A246006 -id:A246006 - cp's OEIS frontend

A241601 Largest divisor of A246006(n) whose prime factors are all >= n+2.

1, 1, 1, 1, 1, 1, 1, 61, 1, 277, 1, 50521, 691, 41581, 1, 199360981, 3617, 228135437, 43867, 2404879675441, 174611, 14814847529501, 77683, 69348874393137901, 236364091, 238685140977801337, 657931, 4087072509293123892361, 3392780147, 454540704683713199807

Offset: 0

Eric Chen, Dec 15 2014

nonn

Notice: Not all a(n) are 1 or primes, the first example is a(11) = 50521, it equals 19*2659.
a(2n) is a product of powers of Bernoulli irregular primes (A000928), with the exception of n = 0,1,2,3,4,5,7.
a(2n+1) is a product of powers of Euler irregular primes (A120337), with the exception of n = 0,1,2.
Conjectures: All terms are squarefree, and there are infinitely many n such that a(n) is prime.
a(n) = 1 iff n is in the set {0, 1, 2, 3, 4, 5, 6, 8, 10, 14}.
a(n) is prime for n = {7, 9, 12, 16, 17, 18, 26, 34, 36, 38, 39, 42, 49, 74, 114, 118, ...}.
All prime factors of a(n) are irregular primes (Bernoulli or Euler) and with an irregular pair to n: (61, 7), (277, 9), (19, 11), (2659, 11), (691, 12), (43, 13), (967, 13), (47, 15), (4241723, 15), (3617, 16), (228135437, 17), (43867, 18), (79, 19), (349, 19), (84224971, 19), ...
Number of ns such that a prime p divides a(n) is the irregular index of p, for example, 67 divides both a(27) and a(58), so it has irregular index two.
a(149) is the first a(n) which is not completely factored (with a 202-digit composite remaining).

Cf. A246006.
Cf. A000928, A120337.
Cf. A001067, A120082, A141590, A060054, A091912.

b[n_] := Numerator[BernoulliB[2 n]/(2 n)];
c[n_] := Numerator[SeriesCoefficient[Log[Tan[x]+1/Cos[x]], {x, 0, 2n+1}]];
a[0] = 1; a[n_] := If[EvenQ[n], b[n/2] // Abs, c[(n-1)/2]];
Table[a[n], {n, 0, 29}] (* Jean-François Alcover, Jul 03 2019 *)

a(2n) = |A001067(n)| = |A120082(2n)| = |A141590(n)| = |A060054(n)|.

a(2n+1) = A091912(n).

A249909 Smallest prime factor of A241601(n), or 1 if A241601(n) = 1.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 61, 1, 277, 1, 19, 691, 43, 1, 47, 3617, 228135437, 43867, 79, 283, 41737, 131, 31, 103, 2137, 657931, 67, 9349, 71, 1721, 15669721, 37, 930157, 151628697551, 4153, 26315271553053477373, 9257, 154210205991661, 23489580527043108252017828576198947741, 137616929, 763601

Offset: 0

Eric Chen, Dec 15 2014

nonn

Also the smallest prime factor of A246006(n) that is >= n+2.
a(n) = A020639(A241601(n)).
a(n) = 1 iff n is in the set {0, 1, 2, 3, 4, 5, 6, 8, 10, 14}.
a(189) is currently unknown; a(190)..a(199) = {5101, 559570609330768709, 40833790860803270336710504624737304862569304959957, 311, 467, 34110029, 461, 26034939865747697437451558982836040663625026070193, 34470847, 1879}.
All terms are Bernoulli or Euler irregular primes.

Eric Chen, Table of n, a(n) for n = 0..188
Samuel Wagstaff, Factorization of Bernoulli and Euler numbers

a246006[n_] := If[EvenQ[n], Abs[Numerator[BernoulliB[n]]], Abs[EulerE[n-1]]];
a241601[n_] := a246006[n]/GCD[a246006[n], n!]
a = {}; Do[a = Append[a, FactorInteger[a241601[n]][[1, 1]]], {n, 0, 99} ]; a

A250220 Numbers k such that A241601(k) is prime.

Original entry on oeis.org

7, 9, 12, 16, 17, 18, 26, 34, 36, 38, 39, 42, 49, 74, 114, 118, 337, 396, 455

Offset: 1

Eric Chen, Dec 24 2014

Is the sequence infinite?

No other terms < 500. - Jinyuan Wang, Apr 02 2020

Cf. A241601, A249909, A246006, A112548, A092132, A103234.

a(17)-a(19) from Jinyuan Wang, Apr 02 2020

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A241601 Largest divisor of A246006(n) whose prime factors are all >= n+2.

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A249909 Smallest prime factor of A241601(n), or 1 if A241601(n) = 1.

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A250220 Numbers k such that A241601(k) is prime.

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