A079612 -id:A079612 - cp's OEIS frontend

A309906 a(n) is the smallest number of divisors of p^n - 1 that may possibly occur for arbitrarily large primes p.

4, 32, 8, 160, 8, 384, 8, 384, 16, 256, 8, 7680, 8, 128, 32, 1792, 8, 4096, 8, 3840, 32, 256, 8, 36864, 16, 128, 32, 2560, 8, 24576, 8, 4096, 32, 128, 32, 327680, 8, 128, 32, 36864, 8, 18432, 8, 2560, 128, 256, 8, 344064, 16, 1024, 32, 2560, 8, 20480, 32

Offset: 1

Jon E. Schoenfield, Aug 21 2019

nonn

The existence of infinitely many primes p such that p^n - 1 has exactly a(n) divisors is conjectured. E.g., although it is known that p-1 has fewer than 4 divisors for only finitely many primes p (see Example section), it is not known whether there exist infinitely many primes p such that p-1 has exactly 4 divisors. (Thanks to Jianing Song, who pointed out the need for this clarification.) - Jon E. Schoenfield, Mar 04 2021
For each prime q, every number k that has exactly q divisors is a prime power k = p^(q-1) for some prime p. As a result, a(q-1) can be useful in identifying numbers of the form p^(q-1) - 1 that are terms of A161460 (see Example section).
From Bernard Schott, Aug 22 2019: (Start)
For n prime >= 3, a(n) = 8;
for n = q^2, q prime >= 3, a(n) = 16. (End)

			a(1) = 4: The only primes p for which p-1 has fewer than 4 divisors are 2, 3, and 5; for all primes p > 5, p-1 has at least 4 divisors, and the terms in A005385 (Safe primes) except 5 are primes p such that p-1 has exactly 4 divisors.
a(2) = 32: p^2 - 1 = (p-1)*(p+1) has fewer than 32 divisors only for p = 2, 3, 5, 7, 11, 13, 17, 19, 23, 31, 37, 47, and 73; for all primes p such that the product of the 3-smooth parts of p-1 and p+1 is 24 and p-1 and p+1 each have one prime factor > 3, p^2 - 1 has exactly 32 divisors (see A341658).
a(4) = 160: primes p such that p^4 - 1 has exactly 160 divisors are plentiful (see A341662), but only p = 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 59, 61, 71, 79, and 101 yield tau(p^4 - 1) < 160. Of these, p = 13, 29, 59, and 61 all give tau(p^4 - 1) = 80; 37 and 101 both give 120 divisors; and 41 and 71 both give 144. For each of the ten remaining primes (p = 2, 3, 5, 7, 11, 17, 19, 23, 31, 79), the value of tau(p^4 - 1) is unique, so each of those ten values of p^4 - 1 is a term in A161460.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A000005, A005385, A006863, A079612, A161460, A341658, A341662, A341670.

f(n) = {if (n%2, 2, res = 1; forprime(p=2, n+1, if (!(n % (p-1)), t = valuation(n, p); if (p==2, if (t, res *= p^(t+2)), res *= p^(t+1)); ); ); res; ); } \\ A079612
a(n) = numdiv(f(n))*2^numdiv(n); \\ Michel Marcus, Aug 22 2019

a(n) = A000005(A079612(n))*2^A000005(n).

a(n) = 2^(A000005(n)+1) for odd n. - Jianing Song, Dec 05 2021

Name edited by Jon E. Schoenfield, Mar 04 2021

A006863 Denominator of B_{2n}/(-4n), where B_m are the Bernoulli numbers.

Original entry on oeis.org

1, 24, 240, 504, 480, 264, 65520, 24, 16320, 28728, 13200, 552, 131040, 24, 6960, 171864, 32640, 24, 138181680, 24, 1082400, 151704, 5520, 1128, 4455360, 264, 12720, 86184, 13920, 1416, 6814407600, 24

