A347096 a(1) = 1; a(n) = -Sum_{d|n, d < n} A341512(n/d) * a(d), where A341512(n) = sigma(n)*A003961(n) - n*sigma(A003961(n)).

1, -1, -2, -10, -2, -32, -4, -64, -42, -54, -2, -214, -4, -112, -112, -316, -2, -469, -4, -412, -232, -168, -6, -792, -90, -262, -612, -860, -2, -1208, -6, -1216, -340, -354, -320, -1655, -4, -484, -532, -1760, -2, -2528, -4, -1438, -1850, -732, -6, 160, -364, -1863, -712, -2210, -6, -4596, -384, -3696, -976, -942, -2

Offset: 1

Author

Antti Karttunen, Aug 19 2021

Keywords

• sign
• look

Comments

Dirichlet inverse of the pointwise sum of A341512 and A063524 (1, 0, 0, 0, ...).

Links

• Antti Karttunen, Table of n, a(n) for n = 1..10000
• Index entries for primes, gaps between
• Index entries for sequences computed from indices in prime factorization
• Index entries for sequences related to sigma(n)

Crossrefs

Cf. A000203, A001223, A003961, A063524, A341512, A347097.

Cf. also A346235, A346239.

Programs

PARI

up_to = 16384;
A003961(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); }; \\ From A003961
A341512(n) = { my(u=A003961(n)); ((sigma(n)*u) - (n*sigma(u))); };
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(dA341512(n));
v347096 = DirInverseCorrect(vector(up_to,n,Aux347096(n)));
A347096(n) = v347096[n];

Formula

a(1) = 1; and for n > 1, a(n) = -Sum_{d|n, d < n} A341512(n/d) * a(d).

For all n >= 1, a(A000040(n)) = -A001223(n).