A051226 Numbers m such that the Bernoulli number B_m has denominator 30.

4, 8, 68, 76, 124, 152, 188, 236, 244, 248, 284, 376, 404, 412, 428, 436, 472, 488, 548, 596, 604, 628, 668, 724, 788, 824, 844, 872, 892, 908, 916, 964, 1028, 1052, 1076, 1084, 1132, 1156, 1208, 1244, 1252, 1256, 1268, 1324, 1336, 1348, 1388

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• nice
• easy

Comments

From the von Staudt-Clausen theorem, denominator(B_{2*m}) = product of primes p such that (p-1)|2*m.

Examples

			The numbers m = 4, 8, 68 are in the list because B_4 = B_8 = -1/30 and B_68 = -78773130858718728141909149208474606244347001/30. - _Petros Hadjicostas_, Jun 06 2020
		

References

• B. C. Berndt, Ramanujan's Notebooks Part IV, Springer-Verlag, see p. 75.
• G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, Th. 118.
• H. Rademacher, Topics in Analytic Number Theory, Springer, 1973, Chap. 1.

Links

• T. D. Noe, Table of n, a(n) for n = 1..1000
• Wikipedia, Von Staudt-Clausen theorem.
• Index entries for sequences related to Bernoulli numbers.

Crossrefs

Cf. A045979, A051222, A051225, A051227, A051228, A051229, A051230.

Programs

Mathematica

Select[Table[n, {n, 4, 1500, 2}], Denominator @ BernoulliB[#] == 30 &] [[1 ;; 47]] (* Jean-François Alcover, Apr 08 2011 *)

PARI

lista(nn) = for (n=1, nn, if (denominator(bernfrac(n)) == 30, print1(n, ", "))); \\ Michel Marcus, Mar 30 2015

Perl

@p=(2,3,5); $p=5; for($n=4; $n<=1388; $n+=4){while($p<$n+1){$p+=2; next if grep$p%$==0,@p; push@p,$p; push@c,$p-1; }print"$n,"if!grep$n%$==0,@c; }print"\n"

Formula

a(n) = 2*A051225(n). - Petros Hadjicostas, Jun 06 2020

Extensions

More terms and Perl program from Hugo van der Sanden

Name edited by Petros Hadjicostas, Jun 06 2020