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%I M2967 N1200 #20 Feb 06 2016 12:10:08
%S 3,14,39,91,173,307,502,779,1150,1651,2280,3090,4090,5313,6787,8564,
%T 10643,13103,15948,19235,23000,27316,32174,37677,43849,50758,58427,
%U 66978,76373,86765,98171,110662,124310,139202,155339,172885
%N Number of partitions into non-integral powers.
%C a(n) counts the solutions to the inequality x_1^(1/2)+x_2^(1/2)<=n for any two distinct integers 1<=x_1<x_2. - _R. J. Mathar_, Jul 03 2009
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H B. K. Agarwala, F. C. Auluck, <a href="http://dx.doi.org/10.1017/S0305004100026505">Statistical mechanics and partitions into non-integral powers of integers</a>, Proc. Camb. Phil. Soc., 47 (1951), 207-216.
%H B. K. Agarwala and F. C. Auluck, <a href="/A000093/a000093.pdf">Statistical mechanics and partitions into non-integral powers of integers</a>, Proc. Camb. Phil. Soc., 47 (1951), 207-216. [Annotated scanned copy]
%p A000263 := proc(n) local a,x1,x2 ; a := 0 ; for x1 from 1 to n^2 do x2 := (n-x1^(1/2))^2 ; if floor(x2) >= x1+1 then a := a+floor(x2-x1) ; fi; od: a ; end: seq(A000263(n),n=3..80) ; # _R. J. Mathar_, Sep 29 2009
%t A000263[n_] := Module[{a, x1, x2 }, a = 0; For[x1 = 1, x1 <= n^2, x1++, x2 = (n-x1^(1/2))^2; If[Floor[x2] >= x1+1, a = a+Floor[x2-x1]]]; a]; Table[ A000263[n], {n, 3, 80}] (* _Jean-François Alcover_, Feb 06 2016, after _R. J. Mathar_ *)
%K nonn
%O 3,1
%A _N. J. A. Sloane_
%E More terms from _R. J. Mathar_, Sep 29 2009
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