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

0,4

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,... .

Links

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

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).

Example

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

Prog

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

Crossrefs

Cf. A053186, A053610.

Sequence in context: A112367 A320801 A035623 * A248949 A325673 A083857

Adjacent sequences: A350175 A350176 A350177 * A350179 A350180 A350181

Keyword

nonn,hear

Author

Thomas Scheuerle, Dec 18 2021

Status

approved