A348341 - OEIS

%I #16 Dec 09 2022 07:08:09

%S 0,0,0,1,0,0,0,0,1,0,0,2,0,0,0,3,0,2,0,2,0,0,0,0,1,0,0,2,0,0,0,2,0,0,

%T 0,5,0,0,0,0,0,0,0,2,2,0,0,6,1,2,0,2,0,0,0,0,0,0,0,4,0,0,2,3,0,0,0,2,

%U 0,0,0,4,0,0,2,2,0,0,0,6,3,0,0,4,0,0,0

%N a(n) is the number of noninfinitary divisors of n.

%H Amiram Eldar, <a href="/A348341/b348341.txt">Table of n, a(n) for n = 1..10000</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/InfinitaryDivisor.html">Infinitary Divisor</a>.

%F a(n) = A000005(n) - A037445(n).

%F a(n) = 0 if and only if the number of divisors of n is a power of 2, (i.e., n is in A036537).

%F a(n) > 0 if and only if the number of divisors of n is not a power of 2, (i.e., n is in A162643).

%F Sum_{k=1..n} a(k) ~ c * n * log(n), where c = (1 - 2 * A327576) = 0.266749... . - _Amiram Eldar_, Dec 09 2022

%e a(4) = 1 since 4 has one noninfinitary divisor, 2.

%t a[1] = 0; a[n_] := DivisorSigma[0, n] - Times @@ Flatten[2^DigitCount[#, 2, 1] & /@ FactorInteger[n][[;; , 2]]]; Array[a, 100]

%o (PARI) A348341(n) = (numdiv(n)-factorback(apply(a -> 2^hammingweight(a), factorint(n)[, 2]))); \\ _Antti Karttunen_, Oct 13 2021

%Y Cf. A000005, A036537, A037445, A077609, A162643, A327576, A348271.

%Y Similar sequences: A034444, A048105, A286324, A049419.

%K nonn

%O 1,12

%A _Amiram Eldar_, Oct 13 2021