A350178 - OEIS

%I #38 Jan 14 2022 07:45:01

%S 0,0,1,3,0,1,3,6,4,0,1,3,6,4,6,9,0,1,3,6,4,6,9,13,12,0,1,3,6,4,6,9,13,

%T 12,9,11,0,1,3,6,4,6,9,13,12,9,11,14,18,0,1,3,6,4,6,9,13,12,9,11,14,

%U 18,17,20,0,1,3,6,4,6,9,13,12,9,11,14,18,17,20,24,16,0,1,3,6,4,6,9,13

%N Take n and subtract the greatest square less than or equal to n. Repeat this process until 0 is reached. a(n) is the sum of all residues after subtractions.

%C Let s_1,s_2,s_3,...,s_m be the greedy partition of n into squares (n = s_1+s_2+s_3+...+s_m) such that s_1 >= s_2 >= s_3 >= ... >= s_m then a(n) = 0*s_1 + 1*s_2 + 2*s_3 + ... + (m-1)*s_m.

%C This sequence contains only numbers which can be written in the form c_1^2 + 2*c_2^2 + ... + m*c_m^2 with c_1 >= c_2 >= c_m. This excludes 2,5,7,8,... .

%H Thomas Scheuerle, <a href="/A350178/b350178.txt">Table of n, a(n) for n = 0..5000</a>

%F a(n) = n - r^2 + a(n - r^2) = a(n - r^2 + (b + r)^2) = a(n + b^2 + 2*b*r), r = floor(sqrt(n)), for any b >= 0. True because a(n) depends only on the distance to the next square <= n.

%F a(n) = Sum_{k>0} A053186^k(n).

%e a(41): 41 - 6^2 = 5; 5 - 2^2 = 1; 1 - 1^2 = 0 -> 5+1 = 6 = a(41).

%o (PARI) A350178(n)={my(r=0); while(n-=sqrtint(n)^2, r+=n); r};

%Y Cf. A053186, A053610.

%K nonn,hear

%O 0,4

%A _Thomas Scheuerle_, Dec 18 2021