A320002 a(0) = 1; for n > 0, a(n) = A002828(n) * a(n-A002828(n)), where A002828(n) is the least number of squares that add up to n.

1, 1, 2, 3, 3, 6, 9, 12, 18, 18, 36, 54, 54, 108, 162, 216, 216, 432, 432, 648, 864, 1296, 1944, 2592, 3888, 3888, 7776, 11664, 15552, 23328, 34992, 46656, 69984, 104976, 139968, 209952, 209952, 419904, 629856, 839808, 1259712, 1679616, 2519424, 3779136, 5038848, 7558272, 11337408, 15116544, 22674816, 22674816, 45349632

Offset: 0

Author

Antti Karttunen, Nov 24 2018

Keywords

• nonn

Comments

Product of A002828(x) computed over all x encountered when map x -> x - A002828(x) is iterated, starting from x = n, until 0 is reached.

Sequence is monotonic because A255131 is monotonic.

All terms are 3-smooth (A003586).

Links

• Antti Karttunen, Table of n, a(n) for n = 0..121

Crossrefs

Cf. A002828, A003586, A255131, A276573.

Cf. also A320008, A320009.

Programs

Mathematica

Nest[Append[#1, #2 #1[[-#2]] ] & @@ {#, If[First@ # > 0, 1, Length@ First@ Split@ # + 1] &@ SquaresR[Range@ 4, Length@ #]} &, {1}, 50] (* Michael De Vlieger, Nov 25 2018, after Harvey P. Dale at A002828 *)

PARI

istwo(n:int) = { my(f); if(n<3, return(n>=0); ); f=factor(n>>valuation(n, 2)); for(i=1, #f[, 1], if(bitand(f[i, 2], 1)==1&&bitand(f[i, 1], 3)==3, return(0))); 1 };
isthree(n:int) = { my(tmp=valuation(n, 2)); bitand(tmp, 1)||bitand(n>>tmp, 7)!=7 };
A002828(n) = if(issquare(n), !!n, if(istwo(n), 2, 4-isthree(n))); \\ From A002828
A255131(n) = (n-A002828(n));
A320002(n) = { my(m=1, v); while(n>0, v = A002828(n); m *= v; n -= v); (m); };

PARI

A320002(n) = if(0==n,1,A002828(n)*A320002(n-A002828(n)));

Formula

a(0) = 1; for n > 0, a(n) = A002828(n) * a(n-A002828(n)).