A260734 a(n) = number of steps needed to reach (n^2)-1 when starting from k = ((n+1)^2)-1 and repeatedly applying the map that replaces k with k - A002828(k), where A002828(k) = the least number of squares that add up to k.

1, 2, 2, 4, 4, 5, 5, 7, 7, 7, 8, 10, 9, 10, 10, 13, 13, 14, 13, 15, 15, 16, 17, 17, 19, 19, 19, 20, 20, 22, 22, 23, 24, 25, 24, 26, 27, 25, 28, 29, 29, 29, 30, 31, 33, 33, 33, 34, 35, 35, 37, 36, 39, 37, 38, 40, 42, 40, 42, 42, 43, 42, 45, 45, 45, 48, 45, 49, 50, 50, 48, 53, 50, 51, 54, 52, 53, 54, 56, 56, 56, 58, 59, 59, 60, 60, 60, 61, 62, 62, 62, 65, 66, 66, 65

Offset: 1

Author

Antti Karttunen, Aug 12 2015

Keywords

• nonn

Links

• Antti Karttunen, Table of n, a(n) for n = 1..1024

Crossrefs

First differences of both A260732 and A260733.

Cf. A000290, A002828, A255131.

Cf. also A261224.

Programs

Mathematica

Table[Length[#] - 1 &@ NestWhileList[# - (If[First@ # > 0, 1, Length[ First@ Split@ #] + 1] &@ SquaresR[Range@ 4, #]) &, ((n + 1)^2) - 1, # != (n^2) - 1 &], {n, 95}] (* Michael De Vlieger, Sep 08 2016, after Harvey P. Dale at A002828 *)

Formula

a(n) = A260731(((n+1)^2)-1) - A260731((n^2)-1). [The definition.]

Equally, for all n >= 1:

a(n) = A260731((n+1)^2) - A260731(n^2).

a(n) = A260732(n+1) - A260732(n).

a(n) = A260733(n+1) - A260733(n).