A350178 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.

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, 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, 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

Offset: 0

Author

Thomas Scheuerle, Dec 18 2021

Keywords

• nonn
• hear

Comments

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.

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,... .

Examples

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

Links

• Thomas Scheuerle, Table of n, a(n) for n = 0..5000

Crossrefs

Cf. A053186, A053610.

Programs

Mathematica

Table[Total[Rest[NestWhileList[#-Floor[Sqrt[#]]^2&,n,#>0&]]],{n,0,90}] (* Harvey P. Dale, Apr 27 2025 *)

PARI

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

Formula

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.

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