# [seqfan] Re: Partitions into non-integral powers

Richard Mathar mathar at strw.leidenuniv.nl
Fri Jul 3 17:46:38 CEST 2009

```ftaw> franktaw at netscape.net franktaw at netscape.net
ftaw> Fri Jul 3 05:20:43 CEST 2009
ftaw> ...
ftaw>
ftaw> All these sequences reference a paper: "B. K. Agarwala and F. C.
ftaw> Auluck, Statistical mechanics and partitions into non-integral powers
ftaw> of integers, Proc. Camb. Phil. Soc., 47 (1951), 207-216.".  Sequence
ftaw> A000093, "[ n^(3/2) ]" (which should really be "floor(n^(3/2))" also
ftaw> references this paper, although there is no obvious connection.

The connection is that the case of restricting the sums (see below) to only
one term, the case x_1^(2/3)<=n becomes degenerate, x_1<=n^(3/2).

%I A000135
%C A000135 a(n) counts the solutions to the inequality sum_{i=1,2,..} x_i^(2/3)<=n for any number of distinct integers 1<=x_1<x_2<x_3<x_4<...
%H A000135 B. K. Agarwala, F. C. Auluck, <a href="http://dx.doi.org/10.1017/S0305004100026505">Statistical mechanics and partitions into non-integral powers of integers</a>, Proc. Camb. Phil. Soc., 47 (1951), 207-216.

%I A000148
%C A000148 a(n) counts the solutions to the inequality x_1^(2/3)+x_2^(2/3)<=n for any two integers 1<=x_1<=x_2.
%H A000148 B. K. Agarwala, F. C. Auluck, <a href="http://dx.doi.org/10.1017/S0305004100026505">Statistical mechanics and partitions into non-integral powers of integers</a>, Proc. Camb. Phil. Soc., 47 (1951), 207-216.

%I A000158
%C A000158 a(n) counts the solutions to the inequality x_1^(2/3)+x_2^(2/3)+x_3^(2/3)<=n for any three integers 1<=x_1<=x_2<=x_3.
%H A000158 B. K. Agarwala, F. C. Auluck, <a href="http://dx.doi.org/10.1017/S0305004100026505">Statistical mechanics and partitions into non-integral powers of integers</a>, Proc. Camb. Phil. Soc., 47 (1951), 207-216.

%I A000160
%C A000160 a(n) counts the solutions to the inequality x_1^(2/3)+x_2^(2/3)+x_3^(2/3)+x_4^(2/3)<=n for any four integers 1<=x_1<=x_2<=x_3<=x_4.
%H A000160 B. K. Agarwala, F. C. Auluck, <a href="http://dx.doi.org/10.1017/S0305004100026505">Statistical mechanics and partitions into non-integral powers of integers</a>, Proc. Camb. Phil. Soc., 47 (1951), 207-216.

%I A000263
%C A000263 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.
%H A000263 B. K. Agarwala, F. C. Auluck, <a href="http://dx.doi.org/10.1017/S0305004100026505">Statistical mechanics and partitions into non-integral powers of integers</a>, Proc. Camb. Phil. Soc., 47 (1951), 207-216.

%I A000298
%C A000298 a(n) counts the solutions to the inequality sum_{i=1,2,..} x_i^(1/2)<=n for any number of distinct integers 1<=x_1<x_2<x_3<....
%H A000298 B. K. Agarwala, F. C. Auluck, <a href="http://dx.doi.org/10.1017/S0305004100026505">Statistical mechanics and partitions into non-integral powers of integers</a>, Proc. Camb. Phil. Soc., 47 (1951), 207-216.

%I A000327
%C A000327 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.
%H A000327 B. K. Agarwala, F. C. Auluck, <a href="http://dx.doi.org/10.1017/S0305004100026505">Statistical mechanics and partitions into non-integral powers of integers</a>, Proc. Camb. Phil. Soc., 47 (1951), 207-216.

%I A000333
%C A000333 a(n) counts the solutions to the inequality sum_{i=1,2,..} x_i^(1/2)<=n for any number of  integers 1<=x_1<=x_2<=x_3<=....
%H A000333 B. K. Agarwala, F. C. Auluck, <a href="http://dx.doi.org/10.1017/S0305004100026505">Statistical mechanics and partitions into non-integral powers of integers</a>, Proc. Camb. Phil. Soc., 47 (1951), 207-216.

%I A000339
%C A000339 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.
%H A000339 B. K. Agarwala, F. C. Auluck, <a href="http://dx.doi.org/10.1017/S0305004100026505">Statistical mechanics and partitions into non-integral powers of integers</a>, Proc. Camb. Phil. Soc., 47 (1951), 207-216.

%I A000345
%C A000345 a(n) counts the solutions to the inequality x_1^(1/2)+x_2^(1/2)+x_3^((1/2)<=n for any three integers 1<=x_1<=x_2<=x_3.
%H A000345 B. K. Agarwala, F. C. Auluck, <a href="http://dx.doi.org/10.1017/S0305004100026505">Statistical mechanics and partitions into non-integral powers of integers</a>, Proc. Camb. Phil. Soc., 47 (1951), 207-216.

%I A000347
%C A000347 a(n) counts the solutions to the inequality x_1^(1/2)+x_2^(1/2)+x_3^((1/2)+x_4^(1/2)<=n for any four integers 1<=x_1<=x_2<=x_3<=x_4.
%H A000347 B. K. Agarwala, F. C. Auluck, <a href="http://dx.doi.org/10.1017/S0305004100026505">Statistical mechanics and partitions into non-integral powers of integers</a>, Proc. Camb. Phil. Soc., 47 (1951), 207-216.

%I A000397
%C A000397 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_1<x_2<x_3.
%H A000397 B. K. Agarwala, F. C. Auluck, <a href="http://dx.doi.org/10.1017/S0305004100026505">Statistical mechanics and partitions into non-integral powers of integers</a>, Proc. Camb. Phil. Soc., 47 (1951), 207-216.

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