A241601 -id:A241601 - cp's OEIS frontend

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

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

A250213 Number of distinct prime factors of A241601(n).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 2, 1, 2, 0, 2, 1, 1, 1, 3, 2, 2, 2, 3, 2, 2, 1, 2, 2, 4, 2, 2, 3, 3, 1, 3, 1, 3, 1, 1, 2, 3, 1, 3, 4, 5, 2, 2, 3, 1, 2, 4, 4, 4, 3, 4, 3, 5, 4, 2, 3, 3, 4, 5, 2, 3, 2, 4, 5, 5, 3, 3, 2, 5, 1, 5, 2, 4, 3, 2, 5, 3, 3, 2, 5, 3, 3, 4

Offset: 0

Eric Chen, Dec 28 2014

nonn

a(n) = 0 iff n is in the set {0, 1, 2, 3, 4, 5, 6, 8, 10, 14};

a(n) = 1 iff n is in A250220.

Cf. A001221, A241601, A249909, A250214, A250220.

b[n_] := Numerator[BernoulliB[2 n]/(2 n)];
c[n_] := Numerator[SeriesCoefficient[Log[Tan[x]+1/Cos[x]], {x, 0, 2n+1}]];
a[0] = 0; a[n_] := PrimeNu[If[EvenQ[n], Abs[b[n/2]], c[(n-1)/2]]];
Table[a[n], {n, 0, 78}] (* Jean-François Alcover, Jul 04 2019 *)

a(n) = A001221(A241601(n)).

More terms from Jean-François Alcover, Jul 04 2019

More terms from Jinyuan Wang, Apr 02 2020

A250214 Number of values of k such that prime(n) divides A241601(k).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 1, 1, 2, 1, 0, 1, 0, 0, 0, 2, 1, 0, 0, 0, 0, 1, 1, 1, 2, 0, 2, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 2, 1, 1, 2, 0, 1, 1, 0, 1, 1, 3, 3, 0, 0, 0, 0, 1, 2, 3, 1, 0, 1, 3, 0, 1, 0, 1, 1, 1, 1, 0, 2, 0, 0, 0, 0, 2, 2, 2, 0, 0, 4, 0, 0, 1, 0, 1, 2, 2, 1, 2, 0, 1, 3, 3, 1, 0, 0, 1, 1, 3, 3, 2, 0, 0, 3, 1, 1

Offset: 1

Eric Chen, Dec 26 2014

nonn

a(n) is called the weak irregular index of n-th prime, that is, the Bernoulli irregular index + Euler irregular index.
Prime(n) is a regular prime if and only if a(n) = 0.
Does every natural number appear in this sequence? For example, for the primes 491 and 1151, a(94) = a(190) = 4. (491 and 1151 are the only primes below 1800 with weak irregular index 4 or more.) However, does a(n) have a limit?

			a(8) = 1 since the 8th prime is 19, which divides A241601(11).
a(13) = 0 since the 13th prime is 41, a regular prime.
a(19) = 2 since the 19th prime is 67, which divides both A241601(27) and A241601(58).

Eric Weisstein's World of Mathematics, Irregular Pair

Cf. A091888, A091887, A250216, A000928, A120337, A128197, A250213.

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

A250215 Least k such that the n-th weak irregular prime (A250216(n)) divides A241601(k).

Original entry on oeis.org

11, 23, 32, 13, 15, 44, 7, 27, 29, 19, 63, 24, 22, 43, 129, 130, 62, 75, 133, 84, 211, 127, 164, 100, 84, 9, 20, 156, 88, 87, 280, 19, 71, 125, 163, 100, 200, 382, 126, 159, 240, 215, 196, 130, 94, 292, 141, 400, 86, 270, 222, 175, 389, 52, 45, 22, 592, 522, 20

Offset: 1

Eric Chen, Dec 26 2014

nonn

A prime p can divide A241601(k) for more than one k; the first few examples are as follows:
p      k
   27, 58
  63, 68
  130, 147
  62, 110
  211, 239
  100, 213
  88, 91, 137
  87, 193, 292
  19, 257
  71, 186, 300
etc.

			19 is the first weak irregular prime and divides A241601(11) = 50521, so a(1) = 11.

Eric Weisstein's World of Mathematics, Irregular Pair

Cf. A000928, A035112, A120337, A128197, A250216.

More terms from Jinyuan Wang, Feb 18 2022

A250216 Weak irregular primes. A prime is weak irregular iff it is a Bernoulli irregular prime or an Euler irregular prime.

Original entry on oeis.org

19, 31, 37, 43, 47, 59, 61, 67, 71, 79, 101, 103, 131, 137, 139, 149, 157, 193, 223, 233, 241, 251, 257, 263, 271, 277, 283, 293, 307, 311, 347, 349, 353, 359, 373, 379, 389, 401, 409, 419, 421, 433, 461, 463, 467, 491, 509, 523, 541, 547, 557, 563, 571, 577, 587, 593

Offset: 1

Eric Chen, Dec 24 2014

nonn

Primes p which divide A241601(k) for some k.

Peter Luschny, Irregular Bernoulli and Euler Primes.
The Prime Pages, Irregular Primes
The Prime Pages, Euler Irregular

Cf. A000928, A120337, A128197.

pmax = 593; m0 = 200; dm = 100;
b[n_] := Numerator[BernoulliB[2 n]/(2 n)];
c[n_] := Numerator[SeriesCoefficient[Log[Tan[x]+1/Cos[x]], {x, 0, 2n+1}]];
(* a1 = A241601 *) a1[0] = 1; a1[n_] := a1[n] = If[EvenQ[n], b[n/2] // Abs, c[(n - 1)/2]];
f[m_] := f[m] = Module[{}, aa = Table[a1[n], {n, 0, m}]; okQ[p_] := AnyTrue[aa, Divisible[#, p] &]; Reap[For[p = 2, p <= pmax, p = NextPrime[p], If[okQ[p], Sow[p]]]][[2, 1]]];
f[m = m0]; f[m = m + dm];
While[Print["m = ", m]; f[m] != f[m - dm], m = m + dm];
A250216 = f[m] (* Jean-François Alcover, Jul 23 2019 *)

cp's OEIS Frontend

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

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A250213 Number of distinct prime factors of A241601(n).

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A250214 Number of values of k such that prime(n) divides A241601(k).

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

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A250215 Least k such that the n-th weak irregular prime (A250216(n)) divides A241601(k).

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A250216 Weak irregular primes. A prime is weak irregular iff it is a Bernoulli irregular prime or an Euler irregular prime.

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