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A000327 Number of partitions into non-integral powers.

%I A000327 M3819 N1563 #32 Jul 01 2022 22:18:09
%S A000327 1,5,12,23,39,62,91,127,171,228,294,370,461,561,677,811,955,1121,1303,
%T A000327 1499,1719,1960,2218,2499,2806,3131,3485,3868,4274,4706,5166,5658,
%U A000327 6175,6725,7309,7923,8572,9256,9972,10728,11521,12349,13218,14126,15072
%N A000327 Number of partitions into non-integral powers.
%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. - _R. J. Mathar_, Jul 03 2009
%D A000327 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A000327 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A000327 Seth A. Troisi, <a href="/A000327/b000327.txt">Table of n, a(n) for n = 3..1000</a>
%H A000327 B. K. Agarwala and 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.
%H A000327 B. K. Agarwala and F. C. Auluck, <a href="/A000093/a000093.pdf">Statistical mechanics and partitions into non-integral powers of integers</a>, Proc. Camb. Phil. Soc., 47 (1951), 207-216. [Annotated scanned copy]
%F A000327 a(n) = A000148(n) - floor((n/2)^(3/2)). - _Seth A. Troisi_, May 25 2022
%p A000327 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
%t A000327 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_ *)
%t A000327 A000327[n_] := Sum[Min[x1 - 1, Floor[(n - x1^(2/3))^(3/2)]], {x1, 2, Floor[n^(3/2)]}];
%t A000327 Table[A000327[n], {n, 3, 80}] (* _Seth A. Troisi_, May 25 2022 *)
%Y A000327 Cf. A000148, A000158, A000160.
%K A000327 nonn
%O A000327 3,2
%A A000327 _N. J. A. Sloane_
%E A000327 More terms from _R. J. Mathar_, Sep 29 2009