A349631 Dirichlet convolution of A003961 with A346479, which is Dirichlet inverse of A250469.

1, 0, 0, 0, 0, 0, 0, 6, 0, -6, 0, 12, 0, -6, 0, 18, 0, 24, 0, 24, 0, -24, 0, 0, 0, -24, 60, 36, 0, 48, 0, 42, -20, -42, 0, -12, 0, -42, -10, 12, 0, 72, 0, 60, 60, -48, 0, -24, 0, 42, -30, 72, 0, -84, 0, 12, -30, -78, 0, -120, 0, -72, 120, 126, 0, 180, 0, 96, -30, 132, 0, -48, 0, -96, 60, 108, 0, 174, 0, -84, 120

Offset: 1

Author

Antti Karttunen, Nov 27 2021

Keywords

• sign

Comments

Note that for n = 2..36, a(n) = -A349632(n).

Dirichlet convolution of this sequence with A347376 is A003972.

Links

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

Crossrefs

Cf. A003961, A250469, A346479, A349632 (Dirichlet inverse).

Cf. also A003972, A347376, A349381.

Cf. also arrays A083221, A246278, A249821, A249822 and permutations A250245, A250246.

Programs

PARI

up_to = 20000;
A020639(n) = if(1==n,n,vecmin(factor(n)[, 1]));
ordinal_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), pt); for(i=1, length(invec), if(mapisdefined(om,invec[i]), pt = mapget(om, invec[i]), pt = 0); outvec[i] = (1+pt); mapput(om,invec[i],(1+pt))); outvec; };
v078898 = ordinal_transform(vector(up_to,n,A020639(n)));
A078898(n) = v078898[n];
A250469(n) = if(1==n,n,my(spn = nextprime(1+A020639(n)), c = A078898(n), k = 0); while(c, k++; if((1==k)||(A020639(k)>=spn),c -= 1)); (k*spn));
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(dA250469(n)));
A346479(n) = v346479[n];
A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
A349631(n) = sumdiv(n,d,A003961(d)*A346479(n/d));

Formula

a(n) = Sum_{d|n} A003961(d) * A346479(n/d).