A193586 Number of attractors under iteration of sum of squares of digits in base n.

1, 5, 1, 6, 9, 13, 10, 8, 9, 9, 20, 13, 12, 35, 7, 15, 7, 21, 27, 37, 24, 36, 32, 26, 10, 36, 27, 28, 10, 56, 22, 26, 23, 63, 39, 27, 19, 67, 9, 36, 40, 54, 54, 48, 18, 73, 52, 75, 18, 117, 52, 74, 22, 65, 48, 53, 45, 44, 43, 18, 30, 67, 39, 49, 87, 111, 15

Offset

2,2

Comments

If b>=2 and a>=b^2 then S(a,2,b)<a. For each positive integer a, there is an positive integer m such that S^m(a,2,b)<b^2. (Grundman/Teeple, 2001, Lemma 6 and Corollary 7)

Links

Martin Renner, Table of n, a(n) for n = 2..300

H. G. Grundman, E. A. Teeple, Generalized Happy Numbers, Fibonacci Quarterly 39 (2001), nr. 5, p. 462-466.

Example

In the decimal system all integers go to (1) or (4, 16, 37, 58, 89, 145, 42, 20) under the iteration of sum of squares of digits, hence there is one fixed point and one 8-cycle. Therefore a(10) = 1 + 8 = 9.

Maple

S:=proc(n, p, b) local Q, k, N, z; Q:=[convert(n, base, b)]; for k from 1 do N:=Q[k]; z:=convert(sum(N['i']^p, 'i'=1..nops(N)), base, b); if not member(z, Q) then Q:=[op(Q), z]; else Q:=[op(Q), z]; break; fi; od; return Q; end:
NumberOfAttractors:=proc(b) local A, i, Q; A:=[]: for i from 1 to b^2 do Q:=S(i, 2, b); A:=[op(A), Q[nops(Q)]]; od: return(nops({op(A)})); end:
seq(NumberOfAttractors(b), b=2..50);

Crossrefs

Cf. A193583, A193585.

Sequence in context: A164105 A262153 A160824 * A007397 A362489 A204203

Adjacent sequences: A193583 A193584 A193585 * A193587 A193588 A193589

Keyword

nonn,base

Author

Martin Renner, Jul 31 2011

Status

approved