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

1,8

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

Formula

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

Prog

(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(d<n, v[n/d]*u[d], 0)))); (u) }; \\ Compute the Dirichlet inverse of the sequence given in input vector v.
v346479 = DirInverseCorrect(vector(up_to, n, A250469(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));

Crossrefs

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

Cf. also A003972, A347376, A349381.

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

Sequence in context: A064373 A195290 A349632 * A347377 A280692 A161419

Adjacent sequences: A349628 A349629 A349630 * A349632 A349633 A349634

Keyword

sign

Author

Antti Karttunen, Nov 27 2021

Status

approved