A081730 -id:A081730 - cp's OEIS frontend

A035163 Composite numbers k, not a power of 2, such that the E(k) == 1 (mod k), where E(k) is the k-th Euler number (A000364).

15, 91, 289, 319, 435, 561, 692, 703, 1016, 1105, 1369, 1495, 1729, 1885, 1891, 2105, 2465, 2701, 2755, 2821, 3367, 4371, 5551, 6409, 6601, 7456, 8224, 8569, 8695, 8911, 9088, 10585, 10621, 11305, 11849, 12121, 12403, 13981, 14065, 15051, 15841, 16471, 17104

Offset: 1

Benoit Cloitre, Apr 06 2003

nonn

Cf. A000364, A081730.

Select[Range[1000], CompositeQ[#] && #/2^IntegerExponent[#, 2] > 1 && Divisible[Abs[EulerE[2*#]] - 1, #] &] (* Amiram Eldar, Nov 26 2020 *)

a000364(n)=subst(bernpol(2*n+1), 'x, 1/4)*4^(2*n+1)*(-1)^(n+1)/(2*n+1);
lista(nn) = {forcomposite(n=1, nn, if ( n != 2^valuation(n, 2), if (Mod(a000364(n), n) == 1, print1(n, ", "));););} \\ Michel Marcus, Apr 18 2015

More terms from Hans Havermann, Apr 07 2003

a(23)-a(43) from Amiram Eldar, Nov 26 2020

A069042 Numbers k such that A000364(k) == 1 (mod k^2).

Original entry on oeis.org

1, 2, 17, 37, 1153, 1303

Offset: 1

Benoit Cloitre, Apr 06 2003

Cf. A000364, A081730.

Select[Range[1, 10000], Divisible[Abs[EulerE[2*#]] - 1, #^2] &] (* Amiram Eldar, Jun 03 2017 *)

is(k) = Mod(abs(eulerfrac(2*k)), k^2) == 1; \\ Jinyuan Wang, Mar 13 2025

More terms from Hans Havermann, Apr 07 2003

a(1) prepended by Jinyuan Wang, Mar 13 2025

A287934 Composite numbers n such that E(n+1)+1 is divisible by n, where E(n) is the n-th Euler number (A122045).

Original entry on oeis.org

289, 341, 561, 1105, 1369, 1387, 1729, 2465, 2821, 4097, 5365, 6179, 6601, 8911, 9105, 9537, 10585, 12673, 14433, 14531, 15457, 15841, 28033, 29341, 33901, 41041, 41905, 42141, 46657, 48705, 52633, 52741, 62745, 63253, 63973, 75361, 80185, 82621, 99937

Offset: 1

Amiram Eldar, Jun 03 2017

nonn

Kummer proved in 1851 that E(2k + p - 1) == E(2k) (mod p) for k > 0 and all odd primes p. This sequence consists of composite numbers for which the congruence, with k=1, also holds. In terms of A000364, the sequence consists of composite odd numbers n that divide A000364((n + 1)/2) + (-1)^((n + 1)/2).

Jozsef Sandor and Borislav Crstici, Handbook of Number theory II, Kluwer Academic Publishers, 2004, Chapter 5, p. 556.

Leonard Carlitz, Congruences for generalized Bell and Stirling numbers, Duke Mathematical Journal, Vol. 22, No. 2 (1955), pp. 193-205.
Ernst Eduard Kummer, Über eine allgemeine Eigenschaft der rationalen Entwickelungscoefficienten einer bestimmten Gattung analytischer Functionen, Journal für die reine und angewandte Mathematik, Vol. 41 (1851), pp. 368-372.
Samuel S. Wagstaff, Jr., Prime divisors of the Bernoulli and Euler numbers, Number Theory for the Millenium III (Urbana, IL, 2000), AK Peters, Natick, MA, 2002, pp. 357-374.

Cf. A000364, A035163, A081730, A122045, A180942.

a={}; For[n = 1, n < 100000, n++; If[!PrimeQ[n] && Divisible[EulerE[n + 1] + 1, n], a=AppendTo[a,n]]];a
Select[Range[100000],CompositeQ[#]&&Divisible[EulerE[#+1]+1,#]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Dec 03 2019 *)

e(n) = 2^n*2^(n+1)*(subst(bernpol(n+1, x), x, 3/4) - subst(bernpol(n+1, x), x, 1/4))/(n+1);
isok(n) = (((e(n+1)+1) % n) == 0);
lista(nn) = forcomposite(n=1, nn, if (isok(n), print1(n, ", "))); \\ Michel Marcus, Jun 10 2017

cp's OEIS Frontend

A035163 Composite numbers k, not a power of 2, such that the E(k) == 1 (mod k), where E(k) is the k-th Euler number (A000364).

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A069042 Numbers k such that A000364(k) == 1 (mod k^2).

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A287934 Composite numbers n such that E(n+1)+1 is divisible by n, where E(n) is the n-th Euler number (A122045).

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