 7 * 11 * 13 * 23 has 6 distinct prime divisors, even though the factor 3 occurs twice in the factorization.
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MATHEMATICA
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Select[Range[3000000], PalindromeQ[#]&&Length[FactorInteger[#]]==6&] (* James C. McMahon, Jun 08 2024 *)
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PROG
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(PARI) A373466_upto(N, start=1, num_fact=6)={ my(L=List()); while(N >= start = nxt_A002113(start), omega(start)==num_fact && listput(L, start)); L}
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CROSSREFS
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Cf. A046332 (same with bigomega = 6: prime factors counted with multiplicity), A046396 (similar, but squarefree terms only), A373465 (same with omega = 5), A373468 (same with bigomega = 7).
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KEYWORD
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nonn,base,new
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AUTHOR
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STATUS
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approved
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A373501
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Size of the collineation group of projective planes of prime power order q.
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168, 5616, 120960, 372000, 5630688, 49448448, 84913920, 212427600, 810534816, 6950204928, 16934047920, 17108582400, 78156525216, 304668000000, 499631102880, 846083360304, 851974934400, 3509844434208, 5492021821440, 7980059337600, 11681731985616, 23800278205248
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OFFSET
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1,1
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COMMENTS
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a(A246655(n)) is the size of the collineation group of the projective plane of order q=p^k. It is also known as the projective semilinear group, PGammaL(3,q), the semidirect product of PGL(3,q) (whose order is probably given by A003800) with the group of field automorphisms of F(q). The latter is the cyclic group of order k. Therefore, |PGammaL(3,p^k)|=|PGL(3,p^k)|*k.
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REFERENCES
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A. Beutelspacher and U. Rosenbaum, Projective Geometry: From Foundations to Applications, Cambridge University Press, 1998, pages 118-132.
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LINKS
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FORMULA
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a(n) = Omega(q)*(q^3-1)*(q^3-q)*(q^3-q^2)/(q-1) where q=A246655(n).
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EXAMPLE
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Take for example the first value 168 which refers to the number of automorphisms of the Fano plane (q=2). Its v=7 (=q^2+q+1) lines are subsets of size 3 (=q+1) of a set of v points. Using 0,1,...,6 to label these points, one way of enumerating the lines is depicted in the first column of the following table:
(0 1 2 3 4 5 6) (0 6)(3 5)
{0,1,3} {1,2,4} {6,1,5}
{1,2,4} {2,3,5} {1,2,4}
{2,3,5} {3,4,6} {2,5,3}
{3,4,6} {4,5,0} {5,4,0}
{4,5,0} {5,6,1} {4,3,6}
{5,6,1} {6,0,2} {3,0,1}
{6,0,2} {0,1,3} {0,6,2}
Note that any two distinct lines have exactly 1 point in common. Applying one of the 7!=5040 possible permutations of the points obviously doesn't change that fact. However, exactly 168 of these permutations lead to the same set of subsets. One such permutation is the full cycle (0,1,2,3,4,5,6) whose action can bee seen in the second column. It also permutes the lines cyclically by mapping line i to line i+1 (mod v). Another one is the cycle product (0 6)(3 5) in the third column. It swaps lines 1 and 6 and lines 4 and 5 and leaves the other three lines fixed.
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PROG
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(PARI) a=(q)->bigomega(q)*(q^3-1)*(q^3-q)*(q^3-q^2)/(q-1) \\ q=A246655(n)
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CROSSREFS
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Cf. A373502 for the size of a complete set of projective planes using a given set of q^2+q+1 points.
Cf. A335866 for the number of projective planes whose lines are cyclic difference sets.
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KEYWORD
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nonn,new
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AUTHOR
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STATUS
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approved
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A373502
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Size of a complete set of projective planes of prime power order q using a given set of q^2+q+1 points.
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0
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30, 1108800, 422378820864000, 22104404984349254886359040000, 7197507570101063450093594584788274920397007398780859842560000000000000, 90399509839271079668491458784005740889517921781547218950513473999637402251071324160000000000000000
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OFFSET
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1,1
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COMMENTS
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Projective planes of order q can be seen as a set of v=q^2+q+1 subsets (the lines) of size q+1 of a set of v points with the property that any two distinct lines have exactly one point in common. Obviously, this also holds for any of the v! permutations of the points. However, some of these permutations map the points of a given line l of the plane to the points of another line l' thereby fixing the set of lines and consequently the whole projective plane. These permutations form a subgroup called the collineation group or semilinear group PGammaL of the projective plane. The size of this group is given by A373501. Therefore, a(q) is the index of the collineation subgroup in the symmetric group of the points where q=A246655(n).
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REFERENCES
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A. Beutelspacher and U. Rosenbaum, Projective Geometry: From Foundations to Applications, Cambridge University Press, 1998, pages 118-132
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LINKS
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FORMULA
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a(n) = (q^2+q+1)!/(Omega(q)*(q^3-1)*(q^3-q)*(q^3-q^2)/(q-1)) where q=A246655(n)
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EXAMPLE
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For the Fano plane (q=2) there are 7 points and 7 lines. Of the 7!=5040 permutations of the points 168 fix the set of lines and thereby the whole plane. Consequently, there are 5040/168=30 different such planes for any given set of points. See A373501 for a more elaborate discussion of this example.
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PROG
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(PARI) a=(q)->(q^2+q+1)!/(bigomega(q)*(q^3-1)*(q^3-q)*(q^3-q^2)/(q-1)) \\ q=A246655(n)
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CROSSREFS
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Cf. A373501 for the size of the collineation groups.
