A325978 a(n) = (1/2)*(A325314(n) + A325814(n)).

1, 2, 3, 1, 5, 6, 7, -1, 3, 10, 11, 0, 13, 14, 15, -5, 17, 0, 19, 2, 21, 22, 23, -12, 10, 26, 3, 4, 29, 30, 31, -13, 33, 34, 35, -24, 37, 38, 39, -14, 41, 42, 43, 8, 9, 46, 47, -36, 21, 5, 51, 10, 53, -18, 55, -16, 57, 58, 59, -12, 61, 62, 15, -29, 65, 66, 67, 14, 69, 70, 71, -72, 73, 74, 15, 16, 77, 78, 79, -46, 3, 82, 83, -12, 85

Offset

1,2

Comments

Question: Are a(12) = 0 and a(18) = 0 the only zeros in this sequence?

Links

Antti Karttunen, Table of n, a(n) for n = 1..20000

Formula

a(n) = (1/2)*(A325314(n) + A325814(n)).

a(n) = n - A325974(n).

a(n) = A033879(n) + A325977(n).

Sum_{k=1..n} a(k) ~ c * n^2, where c = 3/4 - zeta(2)*(1/2 - 1/(4*zeta(3))) = 0.2696411609... . - Amiram Eldar, Feb 22 2024

Mathematica

Table[(1/2) If[n == 1, 2, 2 n - DivisorSigma[1, n] + Times @@ (1 + FactorInteger[n][[;; , 1]]) - DivisorSum[n, # &, ! CoprimeQ[#, n/#] &]], {n, 85}] (* Michael De Vlieger, Jun 06 2019 *)

Prog

(PARI) A325978(n) = ((A325314(n)+A325814(n))/2);
(PARI) A325978(n) = (n-((1/2)*sumdiv(n, d, d*((1-issquarefree(d))+(gcd(d, n/d)>1))))); \\ Antti Karttunen, Sep 30 2025

Crossrefs

Cf. A000203, A033879, A034448, A048146, A162296, A325314, A325814, A325973, A325974, A325975, A325977, A325979, A325981.

Cf. A389211 (positions of nonnegative terms), A389212 (of negative terms), A389213 (of positive terms), A389214 (of terms <= 0).

Cf. A002117, A013661.

Sequence in context: A379457 A160400 A382905 * A326049 A380085 A389455

Adjacent sequences: A325975 A325976 A325977 * A325979 A325980 A325981

Keyword

sign

Author

Antti Karttunen, Jun 02 2019

Status

approved