A000397 Number of partitions into non-integral powers.

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

Author

N. J. A. Sloane

Keywords

• nonn

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_1R. 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

• 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]

Programs

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 *)

Extensions

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

More terms from Sean A. Irvine, Nov 14 2010