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For odd n, a(n) = 2; for even n, a(n) = denominator of Bernoulli(n)/n; The number 2 alternating with the elements of A006953.
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%I #29 Dec 03 2018 18:29:03

%S 2,12,2,120,2,252,2,240,2,132,2,32760,2,12,2,8160,2,14364,2,6600,2,

%T 276,2,65520,2,12,2,3480,2,85932,2,16320,2,12,2,69090840,2,12,2,

%U 541200,2,75852,2,2760,2,564,2,2227680,2,132,2,6360

%N For odd n, a(n) = 2; for even n, a(n) = denominator of Bernoulli(n)/n; The number 2 alternating with the elements of A006953.

%C There is an integer sequence b(n) = A053657(n)/2^(n-1) = 1, 1, 6, 6, 360, 360, 45360, 45360, 5443200, 5443200,... which consists of the duplicated entries of A202367.

%C The ratios of this sequence are b(n+1)/b(n) = 1, 6, 1, 60, 1, 126 .... = a(n)/2, which is a variant of A036283.

%H Antti Karttunen, <a href="/A185633/b185633.txt">Table of n, a(n) for n = 1..16384</a>

%F a(n) = A053657(n+1)/A053657(n).

%F a(2*n) = 2*A036283(n).

%F From _Antti Karttunen_, Dec 03 2018: (Start)

%F a(n) = Product_{d|n} [(1+d)^(1+A286561(n,1+d))]^A010051(1+d) - after _Peter J. Cameron_'s Mar 25 2002 comment in A006863.

%F A007947(a(n)) = A027760(n)

%F A001221(a(n)) = A067513(n).

%F A181819(a(n)) = A322312(n).

%F (End)

%p A185633 := proc(n)

%p A053657(n+1)/A053657(n) ;

%p end proc: # _R. J. Mathar_, Dec 19 2012

%t max = 52; s = Expand[Normal[Series[(-Log[1-x]/x)^z, {x, 0, max}]]]; a[n_, k_] := Denominator[Coefficient[s, x^n*z^k]]; A053657 = Prepend[LCM @@@ Table[a[n, k], {n, max}, {k, n}], 1]; a[n_] := A053657[[n+1]]/A053657[[n]]; Table[a[n], {n, 1, max}] (* _Jean-François Alcover_, Dec 20 2012 *)

%o (PARI) A185633(n) = if(n%2,2,denominator(bernfrac(n)/(n))); \\ _Antti Karttunen_, Dec 03 2018

%o (PARI) A185633(n) = { my(m=1); fordiv(n, d, if(isprime(1+d), m *= (1+d)^(1+valuation(n,1+d)))); (m); }; \\ _Antti Karttunen_, Dec 03 2018

%Y Cf. A006953, A007395 (bisections).

%Y Cf. A006863, A027760, A067513, A322312, A322315 (rgs-transform).

%K nonn

%O 1,1

%A _Paul Curtz_, Dec 18 2012

%E Name edited by _Antti Karttunen_, Dec 03 2018