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

6, 32, 109, 288, 654, 1337, 2506, 4414, 7379, 11822, 18273, 27356, 39938, 56974, 79607, 109267, 147523, 196295, 257715, 334407, 429086, 545034, 685917, 855886, 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

Offset

5,1

Comments

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

References

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Table of n, a(n) for n=5..80.

B. K. Agarwala, F. C. Auluck, Statistical mechanics and partitions into non-integral powers of integers, Proc. Camb. Phil. Soc., 47 (1951), 207-216.

B. K. Agarwala and F. C. Auluck, Statistical mechanics and partitions into non-integral powers of integers, Proc. Camb. Phil. Soc., 47 (1951), 207-216. [Annotated scanned copy]

Maple

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

Mathematica

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-François Alcover, Feb 08 2016, after R. J. Mathar *)

Crossrefs

Sequence in context: A090382 A360839 A102359 * A200765 A239573 A130410

Adjacent sequences: A000394 A000395 A000396 * A000398 A000399 A000400

Keyword

nonn

Author

N. J. A. Sloane

Extensions

More terms from R. J. Mathar, Sep 29 2009

More terms from Sean A. Irvine, Nov 14 2010

Status

approved