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A000397 Number of partitions into non-integral powers.
(Formerly M4212 N1757)
2

%I M4212 N1757

%S 6,32,109,288,654,1337,2506,4414,7379,11822,18273,27356,39938,56974,

%T 79607,109267,147523,196295,257715,334407,429086,545034,685917,855886,

%U 1059360,1301776,1588321,1925620,2320544,2780468,3314007,3930001,4638319,5449943,6376505,7430471,8625369,9976540,11498855,13210238,15128487,17272896,19664754,22326319,25280987,28554486,32173404,36166409,40563607,45397395,50701682,56512012,62866699,69805531,77370606,85607286,94560129,104280410,114819255,126229853,138570284,151899428,166278945,181775849,198456941,216394746,235661505,256338017,278503009,302242623,327644632,354799834,383805368,414759214,447764499,482931051

%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)+x_3^((1/2)<=n for any three distinct integers 1<=x_1<x_2<x_3. - _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 A000397 := proc(n) local a,x1,x2,x3 ; a := 0 ; for x1 from 1 to n^2 do for x2 from x1+1 to floor( (n-x1^(1/2))^2 ) do x3 := (n-x1^(1/2)-x2^(1/2))^2 ; if floor(x3) >= x2+1 then a := a+floor(x3-x2) ; fi; od: od: a ; end: for n from 5 do printf("%d,\n",A000397(n)) ; od: # _R. J. Mathar_, Sep 29 2009

%t A000397[n_] := Module[{a, x1, x2, x3}, a = 0; For[x1 = 1, x1 <= n^2, x1++, For[x2 = x1+1, x2 <= Floor[(n-x1^(1/2))^2], x2++, x3 = (n-x1^(1/2) - x2^(1/2))^2 ; If[Floor[x3] >= x2+1, a = a + Floor[x3-x2]]]]; a]; Reap[ For[n = 5, n <= 40, n++, Print[an = A000397[n]; Sow[an]]]][[2, 1]] (* _Jean-Fran├žois Alcover_, Feb 08 2016, after _R. J. Mathar_ *)

%K nonn

%O 5,1

%A _N. J. A. Sloane_

%E More terms from _R. J. Mathar_, Sep 29 2009

%E More terms from _Sean A. Irvine_, Nov 14 2010

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Last modified April 18 10:56 EDT 2021. Contains 343087 sequences. (Running on oeis4.)