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a(n) = (1/2)*(A034460(n) + A325313(n)).
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%I #24 Feb 22 2024 02:17:41

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

%C Question: Are n = 1, 4, 24, 240, 349440 (A325963) the only positions of zeros in this sequence?

%H Antti Karttunen, <a href="/A325977/b325977.txt">Table of n, a(n) for n = 1..16384</a>

%H Antti Karttunen, <a href="/A325977/a325977.txt">Data supplement: n, a(n) computed for n = 1..65537</a>

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

%F a(n) = A325973(n) - n.

%F a(n) = A325978(n) - A033879(n).

%F 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 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 (PARI)

%o A034448(n) = { my(f=factorint(n)); prod(k=1, #f~, 1+(f[k, 1]^f[k, 2])); }; \\ After code in A034448

%o A034460(n) = (A034448(n) - n);

%o A048250(n) = factorback(apply(p -> p+1,factor(n)[,1]));

%o A325313(n) = (A048250(n) - n);

%o A325977(n) = ((A034460(n)+A325313(n))/2);

%Y Cf. A033879, A034448, A034460, A048250, A306633, A325313, A325963, A325973, A325974, A325975, A325978, A325979, A325981.

%K sign

%O 1,6

%A _Antti Karttunen_, Jun 02 2019