A349627 Numerators of the Möbius transform of ratio A003961(n)/sigma(n).

1, 0, 1, 2, 1, 0, 3, 18, 35, 0, 1, 1, 3, 0, 1, 126, 1, 0, 3, 1, 3, 0, 5, 9, 77, 0, 125, 3, 1, 0, 5, 270, 1, 0, 1, 5, 3, 0, 3, 3, 1, 0, 3, 1, 35, 0, 5, 63, 341, 0, 1, 3, 5, 0, 1, 27, 3, 0, 1, 1, 5, 0, 105, 1674, 1, 0, 3, 1, 5, 0, 1, 9, 5, 0, 77, 3, 1, 0, 3, 21, 1975, 0, 5, 3, 1, 0, 1, 3, 7, 0, 9, 5, 5, 0, 1, 135, 3

Offset

1,4

Comments

Because the ratio A003961(n)/A000203(n) is multiplicative, so is also its Möbius transform. This sequence gives the numerator of that ratio when presented in its lowest terms, while A349628 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

Index entries for sequences related to sigma(n)

Example

The ratio a(n)/A349628(n) for n = 1..15: 1/1, 0/1, 1/4, 2/7, 1/6, 0/1, 3/8, 18/35, 35/52, 0/1, 1/12, 1/14, 3/14, 0/1, 1/24.

Mathematica

f[p_, e_] := NextPrime[p]^e*(p - 1)/(p^(e + 1) - 1); s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; a[n_] := Numerator @ DivisorSum[n, s[#] * MoebiusMu[n/#] &]; Array[a, 100] (* Amiram Eldar, Nov 27 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
A349627(n) = numerator(sumdiv(n, d, moebius(n/d)*(A003961(d)/sigma(d))));

Crossrefs

Cf. A000203, A003961.

Cf. A349628 (denominators).

Cf. also A342671, A349161, A349162, A349633.

Sequence in context: A161552 A366592 A095859 * A300482 A191897 A088850

Adjacent sequences: A349624 A349625 A349626 * A349628 A349629 A349630

Keyword

nonn,frac

Author

Antti Karttunen, Nov 26 2021

Status

approved