A349633 Numerators of the Möbius transform of ratio A003961(n)/n.

1, 1, 2, 3, 2, 1, 4, 9, 10, 1, 2, 1, 4, 2, 4, 27, 2, 5, 4, 3, 8, 1, 6, 3, 14, 2, 50, 3, 2, 2, 6, 81, 4, 1, 8, 5, 4, 2, 8, 9, 2, 4, 4, 3, 4, 3, 6, 9, 44, 7, 4, 3, 6, 25, 4, 9, 8, 1, 2, 1, 6, 3, 40, 243, 8, 2, 4, 3, 4, 4, 2, 5, 6, 2, 28, 3, 8, 4, 4, 27, 250, 1, 6, 2, 4, 2, 4, 9, 8, 2, 16, 9, 4, 3, 8, 27, 4, 22, 20

Offset

1,3

Comments

Because the ratio A003961(n)/n is multiplicative, so is also its Möbius transform. This sequence gives the numerator of that ratio when presented in its lowest terms, while A349634 gives the denominators. See the examples.

Links

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

Index entries for sequences computed from indices in prime factorization

Example

The ratio a(n)/A349634(n) for n = 1..15: 1/1, 1/2, 2/3, 3/4, 2/5, 1/3, 4/7, 9/8, 10/9, 1/5, 2/11, 1/2, 4/13, 2/7, 4/15.

Mathematica

f[p_, e_] := NextPrime[p]^e; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; a[n_] := Numerator @ DivisorSum[n, MoebiusMu[n/#] * s[#]/# &]; Array[a, 100] (* Amiram Eldar, Nov 28 2021 *)

Prog

(PARI)
A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
A349633(n) = numerator(sumdiv(n, d, moebius(n/d)*(A003961(d)/d)));

Crossrefs

Cf. A000010, A003961, A003972, A008683, A349634 (denominators).

Cf. also A349627.

Sequence in context: A216084 A055101 A319108 * A238165 A081316 A226362

Adjacent sequences: A349630 A349631 A349632 * A349634 A349635 A349636

Keyword

nonn,frac

Author

Antti Karttunen, Nov 28 2021

Status

approved