A000339 Number of partitions into non-integral powers.

1, 5, 18, 45, 100, 185, 323, 522, 804, 1180, 1687, 2322, 3139, 4146, 5377, 6859, 8645, 10733, 13203, 16058, 19356, 23132, 27460, 32330, 37846, 44031, 50954, 58637, 67203, 76613, 87021, 98443, 110951, 124616, 139526, 155681, 173246, 192243

Offset: 2

Author

N. J. A. Sloane

Keywords

• nonn

Comments

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

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

• Delbert L. Johnson, Table of n, a(n) for n = 2..7032
• 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.
• 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

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

Mathematica

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 *)

Extensions

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