A189685 -id:A189685 - cp's OEIS frontend

A190572 A189685/2.

6, 8, 9, 10, 10, 11, 11, 12, 12, 13, 14, 14, 15, 15, 16, 16, 16, 17, 18, 19, 20, 20, 21, 22, 22, 22, 22, 23, 23, 24, 24, 24, 25, 25, 26, 26, 26, 26, 27, 27, 27, 28, 28, 28, 29, 29, 29, 29, 30, 30, 30, 31, 31, 31, 31, 32, 32, 33, 33, 34, 34, 34, 34, 35, 35, 35

Offset: 1

Jonathan Sondow, May 12 2011

nonn

a(n) = A189683(2n)/2

A189683 Irregular pairs (p,2k) ordered by increasing k.

Original entry on oeis.org

691, 12, 3617, 16, 43867, 18, 283, 20, 617, 20, 131, 22, 593, 22, 103, 24, 2294797, 24, 657931, 26, 9349, 28, 362903, 28, 1721, 30, 1001259881, 30, 37, 32, 683, 32, 305065927, 32, 151628697551, 34, 26315271553053477373, 36, 154210205991661, 38, 137616929, 40

Offset: 1

Jonathan Sondow, Apr 25 2011

nonn

The subsequence of irregular primes p is A046753.

			The first few irregular pairs are (691,12), (3617,16), (43867,18), (283,20), (617,20), (131,22), (593,22), ...

T. D. Noe, Table of n, a(n) for n = 1..132
Bernd C. Kellner, On irregular prime power divisors of the Bernoulli numbers, Math. Comp. 76 (2007), 405-441; arXiv:0409223 [math.NT], 2004.
B. Mazur, How can we construct abelian Galois extensions of basic number fields?, Bull. Amer. Math. Soc., 48 (2011), 155-209. See footnote 64 on p. 205.
Wikipedia, Regular prime: Irregular pairs.

Cf. A000367, A000928, A035112, A046753, A189684, A189685.

Flatten[Table[p = Select[First /@ FactorInteger[Abs[Numerator[BernoulliB[n]]]], # >= n+3 &]; Transpose[{p, Table[n, {Length[p]}]}], {n, 2, 70, 2}]] (* T. D. Noe, Apr 25 2011 *)

A035112 Smallest even index 2a such that n-th irregular prime p (A000928(n)) divides Bernoulli_{2a} with 0<=2a<=p-3.

Original entry on oeis.org

32, 44, 58, 68, 24, 22, 130, 62, 84, 164, 100, 84, 20, 156, 88, 292, 280, 186, 100, 200, 382, 126, 240, 366, 196, 130, 94, 292, 400, 86, 270, 222, 52, 90, 22, 592, 522, 20, 428, 80, 236, 48, 224, 408, 628, 32, 12, 378, 290, 514, 260, 732, 220, 330, 544, 744, 102

Offset: 1

N. J. A. Sloane

nonn

The ordered pair (p(n),a(n)) where p(n) is the n-th irregular prime is called an irregular pair. Some irregular primes, such as 157, are in more than one pair. See A091887 for the number of pairs for each irregular prime. See A092681 and A092682 for higher-order irregular pairs. - T. D. Noe, Mar 03 2004

			The first irregular prime (37) divides the numerator (-7709321041217) of the 32nd Bernoulli number.

L. C. Washington, Introduction to Cyclotomic Fields, Springer, p. 350.

Eric Weisstein's World of Mathematics, Irregular Pair

Cf. A000928, A000367, A035112, A046753, A189683, A189684, A189685.

Do[ p = Prime[ n ]; k = 1; While[ 2*k < p - 3 && Mod[ Numerator[ BernoulliB[ 2*k ] ], p ] != 0, k++ ]; If[ 2*k != p - 3, Print[ 2*k ] ], { n, 3, 200} ]

More terms from Robert G. Wilson v, May 12 2001

A046753 Prime factors of |numerator(B(2n))| which are >= 2n+3.

Original entry on oeis.org

691, 3617, 43867, 283, 617, 131, 593, 103, 2294797, 657931, 9349, 362903, 1721, 1001259881, 37, 683, 305065927, 151628697551, 26315271553053477373, 154210205991661, 137616929, 1897170067619, 1520097643918070802691, 59

Offset: 1

Bill Gosper

nonn

See A189683 for pairs (p,2n) for the primes p in this sequence.

Robert G. Wilson v, Table of n, a(n) for n = 1..154 (first 66 terms from T. D. Noe)
Hisanori Mishima, WIFC (World Integer Factorization Center)

Cf. A000367, A000928, A189683, A189684, A189685.

for n do for p in map('first,factor_number(abs(num(bern(2*n))))) do if p>=2*n+3 then (?prin1(p),?prin1('?\-));

Flatten[Table[Select[First /@ FactorInteger[Abs[Numerator[BernoulliB[n]]]], # >= n+3 &], {n, 2, 70, 2}]] (* T. D. Noe, Apr 25 2011 *)

Definition modified by Jonathan Sondow, Apr 27 2011

A189684 Number of irregular pairs (p,2n).

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 1, 1, 2, 2, 2, 1, 2, 2, 3, 1, 1, 1, 2, 1, 4, 2, 3, 2, 4, 3, 3, 4, 3, 4, 2, 2, 4, 3, 2, 1, 2, 3, 5, 3, 5, 3, 3, 3, 3, 5, 5, 5, 5, 5, 3, 4, 3, 5, 3, 1, 4, 1, 3, 3, 7, 3, 6, 5, 3, 4, 3, 7, 6, 3, 2, 6, 6, 6, 5, 6, 5, 7, 5, 4, 7, 8, 5, 3, 2, 7

Offset: 1

Jonathan Sondow, Apr 25 2011

nonn

a(n) is the number of primes p >= 2n+3 that divide the numerator of the Bernoulli number B_{2n}.

B. Mazur, How can we construct abelian Galois extensions of basic number fields?, Bull. Amer. Math. Soc., 48 (2011), 155-209. See footnote 64 on p. 205.
Samual S. Wagstaff, Jr., Prime divisors of the Bernoulli and Euler numbers
Wikipedia, Irregular pairs

Cf. A000367, A000928, A035112, A046753, A189683, A189685.

Table[p = Select[First /@ FactorInteger[Abs[Numerator[BernoulliB[2n]]]], # >= 2*n+3 &]; Length[p], {n, 35}] (* T. D. Noe, Apr 25 2011 *)

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A190572 A189685/2.

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A189683 Irregular pairs (p,2k) ordered by increasing k.

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A035112 Smallest even index 2a such that n-th irregular prime p (A000928(n)) divides Bernoulli_{2a} with 0<=2a<=p-3.

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A046753 Prime factors of |numerator(B(2n))| which are >= 2n+3.

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A189684 Number of irregular pairs (p,2n).

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