A193594 Number of attractors under iteration of sum of cubes of digits in base b.

1, 6, 9, 6, 9, 34, 11, 28, 15, 46, 22, 50, 49, 60, 86, 86, 60, 128, 22, 58, 118, 93, 64, 185, 5, 109, 102, 100, 122, 184, 51, 94, 205, 131, 173, 275, 67, 216, 131, 127, 34, 360, 114, 78, 215, 213, 393, 479, 76, 254, 634, 197, 214, 496, 348, 170, 437, 349, 290

Offset

2,2

Comments

If b>=2 and a >= 2*b^3, then S(a,3,b)<a. For each positive integer a, there is a positive integer m such that S^m(a,3,b)<2*b^3. (Grundman/Teeple, 2001, Lemma 8 and Corollary 9.)

Links

Table of n, a(n) for n=2..60.

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); (153); (370); (371); (407) or (55, 250,133); (136, 244); (160, 217, 352); (919, 1459) under the iteration of sum of cubes of digits, hence there are five fixed points, two 2-cycles and two 3-cycles. Therefore a(10) = 5 + 2*2 + 2*3 = 15.

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 2*b^3 do Q:=S(i, 3, b); A:=[op(A), Q[nops(Q)]]; od: return(nops({op(A)})); end:
seq(NumberOfAttractors(b), b=2..20);

Crossrefs

Cf. A193586.

Sequence in context: A118947 A023410 A010725 * A011480 A283743 A339764

Adjacent sequences: A193591 A193592 A193593 * A193595 A193596 A193597

Keyword

nonn,base

Author

Martin Renner, Jul 31 2011

Status

approved