Offset: 0

N. J. A. Sloane, Jeffrey Shallit, Simon Plouffe

Carmichael defines lambda(n) to be the exponent of the group U(n) of units of the integers mod n. He shows that given m there is a number lambda^*(m) such that lambda(n) divides m if and only if n divides lambda^*(m). He gives a formula for lambda^*(m), equivalent to the one I've quoted for even m. (We have lambda^*(m)=2 for any odd m.) The present sequence gives the values of lambda^*(2m) for positive integers m. - Peter J. Cameron, Mar 25 2002
(-1)^n*B_{2n}/(-4n) = Integral_{t>=0} t^(2n-1)/(exp(2*Pi*t) - 1)dt. - Benoit Cloitre, Apr 04 2002
Michael Lugo (see link) conjectures, and Peter McNamara proves, that a(n) = gcd_{ primes p > 2n+1 } (p^(2n) - 1). - Tanya Khovanova, Feb 21 2009 [edited by Charles R Greathouse IV, Dec 03 2014]

Bruce Berndt, Ramanujan's Notebooks Part II, Springer-Verlag; see Integrals and Asymptotic Expansions, p. 220.
F. Hirzebruch et al., Manifolds and Modular Forms, Vieweg, 2nd ed. 1994, p. 130.
J. W. Milnor and J. D. Stasheff, Characteristic Classes, Princeton, 1974, p. 286.
Douglas C. Ravenel, Complex cobordism theory for number theorists, Lecture Notes in Mathematics, 1326, Springer-Verlag, Berlin-New York, 1988, pp. 123-133.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
R. C. Vaughan and T. D. Wooley, Waring's problem: a survey, pp. 285-324 of Surveys in Number Theory (Urbana, May 21, 2000), ed. M. A. Bennett et al., Peters, 2003. (The function K(2n), see p. 303.)

T. D. Noe, Table of n, a(n) for n = 0..10000
P. J. Cameron and D. A. Preece, Notes on primitive lambda-roots
R. D. Carmichael, Note on a new number theory function, Bull. Amer. Math. Soc. 16 (1909-10), 232-238.
G. Everest, Y. Puri and T. Ward, Integer sequences counting periodic points, arXiv:math/0204173 [math.NT], 2002.
Michael Lugo, A little number theory problem (2008)
Eric Weisstein's World of Mathematics, Eisenstein Series.
Index entries for sequences related to Bernoulli numbers.

Numerators are A001067.

Cf. A000367/A002445, A002322, A079612.

Concatenation([1], List([1..35], n-> DenominatorRat(Bernoulli(2*n)/(-4*n)) )); # G. C. Greubel, Sep 19 2019

[1] cat [Denominator(Bernoulli(2*n)/(-4*n)):n in [1..35]]; // G. C. Greubel, Sep 19 2019

1,seq(denom(bernoulli(2*n)/(-4*n)), n=1 .. 100); # Robert Israel, Dec 03 2014

a[n_] := Denominator[BernoulliB[2n]/(-4n)]; Table[a[n], {n, 0, 31}] (* Jean-François Alcover, Mar 20 2011 *)

a(n) = if (n == 0, 1, denominator(bernfrac(2*n)/(-4*n))); \\ Michel Marcus, Sep 10 2013

[1]+[denominator(bernoulli(2*n)/(-4*n)) for n in (1..35)] # G. C. Greubel, Sep 19 2019

B_{2k}/(4k) = -(1/2)*zeta(1-2k). For n > 0, a(n) = gcd k^L (k^{2n}-1) where k ranges over all the integers and L is as large as necessary.

Product of 2^{a+2} (where 2^a exactly divides 2*n) and p^{a+1} (where p is an odd prime such that p-1 divides 2*n and p^a exactly divides 2*n). - Peter J. Cameron, Mar 25 2002

Thanks to Michael Somos for helpful comments.

A075180 Denominators from e.g.f. 1/(1-exp(-x)) - 1/x.

Original entry on oeis.org

2, 12, 1, 120, 1, 252, 1, 240, 1, 132, 1, 32760, 1, 12, 1, 8160, 1, 14364, 1, 6600, 1, 276, 1, 65520, 1, 12, 1, 3480, 1, 85932, 1, 16320, 1, 12, 1, 69090840, 1, 12, 1, 541200, 1, 75852, 1, 2760, 1, 564, 1, 2227680, 1, 132, 1, 6360, 1, 43092, 1, 6960, 1, 708, 1, 3407203800, 1, 12, 1, 32640, 1, 388332, 1, 120, 1, 9372, 1, 10087262640, 1, 12

Offset: 0

Wolfdieter Lang, Sep 06 2002

Denominators of -zeta(-n), n >= 0, where zeta is Riemann's zeta function.

