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

1,2

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

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.

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 *)

Prog

(Scheme, two variants)
(definec (A270418 n) (cond ((= 1 n) 1) (else (* (expt (A020639 n) (* (A010060 (A067029 n)) (A065620 (A067029 n)))) (A270418 (A028234 n))))))
(define (A270418 n) (numerator (A270418perA270419 n)))
(definec (A270418perA270419 n) (cond ((= 1 n) 1) (else (* (expt (A020639 n) (A065620 (A067029 n))) (A270418perA270419 (A028234 n))))))
(PARI) A270418(n)={n=factor(n); n[, 2]=apply(A065620, n[, 2]); numerator(factorback(n))} \\ M. F. Hasler, Apr 16 2018

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.

Sequence in context: A050985 A367168 A367514 * A056192 A255693 A030107

Adjacent sequences: A270415 A270416 A270417 * A270419 A270420 A270421

Keyword

nonn,frac,easy,mult

Author

Antti Karttunen, May 23 2016

Status

approved