|
%I #40 Feb 19 2024 01:57:39
%S 4,8,68,76,124,152,188,236,244,248,284,376,404,412,428,436,472,488,
%T 548,596,604,628,668,724,788,824,844,872,892,908,916,964,1028,1052,
%U 1076,1084,1132,1156,1208,1244,1252,1256,1268,1324,1336,1348,1388
%N Numbers m such that the Bernoulli number B_m has denominator 30.
%C From the von Staudt-Clausen theorem, denominator(B_{2*m}) = product of primes p such that (p-1)|2*m.
%D B. C. Berndt, Ramanujan's Notebooks Part IV, Springer-Verlag, see p. 75.
%D G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, Th. 118.
%D H. Rademacher, Topics in Analytic Number Theory, Springer, 1973, Chap. 1.
%H T. D. Noe, <a href="/A051226/b051226.txt">Table of n, a(n) for n = 1..1000</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Von_Staudt%E2%80%93Clausen_theorem">Von Staudt-Clausen theorem</a>.
%H <a href="/index/Be#Bernoulli">Index entries for sequences related to Bernoulli numbers</a>.
%F a(n) = 2*A051225(n). - _Petros Hadjicostas_, Jun 06 2020
%e 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
%t Select[Table[n, {n, 4, 1500, 2}], Denominator @ BernoulliB[#] == 30 &] [[1 ;; 47]] (* _Jean-François Alcover_, Apr 08 2011 *)
%o (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"
%o (PARI) lista(nn) = for (n=1, nn, if (denominator(bernfrac(n)) == 30, print1(n, ", "))); \\ _Michel Marcus_, Mar 30 2015
%Y Cf. A045979, A051222, A051225, A051227, A051228, A051229, A051230.
%K nonn,nice,easy
%O 1,1
%A _N. J. A. Sloane_
%E More terms and Perl program from _Hugo van der Sanden_
%E Name edited by _Petros Hadjicostas_, Jun 06 2020
|