Numerators are +1, A060054(n+1), n >= 1.

			1/2, 1/12, 0, -1/120, 0, 1/252, 0, -1/240, 0, 1/132, 0, -691/32760, ...

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 807, combined eqs. 23.2.11,14 and 15.

Antti Karttunen, Table of n, a(n) for n = 0..16384
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 807, combined eqs. 23.2.11,14 and 15.

Cf. A060054, A006232 with A006233.

Cf. A006953, A006863, A079612, A242179, A106831, A000142, A300711.

a075180 n = a075180_list !! n
a075180_list = map (denominator . sum) $ zipWith (zipWith (%))
   (zipWith (map . (*)) a000142_list a242179_tabf) a106831_tabf
-- Reinhard Zumkeller, Jul 04 2014

a := n -> denom(bernoulli(n+1,1)/(n+1)); # Peter Luschny, Apr 22 2009

a[m_] := Sum[(-2)^(-k-1) k! StirlingS2[m,k],{k,0,m}]/(2^(m+1)-1); Table[Denominator[a[i]], {i,0,20}] (* Peter Luschny, Apr 29 2009 *)
Table[Denominator[Zeta[-n]], {n, 0, 49}] (* Alonso del Arte, Jan 13 2012 *)
CoefficientList[ Series[ EulerGamma - HarmonicNumber[n] + Log[n], {n, Infinity, 48}], 1/n] // Rest // Denominator (* Jean-François Alcover, Mar 28 2013 *)
With[{nn=50},Denominator[CoefficientList[Series[1/(1-Exp[-x])-1/x,{x,0,nn}],x] Range[0,nn-1]!]] (* Harvey P. Dale, Apr 13 2016 *)

x='x+O('x^66);
egf = 1/(1-exp(-x)) - 1/x;
v=Vec(serlaplace(egf));
vector(#v,n, denominator(v[n]))
/* Joerg Arndt, Mar 28 2013 */

A075180(n) = denominator(bernfrac(n+1)/(n+1)); \\ Antti Karttunen, Dec 19 2018, after Maple-program.

a(n) = denominator(-Zeta(-n)) = denominator(((-1)^(n+1))*B(n+1)/(n+1)), n >= 0, with Riemann's zeta function and the Bernoulli numbers B(n).
a(n) = denominators from e.g.f. (B(-x) - 1)/x, with B(x) = x/(exp(x) - 1), e.g.f. for Bernoulli numbers A027641(n)/A027642(n), n >= 0.
From Jianing Song, Apr 05 2021: (Start)
a(2n-1) = A006863(n)/2 for n > 0. By the comments in A006863, A006863(n) = A079612(2n) for n > 0. Hence a(n) = A079612(n+1)/2 all odd n. For all even n > 0, we have a(n) = 1, which is also equal to A079612(n+1)/2.
For odd n, a(n) is the product of p^(e+1) where p^e*(p-1) divides n+1 but p^(e+1)*(p-1) does not. For example, a(11) = 2^3 * 3^2 * 5^1 * 7^1 * 13^1 = 32760.
a(2n-1) = A002445(n)*(2n)/A300711(n), n > 0. (End)
a(2*n-1) = A006953(n) for n >= 1. - Georg Fischer, Dec 01 2022

More terms from Antti Karttunen, Dec 19 2018

A115000 a(n) = J_2(n) / 24.

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 5, 4, 7, 6, 8, 8, 12, 9, 15, 12, 16, 15, 22, 16, 25, 21, 27, 24, 35, 24, 40, 32, 40, 36, 48, 36, 57, 45, 56, 48, 70, 48, 77, 60, 72, 66, 92, 64, 98, 75, 96, 84, 117, 81, 120, 96, 120, 105, 145, 96, 155, 120, 144, 128, 168, 120, 187, 144, 176, 144, 210, 144

