A079294 -id:A079294 - cp's OEIS frontend

A090947 Largest prime factor of numerator of Bernoulli(2n) (or 1 if the numerator is 1).

1, 1, 1, 1, 1, 5, 691, 7, 3617, 43867, 617, 593, 2294797, 657931, 362903, 1001259881, 305065927, 151628697551, 26315271553053477373, 154210205991661, 1897170067619, 1520097643918070802691, 1798482437, 67568238839737, 153289748932447906241, 47464429777438199, 4534045619429, 1120412849144121779, 19088082706840550550313, 37349583369104129, 109317926249509865753025015237911, 28765594733083851481, 87057315354522179184989699791727, 159562251828620181390358590156239282938769, 5525473366510930028227481

Offset: 0

N. J. A. Sloane, Feb 28 2004

nonn

Amiram Eldar, Table of n, a(n) for n = 0..103

Cf. A000367, A006530, A079294.

PrimeFactors[ n_] := Flatten[ Table[ #[[1]], {1} ] & /@ FactorInteger[ n ]]; A090947[n_] := PrimeFactors[ Numerator[ BernoulliB[2n]]][[ -1]]; Table[ A090947[n], {n, 5, 24}] (* Robert G. Wilson v, Feb 28 2004 *)

a(n) = A006530(abs(A000367(n))). - Amiram Eldar, Feb 09 2020

More terms from Robert G. Wilson v and Hans Havermann, Feb 28 2004

A326727 The prime factorization of abs(numerator(B(2k))) for k >= 5, B(k) the k-th Bernoulli number. Factors sorted by size with the smallest factor negated. a(n) = -1 by convention for 1 <= n <= 5.

Original entry on oeis.org

-1, -1, -1, -1, -1, -5, -691, -7, -3617, -43867, -283, 617, -11, 131, 593, -103, 2294797, -13, 657931, -7, 9349, 362903, -5, 1721, 1001259881, -37, 683, 305065927, -17, 151628697551, -26315271553053477373, -19, 154210205991661, -137616929, 1897170067619

Offset: 1

Peter Luschny, Jul 28 2019

For small Bernoulli numbers the factorizations were computed with SageMath, see the b-file for the script. For larger Bernoulli numbers the values were taken from the table of S. S. Wagstaff, Jr..
The smallest factor was negated only to be able to distinguish the individual factorizations easily. (No general formula for the number of factors is known.)
The factorizations listed in the b-file currently go up to B(204) (the prime factors of numerator(B(206)) are not yet known).

			The data is given as a flatted list of factorizations written with the conventions
stated above. Because it is a list the offset is 1. The list starts:
[[-1], [-1], [-1], [-1], [-1], [-5], [-691], [-7], [-3617], [-43867], [-283, 617], [-11, 131, 593], [-103, 2294797], [-13, 657931], [-7, 9349, 362903], ... ].
.
The first few factorizations are:
B(10) = 5;
B(12) = 691;
B(14) = 7;
B(16) = 3617;
B(18) = 43867;
B(20) = 283 * 617;
B(22) = 11 * 131 * 593;
B(24) = 103 * 2294797;
B(26) = 13 * 657931;
B(28) = 7 * 9349 * 362903;
B(30) = 5 * 1721 * 1001259881;

Peter Luschny, Table of n, a(n) for n = 1..460
factordb, Status of numerator(B(206)).
S. S. Wagstaff, Prime factors of the absolute values of Bernoulli numerators

Cf. A000367, A079294, A090947, A326726.

Sage
```
# See b-file.
```

A332300 The least prime factor of the numerator of Bernoulli(2*n), or 1 if the numerator is 1.

