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%I M3879 N1590 #26 Apr 10 2023 12:50:31
%S 1,5,18,45,100,185,323,522,804,1180,1687,2322,3139,4146,5377,6859,
%T 8645,10733,13203,16058,19356,23132,27460,32330,37846,44031,50954,
%U 58637,67203,76613,87021,98443,110951,124616,139526,155681,173246,192243
%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 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 Delbert L. Johnson, <a href="/A000339/b000339.txt">Table of n, a(n) for n = 2..7032</a>
%H B. K. Agarwala and 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 A000339 := 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 then a := a+floor(x2-x1+1) ; fi; od: a ; end: for n from 2 to 80 do printf("%d,\n",A000339(n)) ; od: # _R. J. Mathar_, Sep 29 2009
%t A000339[n_] := Module[{a, x1, x2}, a = 0; For[x1 = 1 , x1 <= n^2 , x1++, x2 = (n-x1^(1/2))^2; If[Floor[x2] >= x1, a = a+Floor[x2-x1+1]]]; a]; Reap[ For[n = 2, n <= 80, n++, Print[an = A000339[n]]; Sow[an]]][[2, 1]] (* _Jean-François Alcover_, Feb 07 2016, after _R. J. Mathar_ *)
%K nonn
%O 2,2
%A _N. J. A. Sloane_
%E More terms from _R. J. Mathar_, Sep 29 2009
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