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%I #26 Jan 02 2023 12:30:51
%S 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,
%T 24,24,27,27,27,30,30,32,32,35,35,35,35,38,39,39,40,41,43,43,43,45,48,
%U 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
%N n minus the least number of squares that add up to n: a(n) = n - A002828(n).
%C The associated beanstalk-sequence starts from a(0) as: 0, 3, 6, 8, 11, 15, 16, 18, 21, ... (A276573).
%H Robert Israel, <a href="/A255131/b255131.txt">Table of n, a(n) for n = 0..10000</a>
%H <a href="http://list.seqfan.eu/oldermail/seqfan/2013-August/011588.html">Discussion on the SeqFan mailing list</a>
%F a(n) = n - A002828(n).
%F a(n) = A260740(n) + A062535(n).
%e a(0) = 0, because no squares are needed for an empty sum, and 0 - 0 = 0.
%e a(3) = 0, because 3 cannot be represented as a sum of less than three squares (1+1+1), and 3 - 3 = 0.
%e a(4) = 3, because 4 can be represented as a sum of just one square (namely 4 itself), and 4 - 1 = 3.
%p f:= proc(n) local F, x;
%p if issqr(n) then return n-1 fi;
%p if nops(select(t -> t[1] mod 4 = 3 and t[2]::odd, ifactors(n)[2])) = 0 then return n-2 fi;
%p x:= n/4^floor(padic:-ordp(n, 2)/2);
%p if x mod 8 = 7 then n-4 else n-3 fi
%p end proc:
%p f(0):= 0:
%p map(f, [$0..100]); # _Robert Israel_, Mar 27 2018
%t {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 *)
%Y Subsequence: A005563.
%Y Cf. A000290, A002828, A062535, A260731, A260732, A260733, A260734, A262689, A276573.
%Y Cf. also A011371, A236840, A260740.
%K nonn
%O 0,5
%A _Antti Karttunen_, Feb 24 2015
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