

A255131


n minus the least number of squares that add up to n: a(n) = n  A002828(n).


22



0, 0, 0, 0, 3, 3, 3, 3, 6, 8, 8, 8, 9, 11, 11, 11, 15, 15, 16, 16, 18, 18, 19, 19, 21, 24, 24, 24, 24, 27, 27, 27, 30, 30, 32, 32, 35, 35, 35, 35, 38, 39, 39, 40, 41, 43, 43, 43, 45, 48, 48, 48, 50, 51, 51, 51, 53, 54, 56, 56, 56, 59, 59, 59, 63, 63, 63, 64, 66, 66, 67, 67, 70, 71, 72, 72, 73, 74, 75, 75, 78, 80, 80, 80, 81
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OFFSET

0,5


COMMENTS

The associated beanstalksequence starts from a(0) as: 0, 3, 6, 8, 11, 15, 16, 18, 21, ... (A276573).


LINKS

Robert Israel, Table of n, a(n) for n = 0..10000
Discussion on the SeqFan mailing list


FORMULA

a(n) = n  A002828(n).
a(n) = A260740(n) + A062535(n).


EXAMPLE

a(0) = 0, because no squares are needed for an empty sum, and 0  0 = 0.
a(3) = 0, because 3 cannot be represented as a sum of less than three squares (1+1+1), and 3  3 = 0.
a(4) = 3, because 4 can be represented as a sum of just one square (namely 4 itself), and 4  1 = 3.


MAPLE

f:= proc(n) local F, x;
if issqr(n) then return n1 fi;
if nops(select(t > t[1] mod 4 = 3 and t[2]::odd, ifactors(n)[2])) = 0 then return n2 fi;
x:= n/4^floor(padic:ordp(n, 2)/2);
if x mod 8 = 7 then n4 else n3 fi
end proc:
f(0):= 0:
map(f, [$0..100]); # Robert Israel, Mar 27 2018


MATHEMATICA

{0}~Join~Table[n  (If[First@ # > 0, 1, Length[First@ Split@ #] + 1] &@ SquaresR[Range@ 4, n]), {n, 84}] (* Michael De Vlieger, Sep 08 2016, after Harvey P. Dale at A002828 *)


CROSSREFS

Subsequence: A005563.
Cf. A000290, A002828, A062535, A260731, A260732, A260733, A260734, A262689, A276573.
Cf. also A011371, A236840, A260740.
Sequence in context: A142716 A211515 A260740 * A280453 A289903 A285047
Adjacent sequences: A255128 A255129 A255130 * A255132 A255133 A255134


KEYWORD

nonn


AUTHOR

Antti Karttunen, Feb 24 2015


STATUS

approved



