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

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

Author

N. J. A. Sloane

Keywords

• nonn

Comments

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

Examples

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

References

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

Links

• Eric Weisstein's World of Mathematics, Irregular Pair

Crossrefs

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

Programs

Mathematica

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} ]

Extensions

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