Offset: 5

Franklin T. Adams-Watters, Dec 10 2005

nonn

The Jordan function J_m(n) can be defined as multiplicative with J_m(p^e) = (p^m-1)*p^(m*(e-1)). Cf. A059379.
Looking at the sequences J_m(n) for fixed m, one is struck by the fact that all but a few early terms have a common factor, given in A079612. I will refer to this sequence as K(n), following the notation in the paper by Vaughan and Wooley. (The alternate lambda^*(n) in the comment for A006863 is too awkward.)
In fact, K(m) not only divides J_m(n) for all but finitely many n; it also divides Sum_{k=1..n} J_m(k) for all but finitely many n.
J_1(n) = phi(n) and phi(n)/2 and Sum_{k=1..n} phi(n)/2 are A023022 and A046657.
The weight of the n-th elliptic division polynomial -- the analog of cyclotomic polynomials for elliptic divisibility sequences. That is, let e1 = b1, e2 = b2*b1, e3 = b3*b1, e4 = b2*b4*b1, e5 = (b2^4*b4 - b3^3)*b1 = b5*e1 and so on be an elliptic divisibility sequence. Let c2 = b2^4*b4, c3 = b3^3, c4 = b4^2 and cn = bn for n>4. Then c5 = c2 - c3, c6 = c5 - c4, c7 = c6*c3 - c5*c4 and so on. Let the weight of c2, c3, c4 each be 1 and weight of a product is sum of the weights of the factors. The weight of cn is a(n) for n>4. - Michael Somos, Aug 12 2008

			G.f.: x^5 + x^6 + 2*x^7 + 2*x^8 + 3*x^9 + 3*x^10 + 5*x^11 + 4*x^12 + 7*x^13 + ...

G. C. Greubel, Table of n, a(n) for n = 5..5000
Index to divisibility sequences

Cf. A007434.

function a(n) return n lt 5 select 0 else Dimension( ModularForms( Gamma1(n), 2)) - Dimension( ModularForms( Gamma1(n), 1)); end function; /* Michael Somos, Aug 05 2014 */

a[n_] := DivisorSum[n, #^2*MoebiusMu[n/#]&]/24; Table[a[n], {n, 5, 80}] (* Jean-François Alcover, Dec 07 2015, adapted from PARI *)

{a(n) = if( n<5, 0, sumdiv(n, d, d^2 * moebius(n / d)) / 24)}; /* Michael Somos, Aug 12 2008 */

A007434(n) = 24 * a(n) unless n<5. - Michael Somos, Aug 12 2008

More terms from Michael Somos, Aug 12 2008

A143408 Number of numbers k such that the reduced totient function psi(k) = A002174(n).

Original entry on oeis.org

2, 6, 12, 16, 4, 8, 84, 32, 40, 32, 8, 20, 20, 64, 8, 480, 80, 48, 12, 8, 160, 20, 16, 4, 8, 1216, 8, 64, 16, 872, 24, 160, 8, 532, 52, 120, 12, 424, 100, 24, 4, 8, 944, 24, 144, 12, 1912, 272, 8, 16, 276, 24, 64, 144, 1856, 20, 96, 1276, 40, 112, 12, 8, 116, 20, 16, 96, 8

Offset: 1

T. D. Noe, Aug 13 2008

nonn

a(n) is the number of divisors of A143407(n) that are not divisors of A143407(r) for r

			Because A002174(5)=8 and psi(k)=8 for k=32,96,160,480, we have a(5)=4.

T. D. Noe, Table of n, a(n) for n = 1..500
P. J. Cameron and D. A. Preece, Notes on primitive lambda-roots, 2009. See formula in theorem 5.2 (c) p. 8.

Cf. A002322 (reduced totient function), A079612.

a079612(n) = {if (n%2, 2, res = 1; forprime(p=2, n+1, if (!(n % (p-1)), t = valuation(n, p); if (p==2, if (t, res *= p^(t+2)), res *= p^(t+1)););); res;);}
nb(n) = sumdiv(n, d, moebius(n/d)*numdiv(a079612(d)));
lista(nn) = for (n=1, nn, if (nbs = nb(n), print1(nbs, ", "))); \\ Michel Marcus, May 12 2018

A330541 Triangle read by rows: T(n,k) = gcd {x^n - x^k : x is an integer}, 0 < k < n.