Cf. A335866 for the number of projective planes whose lines are cyclic difference sets.
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KEYWORD
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nonn,new
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AUTHOR
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STATUS
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approved
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A373527
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Odd numbers k such that k and k+2 both have at least two divisors with the same value of the Euler totient function (A000010).
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+0
0
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2107, 11275, 42651, 68733, 90153, 99123, 123633, 213003, 226825, 242305, 262143, 272853, 292873, 295405, 308007, 313443, 376675, 376803, 378693, 390115, 427425, 471293, 473263, 524797, 525481, 556983, 579535, 591325, 618469, 638163, 663325, 669123, 699853, 731815
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OFFSET
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1,1
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COMMENTS
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Numbers k such that k and k+2 are both in A359563.
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LINKS
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MATHEMATICA
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q[n_] := q[n] = UnsameQ @@ EulerPhi[Divisors[n]]; Select[Range[1, 10^6, 2], ! q[#] && ! q[# + 2] &]
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PROG
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(PARI) is(k) = k>1 && k%2 && numdiv(k) > #Set(apply(x->eulerphi(x), divisors(k)));
lista(kmax) = {my(q1 = 0, q2); forstep(k = 3, kmax, 2, q2 = is(k); if(q1 && q2, print1(k-2, ", ")); q1 = q2); }
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CROSSREFS
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KEYWORD
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nonn,new
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AUTHOR
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STATUS
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approved
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A373530
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Numbers k such that k, k+1 and k+2 all have at least three divisors with the same value of the Euler totient function (A000010).
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+0
0
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14052608, 83025998, 87703714, 93978520, 117345150, 163338174, 213589088, 218539880, 294321950, 369698434, 401177798, 463425920, 470217824, 497434040, 529524918, 539318438, 554556078, 559474838, 581302358, 584754848, 608842934, 612448640, 617445814, 625591966
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OFFSET
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1,1
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COMMENTS
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Numbers k such that k, k+1 and k+2 are all in A359565.
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LINKS
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MATHEMATICA
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q[n_] := Max[Tally[EulerPhi[Divisors[n]]][[;; , 2]]] > 2; seq[kmax_] := Module[{s = {}, q1 = 0, q2 = 0, q3}, Do[q3 = q[k]; If[q1 && q2 && q3, AppendTo[s, k-2]]; q1=q2; q2=q3, {k, 3, kmax}]; s]; seq[10^8]
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PROG
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(PARI) is(k) = vecmax(matreduce(apply(x->eulerphi(x), divisors(k)))[, 2]) > 2;
lista(kmax) = {my(q1 = 0, q2 = 0, q3); for(k = 3, kmax, q3 = is(k); if(q1 && q2 && q3, print1(k-2, ", ")); q1 = q2; q2 = q3); }
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CROSSREFS
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KEYWORD
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nonn,new
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AUTHOR
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STATUS
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approved
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A373529
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Numbers k such that k and k+1 both have at least three divisors with the same value of the Euler totient function (A000010).
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+0
0
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32319, 111320, 175959, 179360, 191919, 212120, 246519, 254079, 254960, 279279, 319599, 355508, 357399, 398600, 436149, 463239, 512000, 520064, 524799, 542240, 580040, 606879, 657152, 678699, 685880, 701631, 718640, 726920, 739556, 750519, 759759, 775775, 787815
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OFFSET
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1,1
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COMMENTS
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Numbers k such that k and k+1 are both in A359565.
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LINKS
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MATHEMATICA
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q[n_] := q[n] = Max[Tally[EulerPhi[Divisors[n]]][[;; , 2]]] > 2; Select[Range[3*10^6], q[#] && q[# + 1] &]
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PROG
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(PARI) is(k) = vecmax(matreduce(apply(x->eulerphi(x), divisors(k)))[, 2]) > 2;
lista(kmax) = {my(q1 = 0, q2); for(k = 2, kmax, q2 = is(k); if(q1 && q2, print1(k-1, ", ")); q1 = q2); }
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CROSSREFS
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KEYWORD
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nonn,new
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AUTHOR
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STATUS
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approved
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A373528
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Odd numbers k such that k, k+2 and k+4 all have at least two divisors with the same value of the Euler totient function (A000010).
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0
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4142435, 26196331, 77118741, 89690821, 102974571, 196054673, 201060275, 206568171, 277322153, 280039833, 401784953, 402492695, 415097613, 437290371, 515636303, 526721895, 534746581, 549806211, 575090395, 580329603, 625833871, 629588043, 702183625, 710983971, 716133481
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OFFSET
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1,1
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COMMENTS
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Numbers k such that k, k+2 and k+4 are all in A359563.
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LINKS
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MATHEMATICA
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q[n_] := !UnsameQ @@ EulerPhi[Divisors[n]]; seq[kmax_] := Module[{tri = q /@ {1, 3, 5}, s = {}, k = 7}, While[k < kmax, If[And @@ tri, AppendTo[s, k - 6]]; tri = Join[Rest[tri], {q[k]}]; k+=2]; s]; seq[3*10^7]
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PROG
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(PARI) is(k) = k>1 && k%2 && numdiv(k) > #Set(apply(x->eulerphi(x), divisors(k)));
lista(kmax) = {my(q1 = 0, q2 = 0, q3); forstep(k = 5, kmax, 2, q3 = is(k); if(q1 && q2 && q3, print1(k-4, ", ")); q1 = q2; q2 = q3); }
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CROSSREFS
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KEYWORD
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nonn,new
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AUTHOR
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STATUS
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approved
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