A000148 Number of partitions into non-integral powers.

1, 2, 7, 15, 28, 45, 70, 100, 138, 183, 242, 310, 388, 481, 583, 701, 838, 984, 1152, 1337, 1535, 1757, 2001, 2262, 2545, 2855, 3183, 3540, 3926, 4335, 4770, 5233, 5728, 6248, 6801, 7388, 8005, 8658, 9345, 10064, 10824, 11620, 12452, 13324, 14236

Offset: 2

Author

N. J. A. Sloane

Keywords

• nonn

Comments

a(n) is the number of solutions to the inequality x_1^(2/3) + x_2^(2/3) <= n where 1 <= x_1 <= x_2 are any two integers. If the number of terms in the sum is not restricted to 2, we get A000234. - 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

• Seth A. Troisi, Table of n, a(n) for n = 2..1000
• 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]

Crossrefs

Cf. A000158, A000160, A000327.

Programs

Mathematica

A000148[n_] := Sum[Min[xi, Floor[(n - xi^(2/3))^(3/2)]], {xi, 1, Floor[n^(3/2)]}];
Table[A000148[n], {n, 2, 100}] (* Seth A. Troisi, May 25 2022 *)

Extensions

More terms from Sean A. Irvine, Oct 08 2009