A326727 -id:A326727

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A332300

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

+0
3

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,6

COMMENTS

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)

LINKS

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.

FORMULA

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

EXAMPLE

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

MATHEMATICA

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

PROG

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

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

(Python)

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

CROSSREFS

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

KEYWORD

nonn

AUTHOR

Amiram Eldar, Feb 09 2020

STATUS

approved

A326726

The prime factorization of abs(E(2k)) for k >= 2, E(k) the k-th Euler number. Factors sorted by size with the smallest factor negated. a(n) = -1 by convention for n = 1, 2.

+0
2

-1, -1, -5, -61, -5, 277, -19, 2659, -5, 13, 43, 967, -47, 4241723, -5, 17, 228135437, -79, 349, 87224971, -5, 5, 41737, 354957173, -31, 1567103, 1427513357, -5, 13, 2137, 111691689741601, -67, 61001082228255580483, -5, 19, 29, 71, 30211, 2717447, 77980901

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

COMMENTS

For small Euler numbers the factorizations were computed with SageMath, see the b-file for the script. For larger Euler 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 E(164) (the prime factors of E(166) are not yet known).

LINKS

Peter Luschny, Table of n, a(n) for n = 1..428

factordb, Status of E(166).

S. S. Wagstaff, Prime factors of the absolute values of Euler numbers

EXAMPLE

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], [-5], [-61], [-5, 277], [-19, 2659], [-5, 13, 43, 967], [-47, 4241723], [-5, 17, 228135437], [-79, 349, 87224971], [-5, 5, 41737, 354957173], ... ].

The first few factorizations are:

E(4) = 5;

E(6) = 61;

E(8) = 5 * 277;

E(10) = 19 * 2659;

E(12) = 5 * 13 * 43 * 967;

E(14) = 47 * 4241723;

E(16) = 5 * 17 * 228135437;

E(18) = 79 * 349 * 87224971;

E(20) = 5 * 5 * 41737 * 354957173;

E(22) = 31 * 1567103 * 1427513357;

E(24) = 5 * 13 * 2137 * 111691689741601;

PROG

(Sage) # See b-file.

CROSSREFS

Cf. A122045, A000364, A326727.

KEYWORD

sign,tabf,hard

AUTHOR

Peter Luschny, Jul 29 2019

STATUS

approved

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