A270418 Numerator of the rational number obtained when exponents in prime factorization of n are reinterpreted as alternating binary sums (A065620).

1, 2, 3, 4, 5, 6, 7, 1, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 3, 25, 26, 1, 28, 29, 30, 31, 1, 33, 34, 35, 36, 37, 38, 39, 5, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 2, 55, 7, 57, 58, 59, 60, 61, 62, 63, 1, 65, 66, 67, 68, 69, 70, 71, 9, 73, 74, 75, 76, 77, 78, 79, 80, 81

Offset: 1

Author

Antti Karttunen, May 23 2016

Keywords

• nonn
• frac
• easy
• 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. A270419 (gives the denominators).

Cf. A262675 (indices of ones).

Cf. A000069, A001969, A010060, A020639, A028234, A065620, A067029.

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

Differs from A056192 for the first time at n=32, which here a(32)=1, while A056192(32)=4.

Programs

Mathematica

s[0] = 0; s[n_]:= s[n]= If[OddQ[n], 1 - 2*s[(n-1)/2], 2*s[n/2]]; f[p_, e_] := p^(ThueMorse[e] * s[e]); a[n_] := Times @@ f @@@ FactorInteger[n]; a[1] = 1; Array[a, 100] (* Amiram Eldar, Sep 05 2023 *)

PARI

A270418(n)={n=factor(n);n[,2]=apply(A065620,n[,2]);numerator(factorback(n))} \\ M. F. Hasler, Apr 16 2018

Formula

Multiplicative with a(p^e) = p^A065620(e) for odious e, a(p^e)=1 for evil e, or equally, a(p^e) = p^(A010060(e)*A065620(e)).

a(1) = 1, for n > 1, a(n) = a(A028234(n)) * A020639(n)^( A010060(A067029(n)) * A065620(A067029(n)) ).

Other identities. For all n >= 1:

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