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

%I A270419 #22 Nov 17 2024 07:17:14
%S A270419 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,
%T A270419 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,
%U A270419 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
%N A270419 Denominator of the rational number obtained when the exponents in prime factorization of n are reinterpreted as alternating binary sums (A065620).
%C A270419 Map n -> A270418(n)/A270419(n) is a bijection from N (1, 2, 3, ...) to the set of positive rationals.
%H A270419 Antti Karttunen, <a href="/A270419/b270419.txt">Table of n, a(n) for n = 1..12167</a>
%F A270419 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)).
%F A270419 a(1) = 1, for n > 1, a(n) = a(A028234(n)) * A020639(n)^( A010059(A067029(n)) * -A065620(A067029(n)) ).
%F A270419 Other identities. For all n >= 1:
%F A270419 a(A270436(n)) = 1, a(A270437(n)) = n.
%t A270419 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 *)
%o A270419 (Scheme, two variants)
%o A270419 (definec (A270419 n) (cond ((= 1 n) 1) (else (* (expt (A020639 n) (* (A010059 (A067029 n)) (- (A065620 (A067029 n))))) (A270419 (A028234 n))))))
%o A270419 (define (A270419 n) (denominator (A270418perA270419 n)))
%o A270419 (definec (A270418perA270419 n) (cond ((= 1 n) 1) (else (* (expt (A020639 n) (A065620 (A067029 n))) (A270418perA270419 (A028234 n))))))
%o A270419 (PARI) A270419(n)={n=factor(n);n[,2]=apply(A065620,n[,2]);denominator(factorback(n))} \\ _M. F. Hasler_, Apr 16 2018
%Y A270419 Cf. A270418 (gives the numerators).
%Y A270419 Cf. A270428 (indices of ones).
%Y A270419 Cf. A000069, A001969, A010059, A020639, A028234, A065620, A067029.
%Y A270419 Cf. also A270420, A270421, A270436, A270437 and permutation pair A273671/A273672.
%Y A270419 Differs from A055229 for the first time at n=32, where a(32)=8, while A055229(32)=2.
%K A270419 nonn,easy,frac,mult
%O A270419 1,8
%A A270419 _Antti Karttunen_, May 23 2016