A392629 - OEIS

%I #7 Jan 21 2026 21:24:55

%S 1,1,1,0,1,1,1,2,0,1,1,0,1,1,1,0,1,0,1,0,1,1,1,2,0,1,2,0,1,1,1,1,1,1,

%T 1,0,1,1,1,2,1,1,1,0,0,1,1,0,0,0,1,0,1,2,1,2,1,1,1,0,1,1,0,0,1,1,1,0,

%U 1,1,1,0,1,1,0,0,1,1,1,0,0,1,1,0,1,1,1,2,1

%N The number of coreful infinitary divisors of n that are numbers whose number of divisors is a power of 2.

%C For the definition of a coreful divisor see A284318 and A307958, and for the definition of an infinitary divisor see A037445.

%H Amiram Eldar, <a href="/A392629/b392629.txt">Table of n, a(n) for n = 1..10000</a>

%F Multiplicative with a(p^e) = A007814(e+1).

%F a(n) = 0 if and only if n is not an exponentially odd number (A268335, i.e., n is in A072587).

%F a(n) <= A363329(n), with equality if and only if n is squarefree (A005117).

%F Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} f(1/p) = 0.79378624236775721631..., where f(x) = (1-x) * (1 + Sum_{k>=1} x^(2^k-1)/(1-x^(2^k))).

%t f[p_, e_] := IntegerExponent[e + 1, 2]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

%o (PARI) a(n) = vecprod(apply(x -> valuation(x+1, 2), factor(n)[, 2]));

%Y Cf. A005117, A007814, A036537, A037445, A072587, A077609, A268335, A284318, A307958, A363329, A392628, A392630.

%K nonn,mult,easy

%O 1,8

%A _Amiram Eldar_, Jan 18 2026