cp's OEIS Frontend

A325977 a(n) = (1/2)*(A034460(n) + A325313(n)).

%I A325977 #24 Feb 22 2024 02:17:41
%S A325977 0,1,1,0,1,6,1,-2,-2,8,1,4,1,10,9,-6,1,3,1,4,11,14,1,0,-9,16,-11,4,1,
%T A325977 42,1,-14,15,20,13,-5,1,22,17,-4,1,54,1,4,-3,26,1,-8,-20,-2,21,4,1,-6,
%U A325977 17,-8,23,32,1,36,1,34,-7,-30,19,78,1,4,27,74,1,-21,1,40,-11,4,19,90,1,-20,-38,44,1,44,23,46,33,-16,1,36,21,4
%N A325977 a(n) = (1/2)*(A034460(n) + A325313(n)).
%C A325977 Question: Are n = 1, 4, 24, 240, 349440 (A325963) the only positions of zeros in this sequence?
%H A325977 Antti Karttunen, <a href="/A325977/b325977.txt">Table of n, a(n) for n = 1..16384</a>
%H A325977 Antti Karttunen, <a href="/A325977/a325977.txt">Data supplement: n, a(n) computed for n = 1..65537</a>
%F A325977 a(n) = (1/2)*(A034460(n) + A325313(n)).
%F A325977 a(n) = A325973(n) - n.
%F A325977 a(n) = A325978(n) - A033879(n).
%F A325977 Sum_{k=1..n} a(k) ~ c * n^2, where c = (zeta(2)/zeta(3) - 1)/4 = 0.0921081944... . - _Amiram Eldar_, Feb 22 2024
%t A325977 Array[(1/2) If[# == 1, 2, Times @@ (1 + Power @@@ #2) - 2 #1 + Times @@ (1 + #2[[;; , 1]]) & @@ {#, FactorInteger[#]}] &, 90] (* _Michael De Vlieger_, Jun 06 2019, after _Giovanni Resta_ at A034448 and _Amiram Eldar_ at A048250. *)
%o A325977 (PARI)
%o A325977 A034448(n) = { my(f=factorint(n)); prod(k=1, #f~, 1+(f[k, 1]^f[k, 2])); }; \\ After code in A034448
%o A325977 A034460(n) = (A034448(n) - n);
%o A325977 A048250(n) = factorback(apply(p -> p+1,factor(n)[,1]));
%o A325977 A325313(n) = (A048250(n) - n);
%o A325977 A325977(n) = ((A034460(n)+A325313(n))/2);
%Y A325977 Cf. A033879, A034448, A034460, A048250, A306633, A325313, A325963, A325973, A325974, A325975, A325978, A325979, A325981.
%K A325977 sign
%O A325977 1,6
%A A325977 _Antti Karttunen_, Jun 02 2019