A349381 Dirichlet convolution of A003961 with A349125 (Dirichlet inverse of A064989), where A003961 and A064989 are fully multiplicative sequences that shift the prime factorization of n one step towards larger and smaller primes respectively.

1, 2, 3, 6, 4, 6, 6, 18, 15, 8, 6, 18, 6, 12, 12, 54, 6, 30, 6, 24, 18, 12, 10, 54, 28, 12, 75, 36, 8, 24, 8, 162, 18, 12, 24, 90, 10, 12, 18, 72, 6, 36, 6, 36, 60, 20, 10, 162, 66, 56, 18, 36, 12, 150, 24, 108, 18, 16, 8, 72, 8, 16, 90, 486, 24, 36, 10, 36, 30, 48, 6, 270, 8, 20, 84, 36, 36, 36, 10, 216, 375, 12

Offset

1,2

Comments

Multiplicative because both A003961 and A349125 are.

Convolving this with A349127 gives A003972.

Links

Antti Karttunen, Table of n, a(n) for n = 1..20000

Index entries for sequences computed from indices in prime factorization

Formula

a(n) = Sum_{d|n} A003961(n/d) * A349125(d).

a(n) = A349383(n) - A349382(n).

Prog

(PARI)
A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A064989(n) = { my(f = factor(n)); if((n>1 && f[1, 1]==2), f[1, 2] = 0); for (i=1, #f~, f[i, 1] = precprime(f[i, 1]-1)); factorback(f); };
A349125(n) = (moebius(n)*A064989(n));
A349381(n) = sumdiv(n, d, A003961(n/d)*A349125(d));

Crossrefs

Cf. A003961, A064989, A349125, A349382 (Dirichlet inverse), A349383 (sum with it).

Cf. also A003972, A349127, and A349355, A349356 and A349384, A349385, and A349387.

Sequence in context: A360599 A306231 A125703 * A156688 A348243 A019567

Adjacent sequences: A349378 A349379 A349380 * A349382 A349383 A349384

Keyword

nonn,mult

Author

Antti Karttunen, Nov 17 2021

Status

approved