

A000234


Partitions into nonintegral powers (see Comments for precise definition).
(Formerly M2730 N1095)


2



1, 3, 8, 18, 37, 72, 136, 251, 445, 770, 1312, 2202, 3632, 5908, 9501, 15111, 23781, 37083, 57293, 87813, 133530, 201574, 302265, 450317, 666743, 981488, 1437003, 2092976, 3033253, 4375104, 6282026, 8981046, 12786327, 18131492, 25612628
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OFFSET

1,2


COMMENTS

This sequence gives the number of solutions to the inequality Sum_{i=1,2,...} xi^(2/3) <= n with the constraint that 1 <= x1 <= x2 <= x3 <= ... is a list of at least 1 and no more than n integers.  R. J. Mathar, Oct 19 2007


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=1..35.
B. K. Agarwala and F. C. Auluck, Statistical mechanics and partitions into nonintegral powers of integers, Proc. Camb. Phil. Soc., 47 (1951), 207216.
B. K. Agarwala and F. C. Auluck, Statistical mechanics and partitions into nonintegral powers of integers, Proc. Camb. Phil. Soc., 47 (1951), 207216. [Annotated scanned copy]


EXAMPLE

a(3)=8 counts 5 partitions with 1 term, explicitly { 1^(2/3), 2^(2/3), 3^(2/3), 4^(2/3), 5^(2/3) }, 2 partitions into sums of 2 terms { 1^(2/3) + 1^(2/3), 1^(2/3) + 2^(2/3) } and one partition into a sum of three terms { 1^(2/3) + 1^(2/3) + 1^(2/3) }.


MAPLE

fs:=n>floor(simplify(n)): a:=proc(i, m, k) options remember: local s, l, j, m2: if(k=1) then RETURN(1) else s:=0: l:=fs(m^(3/2)): for j from 1 to min(l, i) do m2:=mj^(2/3): if(fs(m2)>=1) then s:=s+a(j, m2, k1) fi: s:=s+1 od: RETURN(s) fi: end: seq(a(fs(n^(3/2)), n, n), n=1..19); # Herman Jamke (hermanjamke(AT)fastmail.fm), May 03 2008


MATHEMATICA

fs[n_] := Floor[Simplify[n]]; a[i_, m_, k_] := a[i, m, k] = Module[{s, l, j, m2}, If[k == 1, Return[1], s = 0; l = fs[m^(3/2)]; For[j = 1, j <= Min[l, i], j++, m2 = m  j^(2/3); If[fs[m2] >= 1, s = s + a[j, m2, k1] ]; s = s+1]; Return[s]]]; A000234 = Table[an = a[fs[n^(3/2)], n, n]; Print["a(", n, ") = ", an]; an, {n, 1, 19}] (* JeanFrançois Alcover, Feb 06 2016, after Herman Jamke *)


CROSSREFS

Sequence in context: A227161 A241080 A332706 * A136376 A099845 A036635
Adjacent sequences: A000231 A000232 A000233 * A000235 A000236 A000237


KEYWORD

nonn


AUTHOR

N. J. A. Sloane


EXTENSIONS

More terms from R. J. Mathar, Oct 19 2007
One more term from Herman Jamke (hermanjamke(AT)fastmail.fm), May 03 2008
a(20)a(35) from Jon E. Schoenfield, Jan 17 2009


STATUS

approved



