This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A299150 #42 May 09 2025 04:36:03
%S A299150 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,
%T A299150 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,
%U A299150 4,2,4,4,4,2,16,2,2,16,4,4,4,2,16,128,2,2
%N A299150 Denominators of the positive solution to n = Sum_{d|n} a(d) * a(n/d).
%H A299150 Antti Karttunen, <a href="/A299150/b299150.txt">Table of n, a(n) for n = 1..65537</a> (first 1000 terms from Andrew Howroyd)
%F A299150 a(n) = denominator(n*A317848(n)/A165825(n)) = A165825(n)/(A037445(n) * A006519(n)). - _Andrew Howroyd_, Aug 09 2018
%F A299150 a(n) = A046644(n)/A006519(n). - _Andrew Howroyd_ and _Antti Karttunen_, Aug 30 2018
%F A299150 From _Antti Karttunen_, Sep 03 2018: (Start)
%F A299150 a(n) = 2^A318440(n).
%F A299150 Multiplicative with a(2^e) = 2^A011371(e), a(p^e) = 2^A005187(e) for odd primes p.
%F A299150 Multiplicative with a(p^e) = 2^(((1+A000035(p))*e)-A000120(e)) for all primes p.
%F A299150 (End)
%e A299150 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.
%t A299150 nn=50;
%t A299150 sys=Table[n==Sum[a[d]*a[n/d],{d,Divisors[n]}],{n,nn}];
%t A299150 Denominator[Array[a,nn]/.Solve[sys,Array[a,nn]][[2]]]
%t A299150 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 *)
%o A299150 (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
%o A299150 (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
%o A299150 (PARI) for(n=1, 100, print1(denominator(direuler(p=2, n, 1/(1-p*X)^(1/2))[n]), ", ")) \\ _Vaclav Kotesovec_, May 08 2025
%Y A299150 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.
%Y A299150 Cf. A318440 (the 2-adic valuation).
%K A299150 nonn,frac,mult
%O A299150 1,3
%A A299150 _Gus Wiseman_, Feb 03 2018
%E A299150 Keyword:mult added by _Andrew Howroyd_, Aug 09 2018