A299150 Denominators of the positive solution to n = Sum_{d|n} a(d) * a(n/d).

1, 1, 2, 2, 2, 2, 2, 2, 8, 2, 2, 4, 2, 2, 4, 8, 2, 8, 2, 4, 4, 2, 2, 4, 8, 2, 16, 4, 2, 4, 2, 8, 4, 2, 4, 16, 2, 2, 4, 4, 2, 4, 2, 4, 16, 2, 2, 16, 8, 8, 4, 4, 2, 16, 4, 4, 4, 2, 2, 8, 2, 2, 16, 16, 4, 4, 2, 4, 4, 4, 2, 16, 2, 2, 16, 4, 4, 4, 2, 16, 128, 2, 2

Offset

1,3

Links

Antti Karttunen, Table of n, a(n) for n = 1..65537 (first 1000 terms from Andrew Howroyd)

Formula

a(n) = denominator(n*A317848(n)/A165825(n)) = A165825(n)/(A037445(n) * A006519(n)). - Andrew Howroyd, Aug 09 2018

a(n) = A046644(n)/A006519(n). - Andrew Howroyd and Antti Karttunen, Aug 30 2018

From Antti Karttunen, Sep 03 2018: (Start)

a(n) = 2^A318440(n).

Multiplicative with a(2^e) = 2^A011371(e), a(p^e) = 2^A005187(e) for odd primes p.

Multiplicative with a(p^e) = 2^(((1+A000035(p))*e)-A000120(e)) for all primes p.

(End)

Example

Sequence begins: 1, 1, 3/2, 3/2, 5/2, 3/2, 7/2, 5/2, 27/8, 5/2, 11/2, 9/4, 13/2, 7/2.

Mathematica

nn=50;
sys=Table[n==Sum[a[d]*a[n/d], {d, Divisors[n]}], {n, nn}];
Denominator[Array[a, nn]/.Solve[sys, Array[a, nn]][[2]]]
f[p_, e_] := 2^((1 + Mod[p, 2])*e - DigitCount[e, 2, 1]); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Apr 28 2023 *)

Prog

(PARI) a(n)={my(v=factor(n)[, 2]); denominator(n*prod(i=1, #v, my(e=v[i]); binomial(2*e, e)/4^e))} \\ Andrew Howroyd, Aug 09 2018
(PARI) A299150(n) = { my(f = factor(n), m=1); for(i=1, #f~, m *= 2^(((1+(f[i, 1]%2))*f[i, 2]) - hammingweight(f[i, 2]))); (m); }; \\ Antti Karttunen, Sep 03 2018

Crossrefs

Cf. A000010, A003958, A005187, A006519, A007431, A011371, A018804, A023900, A029935, A037445, A046643, A046644, A165825, A257098, A298971, A299119, A299149 (numerators), A299151, A317848, A317929, A317932, A318314, A318512, A318653, A318681.

Cf. A318440 (the 2-adic valuation).

Sequence in context: A140818 A139813 A172009 * A202448 A339165 A129381

Adjacent sequences: A299147 A299148 A299149 * A299151 A299152 A299153

Keyword

nonn,frac,mult

Author

Gus Wiseman, Feb 03 2018

Extensions

Keyword:mult added by Andrew Howroyd, Aug 09 2018

Status

approved