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URL: https://oeis.org/A000327

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A000327
Number of partitions into non-integral powers.
3
1, 5, 12, 23, 39, 62, 91, 127, 171, 228, 294, 370, 461, 561, 677, 811, 955, 1121, 1303, 1499, 1719, 1960, 2218, 2499, 2806, 3131, 3485, 3868, 4274, 4706, 5166, 5658, 6175, 6725, 7309, 7923, 8572, 9256, 9972, 10728, 11521, 12349, 13218, 14126, 15072
OFFSET
3,2
COMMENTS
a(n) counts the solutions to the inequality x_1^(2/3) + x_2^(2/3) <= 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
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]
FORMULA
a(n) = A000148(n) - floor((n/2)^(3/2)). - Seth A. Troisi, May 25 2022
MAPLE
A000327 := proc(n) local a, x1, x2 ; a := 0 ; for x1 from 1 to floor(n^(3/2)) do x2 := (n-x1^(2/3))^(3/2) ; if floor(x2) >= x1+1 then a := a+floor(x2-x1) ; fi; od: a ; end: seq(A000327(n), n=3..80) ; # R. J. Mathar, Sep 29 2009
MATHEMATICA
A000327[n_] := Module[{a, x1, x2 }, a = 0; For[x1 = 1, x1 <= Floor[ n^(3/2)], x1++, x2 = (n - x1^(2/3))^(3/2); If[Floor[x2] >= x1+1, a = a + Floor[x2 - x1]]]; a ]; Table[A000327[n], {n, 3, 80}] (* Jean-François Alcover, Feb 07 2016, after R. J. Mathar *)
A000327[n_] := Sum[Min[x1 - 1, Floor[(n - x1^(2/3))^(3/2)]], {x1, 2, Floor[n^(3/2)]}];
Table[A000327[n], {n, 3, 80}] (* Seth A. Troisi, May 25 2022 *)
CROSSREFS
KEYWORD
nonn
EXTENSIONS
More terms from R. J. Mathar, Sep 29 2009
STATUS
approved