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Number of divisors d of 2n satisfying (d+1) = prime or number of prime factors of the denominator of the even Bernoulli numbers.
7

%I #15 Aug 07 2022 08:14:32

%S 2,3,3,3,3,5,2,4,4,4,3,5,2,4,5,4,2,7,2,5,4,4,3,6,3,4,4,4,3,8,2,4,5,3,

%T 4,8,2,3,4,6,3,7,2,5,6,4,2,7,2,5,4,4,3,8,4,6,3,4,2,9,2,3,6,4,4,7,2,4,

%U 5,6,2,9,2,4,6,3,3,8,2,6,5,4,3,7,3,4,4,6,3,11,2,4,3,3,4,8,2,5,7,6,2,6,2,5

%N Number of divisors d of 2n satisfying (d+1) = prime or number of prime factors of the denominator of the even Bernoulli numbers.

%C From von Staudt-Clausen theorem.

%D G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, Th. 118.

%D Hans Rademacher, Topics in Analytic Number Theory, Springer, 1973, Chap. 1.

%H Amiram Eldar, <a href="/A046886/b046886.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = A067513(2n). - _R. J. Mathar_, Aug 07 2022

%p A046886 := proc(n)

%p A067513(2*n) ;

%p end proc:

%p seq(A046886(n),n=1..100) ; # _R. J. Mathar_, Aug 07 2022

%t Length[ Select[ Divisors[ 2n ], PrimeQ[ #+1 ]& ] ] or Length[ FactorInteger[ Denominator@BernoulliB[ 2k ] ] ]

%t Table[Count[Divisors[2n],_?(PrimeQ[#+1]&)],{n,110}] (* or *) PrimeOmega/@ Denominator[BernoulliB[2*Range[110]]] (* _Harvey P. Dale_, Mar 19 2015 *)

%Y Cf. A000146, A027642.

%K nonn

%O 1,1

%A _Wouter Meeussen_, Jan 23 2001