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

Author

Antti Karttunen, Nov 26 2021

Keywords

• nonn
• frac

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.

Examples

			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.
		

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)

Crossrefs

Cf. A000203, A003961.

Cf. A349628 (denominators).

Cf. also A342671, A349161, A349162, A349633.

Programs

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 *)

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))));