A000263 Number of partitions into non-integral powers. (Formerly M2967 N1200)

3, 14, 39, 91, 173, 307, 502, 779, 1150, 1651, 2280, 3090, 4090, 5313, 6787, 8564, 10643, 13103, 15948, 19235, 23000, 27316, 32174, 37677, 43849, 50758, 58427, 66978, 76373, 86765, 98171, 110662, 124310, 139202, 155339, 172885

Offset

3,1

Comments

a(n) counts the solutions to the inequality x_1^(1/2)+x_2^(1/2)<=n for any two distinct 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

Table of n, a(n) for n=3..38.

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

A000263 := 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+1 then a := a+floor(x2-x1) ; fi; od: a ; end: seq(A000263(n), n=3..80) ; # R. J. Mathar, Sep 29 2009

Mathematica

A000263[n_] := Module[{a, x1, x2 }, a = 0; For[x1 = 1, x1 <= n^2, x1++, x2 = (n-x1^(1/2))^2; If[Floor[x2] >= x1+1, a = a+Floor[x2-x1]]]; a]; Table[ A000263[n], {n, 3, 80}] (* Jean-François Alcover, Feb 06 2016, after R. J. Mathar *)

Crossrefs

Sequence in context: A162147 A319791 A027444 * A333293 A102590 A174517

Adjacent sequences: A000260 A000261 A000262 * A000264 A000265 A000266

Keyword

nonn

Author

N. J. A. Sloane

Extensions

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

Status

approved