A389074 - OEIS

A389074

Numerator of ratio A034448(n) / A048250(n), where A034448 is the sum of unitary divisors and A048250 is the sum of squarefree divisors.

1, 1, 1, 5, 1, 1, 1, 3, 5, 1, 1, 5, 1, 1, 1, 17, 1, 5, 1, 5, 1, 1, 1, 3, 13, 1, 7, 5, 1, 1, 1, 11, 1, 1, 1, 25, 1, 1, 1, 3, 1, 1, 1, 5, 5, 1, 1, 17, 25, 13, 1, 5, 1, 7, 1, 3, 1, 1, 1, 5, 1, 1, 5, 65, 1, 1, 1, 5, 1, 1, 1, 15, 1, 1, 13, 5, 1, 1, 1, 17, 41, 1, 1, 5, 1, 1, 1, 3, 1, 5, 1, 5, 1, 1, 1, 11, 1, 25, 5, 65

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OFFSET

1,4

COMMENTS

The sequence of fractions in their reduced form begins as r(n) = A389074(n)/A389075(n): 1, 1, 1, 5/3, 1, 1, 1, 3, 5/2, 1, 1, 5/3, 1, 1, 1, 17/3, 1, 5/2, 1, 5/3, 1, 1, 1, 3, 13/3, 1, 7, 5/3, 1, 1, 1, 11, 1, 1, 1, 25/6, etc. As both A034448 and A048250 are multiplicative sequences, r(n) is also. Thus, if gcd(x,y)=1, then r(x*y) = r(x)*r(y), as for example, r(36) = r(4)*r(9) = 5/3 * 5/2 = 25/6.

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..65537

FORMULA

a(n) = A034448(n) / A323159(n).

MATHEMATICA

A389074[n_] := Numerator[Times @@ (Power @@@ # + 1)/Times @@ (#[[All, 1]] + 1)] & [FactorInteger[n]];

Array[A389074, 100] (* Paolo Xausa, Oct 07 2025 *)

PROG

(PARI)

A034448(n) = { my(f=factorint(n)); prod(k=1, #f~, 1+(f[k, 1]^f[k, 2])); };

A048250(n) = factorback(apply(p -> p+1, factor(n)[, 1]));

A389074(n) = numerator(A034448(n)/A048250(n));