A349384 Dirichlet convolution of A003961 with the Dirichlet inverse of A048673, where A003961 is fully multiplicative with a(p) = nextprime(p), and A048673(n) = (1+A003961(n))/2.

1, 1, 2, 2, 3, 0, 5, 4, 6, 0, 6, -2, 8, 0, 0, 8, 9, -4, 11, -3, 0, 0, 14, -8, 12, 0, 18, -5, 15, -12, 18, 16, 0, 0, 0, -14, 20, 0, 0, -12, 21, -20, 23, -6, -12, 0, 26, -24, 30, -9, 0, -8, 29, -24, 0, -20, 0, 0, 30, -24, 33, 0, -20, 32, 0, -24, 35, -9, 0, -30, 36, -36, 39, 0, -18, -11, 0, -32, 41, -36, 54, 0, 44

Offset: 1

Author

Antti Karttunen, Nov 17 2021

Keywords

• sign

Comments

Convolving this with A336840 gives A003973.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..20000
• Index entries for sequences computed from indices in prime factorization

Crossrefs

Cf. A003961, A048673, A323893, A349385 (Dirichlet inverse), A349386 (sum with it).

Cf. also A003973, A336840, A349572.

Programs

PARI

A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A048673(n) = (A003961(n)+1)/2;
memoA323893 = Map();
A323893(n) = if(1==n,1,my(v); if(mapisdefined(memoA323893,n,&v), v, v = -sumdiv(n,d,if(dA048673(n/d)*A323893(d),0)); mapput(memoA323893,n,v); (v)));
A349384(n) = sumdiv(n,d,A003961(n/d)*A323893(d));

Formula

a(n) = Sum_{d|n} A003961(n/d) * A323893(d).

a(n) = A349386(n) - A349385(n).