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

Author

Antti Karttunen, Nov 28 2021

Keywords

• nonn
• frac

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.

Examples

			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.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..20000
• Index entries for sequences computed from indices in prime factorization

Crossrefs

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

Cf. also A349627.

Programs

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 *)

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)));