A324920 a(n) is the number of iterations of the integer splitting function (A056737) necessary to reach zero.

0, 1, 2, 3, 1, 2, 2, 3, 3, 1, 4, 5, 2, 3, 3, 3, 1, 2, 4, 5, 2, 2, 2, 3, 3, 1, 6, 3, 4, 5, 2, 3, 2, 4, 4, 3, 1, 2, 3, 5, 4, 5, 2, 3, 4, 2, 3, 4, 3, 1, 3, 4, 2, 3, 4, 3, 2, 2, 4, 5, 2, 3, 6, 3, 1, 4, 3, 4, 4, 3, 4, 5, 2, 3, 4, 5, 4, 2, 4, 5, 3, 1, 6, 7, 3, 3, 6, 7, 4, 5, 2, 3, 6, 5, 3, 4, 2, 3, 4, 3, 1, 2, 6, 7, 3

Offset: 0

Author

Robert G. Wilson v, Mar 20 2019

Keywords

• nonn

Comments

The iterations always fall to zero, proof by induction: 0 is 0; 1 -> 0; 2 -> 1; 3 -> 2; 4 -> 2; n -> some number less than n.

First occurrence of k >= 0: 0, 1, 2, 3, 10, 11, 26, 83, 178, ... see A324921.

Examples

			a(0) = 0;
a(1) = 1 since 1 -> 0;
a(2) = 2 since 2 -> 1 -> 0;
a(3) = 3 since 3 -> 2 -> 1 -> 0;
a(4) = 1 since 4 -> 0; etc.
		

Crossrefs

Cf. A056737, A139693, A324921.

Programs

Mathematica

g[n_] := Block[{d = Divisors@n}, len = Length@d; If[ OddQ@ len, 0, d[[1 + len/2]] - d[[len/2]]]]; f[n_] := Length@ NestWhileList[f, n, # > 0 &] -1; Array[f, 105, 0]

PARI

a056737(n)=n=divisors(n); n[(2+#n)\2]-n[(1+#n)\2] \\ after M. F. Hasler in A056737
a(n) = my(x=n, i=0); while(x!=0, i++; x=a056737(x)); i \\ Felix Fröhlich, Mar 20 2019

Formula

a(n) = 1 iff n is a perfect square (A000290).