A270419 - OEIS

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,8

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 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.

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 *)

PROG

(Scheme, two variants)

(definec (A270419 n) (cond ((= 1 n) 1) (else (* (expt (A020639 n) (* (A010059 (A067029 n)) (- (A065620 (A067029 n))))) (A270419 (A028234 n))))))

(define (A270419 n) (denominator (A270418perA270419 n)))

(definec (A270418perA270419 n) (cond ((= 1 n) 1) (else (* (expt (A020639 n) (A065620 (A067029 n))) (A270418perA270419 (A028234 n))))))

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

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.

Sequence in context: A367698 A367931 A365297 * A275216 A062379 A325625

Adjacent sequences: A270416 A270417 A270418 * A270420 A270421 A270422

KEYWORD

nonn,easy,frac,mult

AUTHOR

Antti Karttunen, May 23 2016

STATUS

approved