A261222 a(n) = number of steps to reach 0 when starting from k = n*n and repeatedly applying the map that replaces k with k - A053610(k), where A053610(k) = the number of positive squares that sum to k using the greedy algorithm.

0, 1, 2, 4, 6, 9, 12, 15, 19, 24, 29, 35, 41, 48, 55, 62, 70, 78, 87, 97, 107, 118, 129, 141, 153, 165, 178, 191, 205, 219, 234, 249, 265, 282, 299, 317, 335, 354, 373, 392, 412, 433, 454, 475, 497, 519, 542, 565, 589, 613, 638, 664, 690, 717, 744, 772, 800, 828, 857, 887, 917, 948, 979, 1010, 1042, 1074, 1107, 1140, 1174, 1208, 1243, 1278, 1314, 1351, 1388, 1426, 1464, 1503

Offset: 0

Author

Antti Karttunen, Aug 12 2015

Keywords

• nonn

Links

• Antti Karttunen, Table of n, a(n) for n = 0..6000

Crossrefs

Essentially one more than A261223.

First differences: A261224.

Cf. A000290, A053610, A260740, A261221.

Cf. also A260732, A261227.

Programs

Mathematica

Table[-1 + Length@ NestWhileList[# - Block[{m = #, c = 1}, While[a = (# - Floor[Sqrt@ #]^2) &@ m; a != 0, c++; m = a]; c] &, n^2, # != 0 &], {n, 0, 77}] (* Michael De Vlieger, Sep 08 2016, after Jud McCranie at A053610 *)

Formula

a(n) = A261221(n^2).

Other identities. For all n >= 1:

a(n) = 1 + A261223(n).