Original entry on oeis.org

2, 6, 2, 2, 12, 2, 30, 2, 24, 2, 2, 60, 2, 24, 2, 42, 2, 120, 2, 24, 2, 2, 252, 2, 240, 2, 24, 2, 30, 2, 504, 2, 240, 2, 24, 2, 2, 60, 2, 504, 2, 240, 2, 24, 2, 66, 2, 120, 2, 504, 2, 240, 2, 24, 2, 2, 132, 2, 240, 2, 504, 2, 240, 2, 24, 2

Offset: 2

Peter Kagey, Dec 17 2019

All diagonals are weakly increasing, T(n,k) divides T(n+1,k+1), and the m-th diagonal converges to A079612(m).
First column is A027760.
First value where T(n,k) < gcd(2^n - 2^k, 3^n - 3^k) is T(12,1) = 2 < 46.
Maximum value in the n-th row is given by A330542(n).

			Table begins:
  n\k|  1    2    3    4    5    6    7    8   9  10 11
  ---+-------------------------------------------------
   2 |  2;
   3 |  6,   2;
   4 |  2,  12,   2;
   5 | 30,   2,  24,   2;
   6 |  2,  60,   2,  24,   2;
   7 | 42,   2, 120,   2,  24,   2;
   8 |  2, 252,   2, 240,   2,  24,   2;
   9 | 30,   2, 504,   2, 240,   2,  24,   2;
  10 |  2,  60,   2, 504,   2, 240,   2,  24,  2;
  11 | 66,   2, 120,   2, 504,   2, 240,   2, 24,  2;
  12 |  2, 132,   2, 240,   2, 504,   2, 240,  2, 24, 2.

Peter Kagey, Table of n, a(n) for n = 2..10012 (first 141 rows, flattened)
Mathematics Stack Exchange, Computing gcd {n^k - n^l : n in Z}.

Cf. A027760, A079612, A330542.

A358625 a(n) = numerator of Bernoulli(n, 1) / n for n >= 1, a(0) = 1.

Original entry on oeis.org

1, 1, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -691, 0, 1, 0, -3617, 0, 43867, 0, -174611, 0, 77683, 0, -236364091, 0, 657931, 0, -3392780147, 0, 1723168255201, 0, -7709321041217, 0, 151628697551, 0, -26315271553053477373, 0, 154210205991661, 0, -261082718496449122051

Offset: 0

Peter Luschny, Dec 02 2022

The rational numbers r(n) = Bernoulli(n, 1) / n are called the 'divided Bernoulli numbers'. r(n) is a p-integer for all primes p if p - 1 does not divide n. This is sometimes called 'Adams's theorem' (Ireland and Rosen). The important Kummer congruences for the Bernoulli numbers (1851) are stated in terms of the r(n).

			Rationals: 1, 1/2, 1/12, 0, -1/120, 0, 1/252, 0, -1/240, 0, 1/132, ...
Note that a(68) = -4633713579924631067171126424027918014373353 but A120082(68) = -125235502160125163977598011460214000388469.

Kenneth Ireland and Michael Rosen, A classical introduction to modern number theory, Vol. 84, Graduate Texts in Mathematics, Springer-Verlag, 2nd edition, 1990. [Prop. 15.2.4, p. 238]

Peter Luschny, Table of n, a(n) for n = 0..300
Arnold Adelberg, Shaofang Hong and Wenli Ren, Bounds of Divided Universal Bernoulli Numbers and Universal Kummer Congruences, Proceedings of the American Mathematical Society, Vol. 136(1), 2008, p. 61-71.
Bernd C. Kellner, The structure of Bernoulli numbers, arXiv:math/0411498 [math.NT], 2004.

Cf. A001067, A342318, A120080, A120082, A120084, A120086, A079612, A075180, A060054.
Cf. A164555 / A027642.
For the denominators cf. A006953, A036283, A053657, A202318.