Original entry on oeis.org

1, 1, 1, 1, 1, 5, 691, 7, 3617, 43867, 283, 11, 103, 13, 7, 5, 37, 17, 26315271553053477373, 19, 137616929, 1520097643918070802691, 11, 23, 653, 5, 13, 39409, 7, 29, 2003, 31, 1226592271, 11, 17, 5, 3112655297839, 37, 19, 13, 631, 41, 233, 43, 11, 5, 23, 47, 7823741903

Offset: 0

Amiram Eldar, Feb 09 2020

nonn

a(n)=5 if and only if n is in A017329. - Robert Israel, Feb 09 2020
From Chai Wah Wu, Feb 10 2020: (Start)
For n > 1, clearly if a(n) = n, then n is prime. However, the converse is not true. Prime numbers p such that a(p) != p are: 2, 3, 109, 167, 211, 227, 271, ...
Conjecture: for prime p > 3, p is a prime factor of the numerator of Bernoulli(2*p), thus the conjecture implies that a(p) <= p for prime p.
(End)

			a(10) = 283, since Bernoulli(2*10) = -174611/330, and 283 is the least prime factor of its numerator, 174611 = 283 * 617.

Chai Wah Wu, Table of n, a(n) for n = 0..191 (n = 0..103 from Amiram Eldar)
S. S. Wagstaff, Jr., Factors of Bernoulli numbers.

Cf. A000367, A017329, A020639, A079294, A090947, A242193, A326727.

[n le 4 select 1 else Min(PrimeDivisors(Abs(Numerator(Bernoulli(2*n))))):n in [0..48]]; // Marius A. Burtea, Feb 09 2020

Array[FactorInteger[Abs @ Numerator @  BernoulliB[2*#]][[1, 1]] &, 30, 0]

a(n) = my(x=abs(numerator(bernfrac(2*n)))); if (x==1, 1, vecmin(factor(x)[,1])); \\ Michel Marcus, Feb 09 2020

from sympy import bernoulli, primefactors
def A332300(n):
    x = abs(bernoulli(2*n).p)
    return 1 if x == 1 else min(primefactors(x)) # Chai Wah Wu, Feb 10 2020

a(n) = A020639(abs(A000367(n))).

A110841 a(n) is the number of prime factors, with multiplicity, of abs(A014509(n)).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 4, 2, 7, 7, 2, 2, 4, 3, 3, 7, 1, 6, 4, 5, 14, 4, 9, 5, 10, 3, 11, 2, 5, 3, 7, 11, 5, 3, 4, 15, 6, 5, 19, 10, 6, 13, 15, 5, 10, 5, 5, 6, 7, 5, 15, 7, 5, 2, 13, 4, 3, 10, 5, 9, 7, 5, 4, 9, 5, 4, 1, 7, 4, 4, 5, 3, 11, 13, 10, 5, 5, 7, 6

Offset: 0

Jonathan Vos Post, Sep 16 2005

			a(10) = 2 because A014509(10) = 529 = 23^2.
a(8) = a(19) = 1 since A014509(8) and A014509(19) are prime.

Sean A. Irvine, Table of n, a(n) for n = 0..91

Cf. A001222, A073276, A075178, A076547, A076549, A079294, A083687, A084217, A085092, A085737, A089170, A089644, A089655.

a(n) = my(b=bernfrac(2*n), c=floor(abs(b))*sign(b)); if (c==0, 0, bigomega(c)); \\ Michel Marcus, Mar 29 2020

a(n) = A001222(abs(A014509(n))).

More terms from Michel Marcus, Mar 29 2020
a(51)-a(65) from Jinyuan Wang, Apr 02 2020
More terms from Sean A. Irvine, Jul 29 2024

cp's OEIS Frontend

A090947 Largest prime factor of numerator of Bernoulli(2n) (or 1 if the numerator is 1).

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A326727 The prime factorization of abs(numerator(B(2k))) for k >= 5, B(k) the k-th Bernoulli number. Factors sorted by size with the smallest factor negated. a(n) = -1 by convention for 1 <= n <= 5.

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Sage

A332300 The least prime factor of the numerator of Bernoulli(2*n), or 1 if the numerator is 1.

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Magma

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A110841 a(n) is the number of prime factors, with multiplicity, of abs(A014509(n)).

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