A323893 Dirichlet inverse of A048673, where A048673(n) = (A003961(n)+1) / 2, and A003961 is fully multiplicative with a(prime(i)) = prime(i+1).

1, -2, -3, -1, -4, 4, -6, -2, -4, 5, -7, 3, -9, 7, 6, -4, -10, 8, -12, 4, 8, 8, -15, 8, -9, 10, -12, 6, -16, 5, -19, -8, 9, 11, 9, 8, -21, 13, 11, 11, -22, 11, -24, 7, 16, 16, -27, 20, -25, 18, 12, 9, -30, 32, 10, 17, 14, 17, -31, 6, -34, 20, 24, -16, 12, 14, -36, 10, 17, 20, -37, 16, -40, 22, 27, 12, 12, 20, -42, 28, -36, 23, -45, 12, 13

Offset: 1

Author

Antti Karttunen, Feb 08 2019

Keywords

• sign

Links

• Antti Karttunen, Table of n, a(n) for n = 1..16383
• Index entries for sequences related to prime indices in the factorization of n.

Crossrefs

Cf. A003961, A048673, A323894, A349134, A378520 (Möbius transform).

Programs

PARI

up_to = 20000;
DirInverse(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(dA003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
A048673(n) = (A003961(n)+1)/2;
v323893 = DirInverse(vector(up_to,n,A048673(n)));
A323893(n) = v323893[n];

PARI

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))); \\ Antti Karttunen, Nov 30 2024

Formula

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, dA048673(n/d) * a(d).

a(n) = A349134(A003961(n)). - Antti Karttunen, Nov 30 2024