A349633 -id:A349633 - cp's OEIS frontend

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

Antti Karttunen, Nov 26 2021

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.

			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.

Cf. A000203, A003961.
Cf. A349628 (denominators).
Cf. also A342671, A349161, A349162, A349633.

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 *)

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))));

A349634 Denominators of the Möbius transform of ratio A003961(n)/n.

Original entry on oeis.org

1, 2, 3, 4, 5, 3, 7, 8, 9, 5, 11, 2, 13, 7, 15, 16, 17, 9, 19, 10, 21, 11, 23, 4, 25, 13, 27, 7, 29, 15, 31, 32, 33, 17, 35, 6, 37, 19, 39, 20, 41, 21, 43, 22, 9, 23, 47, 8, 49, 25, 51, 13, 53, 27, 55, 14, 57, 29, 59, 5, 61, 31, 63, 64, 65, 33, 67, 34, 23, 35, 71, 4, 73, 37, 75, 19, 77, 39, 79, 40, 81, 41, 83, 7

Offset: 1

Antti Karttunen, Nov 28 2021

See comments in A349633.

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

Cf. A000010, A003961, A003972, A008683, A349633 (numerators).

Cf. also A349628.

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

A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
A349634(n) = denominator(sumdiv(n,d,moebius(n/d)*(A003961(d)/d)));

cp's OEIS Frontend

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

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