A174688 - OEIS

%I #28 Nov 21 2020 06:59:00

%S 1,3,5,9,15,17,25,27,45,51,75,81,85,125,135,153,225,243,255,257,289,

%T 375,405,425,459,625,675,729,765,771,867,1125,1215,1275,1285,1377,

%U 1445,1875,2025,2125,2187,2295,2313,2601,3125,3375,3645,3825,3855,4131,4335,4369

%N All different products of not necessarily distinct terms of A001317.

%C Sequence differs from A143512 beginning with a(970).

%H Amiram Eldar, <a href="/A174688/b174688.txt">Table of n, a(n) for n = 1..10000</a>

%F Sum_{n>=1} 1/a(n) = 2.

%F Let m_a(n) = (-1)^A010060(n), if n is squarefree, and 0, otherwise (a-analog of Möbius function). Then Sum_{n>=1} m_a(n)/a(n) = 1/2.

%F A generalization: Sum_{n>=1} 1/(a(n))^s = Product_{Fermat numbers F} (1-F^(-s))^(-1), where s>0 (an analog of Euler identity for primes, where, for real s, s>1).

%e 9 = 3^2 is a term since 3 is in A001317.

%t f[n_] := FromDigits[Table[Mod[Binomial[n, k], 2], {k, 0, n}], 2]; n = 13; v = Array[f, n, 0]; vmax = v[[-1]]; s = {1}; Do[v1 = v[[k]]; rmax = Floor[Log[v1, vmax]]; s1 = v1^Range[0, rmax]; s2 = Select[Union[Flatten[Outer[Times, s, s1]]], # <= vmax &]; s = Union[s, s2], {k, 2, n}]; s (* _Amiram Eldar_, Sep 27 2020 *)

%Y Cf. A001317, A000215, A143512.

%K nonn

%O 1,2

%A _Vladimir Shevelev_, Dec 01 2010

%E Offset corrected and more terms added by _Amiram Eldar_, Sep 27 2020