Concatenation([1, 1], List([2..45], n-> NumeratorRat(Bernoulli(n)/(n)))); # G. C. Greubel, Sep 19 2019

[1, 1] cat [Numerator(Bernoulli(n)/(n)): n in [2..45]]; // G. C. Greubel, Sep 19 2019

A358625 := n -> ifelse(n = 0, 1, numer(bernoulli(n, 1) / n)):
seq(A358625(n), n = 0.. 40);
# Alternative:
egf := 1 + x + log(1 - exp(-x)) - log(x): ser := series(egf, x, 42):
seq(numer(n! * coeff(ser, x, n)), n = 0..40);

Join[{1, 1}, Table[Numerator[BernoulliB[n] / n], {n, 2, 45}]]

a(n) = if (n<=1, 1, numerator(bernfrac(n)/n)); \\ Michel Marcus, Feb 24 2015

a(n) = numerator(n! * [x^n](1 + x + log(1 - exp(-x)) - log(x))).
a(n) = numerator(-zeta(1 - n)) for n >= 1.
a(n) = numerator(Euler(n-1, 1) / (2*(2^n - 1))) for n >= 1.
denominator(r(2*n)) = A006953(n) for n >= 1.
denominator(r(2*n)) / 2 = A036283(n) for n >= 1.
denominator(r(2*n)) / 12 = A202318(n) for n >= 1.
denominator(r(2*n)) = (1/2) * A053657(2*n+1) / A053657(2*n-1) for n >= 1.

A308357 Smallest k such that k! can be represented as the sum of the n-th powers of two or more distinct primes; or -1 if no such k exists.

Original entry on oeis.org

2, 4, 8, 10, 12, 15, 19, 20, 24, 25

Offset: 0

Dmitry Kamenetsky, May 21 2019

If a(10)..a(14) exist then a(10) > 26, a(11) > 28, a(12) > 30, a(13) > 32, a(14) > 33.
From Jon E. Schoenfield, Jun 07 2019: (Start)
If such a number k exists for n=8, then a(8) > 23.
If a set of distinct primes whose 8th powers sum to 24! exists (i.e., if a(8)=24), it must consist of exactly 96 primes: p=5 and exactly 95 primes p > 5. Additionally, if we let j be the number of primes p satisfying p^8 mod 17 = 1 among those 95 primes > 5, it can be shown that:
- if 17 is one of the 95 primes, j must be either 39 or 56;
- otherwise, j must be either 31 or 48.
If such a number k exists for n=12, a(12) > 43.
(For proofs and additional notes, see the Links.) (End)

			a(0) = 2, because 2! = 2 = 2^0 + 3^0.
a(1) = 4, because 4! = 24 = 11^1 + 13^1.
a(2) = 8, because 8! = 40320 = 2^2 + 3^2 + 5^2 + 7^2 + 11^2 + 13^2 + 17^2 + 19^2 + 23^2 + 29^2 + 41^2 + 59^2 + 181^2.
a(3) = 10, because 10! = 3628800 = 5^3 + 19^3 + 29^3 + 37^3 + 47^3 + 151^3.
a(4) = 12, because 12! = 479001600 = 3^4 + 5^4 + 7^4 + 11^4 + 17^4 + 19^4 + 29^4 + 31^4 + 37^4 + 47^4 + 53^4 + 59^4 + 73^4 + 79^4 + 97^4 + 131^4.
a(5) = 15, because 15! = 13^5 + 17^5 + 19^5 + 31^5 + 37^5 + 41^5 + 53^5 + 61^5 + 89^5 + 97^5 + 139^5 + 163^5 + 199^5 + 241^5.
a(6) = 19, because 19! is the sum of the 6th powers of the primes in {3, 7, 17, 23, 37, 43, 47, 53, 61, 71, 73, 79, 89, 101, 103, 107, 113, 127, 137, 157, 167, 193, 211, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463}.
a(7) = 20, because 20! is the sum of the 7th powers of the primes in {5, 13, 31, 43, 59, 67, 71, 83, 97, 103, 109, 113, 137, 149, 167, 179, 181, 191, 193, 197, 227, 229, 233, 239, 241, 257, 263, 269, 277, 281, 283, 293, 311, 313, 317, 331}.
Note that these are the smallest k for which such a representation is possible.

dxdy forum, post (in Russian).
Dmitry Kamenetsky, Solutions for n >= 8
Carlos Rivera, Puzzle 964: A308357, The Prime Puzzles and Problems Connection.
Carlos Rivera, Puzzle 988: Another puzzle about factorials, The Prime Puzzles and Problems Connection.
Jon E. Schoenfield, A proof that a(8) > 23 (if a(8) != -1) and additional notes

Cf. A308459, A079612.

