A345065 Sum of A011772 and its Dirichlet inverse.

2, 0, 0, 9, 0, 12, 0, 15, 4, 24, 0, -8, 0, 36, 16, 31, 0, 24, 0, -28, 24, 60, 0, 33, 16, 72, 24, -36, 0, -74, 0, 63, 40, 96, 48, 9, 0, 108, 48, 86, 0, -116, 0, -64, 36, 132, 0, 14, 36, 32, 64, -84, 0, 8, 80, 23, 72, 168, 0, 341, 0, 180, 48, 127, 96, -190, 0, -112, 88, -244, 0, -77, 0, 216, 40, -120, 120, -240, 0, -73, 88

Offset: 1

Author

Antti Karttunen, Jun 20 2021

Keywords

• sign

Links

• Antti Karttunen, Table of n, a(n) for n = 1..10000
• Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Crossrefs

Cf. A011772, A345053, A345055.

Programs

PARI

up_to = 65537;
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1])*sumdiv(n, d, if(dA011772(n) = { if(n==1, return(1)); my(f=factor(if(n%2, n, 2*n)), step=vecmax(vector(#f~, i, f[i, 1]^f[i, 2]))); forstep(m=step, 2*n, step, if(m*(m-1)/2%n==0, return(m-1)); if(m*(m+1)/2%n==0, return(m))); }; \\ From A011772
v345055 = DirInverseCorrect(vector(up_to,n,A011772(n)));
A345055(n) = v345055[n];
A345065(n) = (A011772(n)+A345055(n));

Formula

a(n) = A011772(n) + A345055(n).

a(2^i) = 2^(i+1)-1 for i >= 3. See A345053. - Chai Wah Wu, Jul 05 2021