A270419 Denominator of the rational number obtained when the exponents in prime factorization of n are reinterpreted as alternating binary sums (A065620).

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 3, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 2, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2

Offset: 1

Author

Antti Karttunen, May 23 2016

Keywords

• nonn
• easy
• frac
• mult

Comments

Map n -> A270418(n)/A270419(n) is a bijection from N (1, 2, 3, ...) to the set of positive rationals.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..12167

Crossrefs

Cf. A270418 (gives the numerators).

Cf. A270428 (indices of ones).

Cf. A000069, A001969, A010059, A020639, A028234, A065620, A067029.

Cf. also A270420, A270421, A270436, A270437 and permutation pair A273671/A273672.

Differs from A055229 for the first time at n=32, where a(32)=8, while A055229(32)=2.

Programs

Mathematica

s[n_] := s[n] = If[OddQ[n], -2*s[(n - 1)/2] - 1, 2*s[n/2]]; s[0] = 0; f[p_, e_] := p^If[OddQ[DigitCount[e, 2, 1]], 0, s[e]]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 01 2023 *)

PARI

A270419(n)={n=factor(n);n[,2]=apply(A065620,n[,2]);denominator(factorback(n))} \\ M. F. Hasler, Apr 16 2018

Formula

Multiplicative with a(p^e) = p^(-A065620(e)) for evil e, a(p^e)=1 for odious e, or equally, a(p^e) = p^(A010059(e) * -A065620(e)).

a(1) = 1, for n > 1, a(n) = a(A028234(n)) * A020639(n)^( A010059(A067029(n)) * -A065620(A067029(n)) ).

Other identities. For all n >= 1:

a(A270436(n)) = 1, a(A270437(n)) = n.