A349937 Odd numbers k > 1 such that A309906(k-1) < A309906(k) > A309906(k+1).

Original entry on oeis.org

315, 525, 693, 765, 825, 945, 1125, 1155, 1323, 1395, 1575, 1683, 1725, 1755, 1785, 1815, 1845, 1995, 2205, 2275, 2277, 2415, 2457, 2475, 2535, 2565, 2691, 2695, 2793, 2805, 2835, 2907, 3003, 3045, 3285, 3315, 3375, 3465, 3591, 3645, 3675, 3705, 3735, 3825, 3885

Offset: 1

Jianing Song, Dec 05 2021

nonn

Conjecturally, odd numbers k > 1 such that liminf_{n->oo} d(p(n)^(k-1)-1) < liminf_{n->oo} d(p(n)^k-1) > liminf_{n->oo} d(p(n)^(k+1)-1), where p(n) = prime(n), d = A000005.

If k is odd, then A079612(k) = 2, so A309906(k) is usually smaller than either A309906(k-1) or A309906(k+1) (or both). This sequence lists the exceptions.

			A309906(314) = 128 < A309906(315) = 8192 > A309906(316) = 2560, so 315 is a term.

Cf. A309906, A349938, A349941.

isA349937(k) = (k%2&&k>1) && A309906(k)>A309906(k-1) && A309906(k)>A309906(k+1) \\ See A309906 for its program

A345262 a(n) is the order of the image of the J-homomorphism in the stable homotopy groups of spheres.

Original entry on oeis.org

1, 2, 1, 24, 1, 1, 1, 240, 2, 2, 1, 504, 1, 1, 1, 480, 2, 2, 1, 264, 1, 1, 1, 65520, 2, 2, 1, 24, 1, 1, 1, 16320, 2, 2, 1, 28728, 1, 1, 1, 13200, 2, 2, 1, 552, 1, 1, 1, 131040, 2, 2, 1, 24, 1, 1, 1, 6960, 2, 2, 1, 171864, 1, 1, 1, 32640, 2, 2, 1, 24, 1, 1, 1

Offset: 0

Tom Harris, Jun 12 2021

Im(J) is a finite cyclic subgroup of Pi_n^S and has known order a(n) calculated by Adams using the Adams conjecture, subsequently proven by Quillen. When n is 3 or 7 mod 8 the value a(n) is related to the Bernoulli numbers; the other values of a(n) are 8-periodic (after an exceptional n=0).

D. Ravenel, Complex cobordism and stable homotopy groups of spheres (2ed), AMS Chelsea Publishing, (2003), ISBN: 978-0-8218-2967-7.

J.F. Adams, On the groups J(x), IV, Topology 5 (1966), 21-71.
D.G. Quillen, The Adams conjecture, Topology 10 (1971), 1-10.

Cf. A006863, A079612. Divides A048648.

from sympy import bernoulli
def a(n):
    if n == 0:
        return 1
    n_ = n % 8
    d = {0:2, 1:2, 2:1, 4:1, 5:1, 6:1}
    if n_ in [3, 7]:
        k = (n+1)//4
        return (bernoulli(2*k)/(4*k)).denominator
    else:
        return d[n_]

a(n) is:
                 2 if n = 0 or 1 mod 8 (except a(0) = 1)
                 1 if n = 2, 4, 5 or 6 mod 8
  A006863((n+1)/4) if n = 3 or 7 mod 8.
(A006863(k) = denominator of B_2k/4k, where B_m are the Bernoulli numbers.)

Showing 1-10 of 10 results.

cp's OEIS Frontend

A309906 a(n) is the smallest number of divisors of p^n - 1 that may possibly occur for arbitrarily large primes p.

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A006863 Denominator of B_{2n}/(-4n), where B_m are the Bernoulli numbers.

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A075180 Denominators from e.g.f. 1/(1-exp(-x)) - 1/x.

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A115000 a(n) = J_2(n) / 24.

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A143408 Number of numbers k such that the reduced totient function psi(k) = A002174(n).

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A330541 Triangle read by rows: T(n,k) = gcd {x^n - x^k : x is an integer}, 0 < k < n.

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A358625 a(n) = numerator of Bernoulli(n, 1) / n for n >= 1, a(0) = 1.