A194281 Number of cycles under iteration of sum of cubes of digits in base b.

0, 1, 0, 1, 1, 8, 1, 4, 4, 6, 2, 12, 3, 7, 8, 7, 3, 16, 3, 6, 7, 7, 4, 14, 1, 8, 11, 7, 2, 20, 7, 5, 16, 9, 7, 18, 4, 7, 10, 6, 4, 24, 5, 5, 13, 6, 7, 25, 2, 10, 20, 6, 5, 23, 7, 7, 17, 9, 7, 29, 3, 10, 14, 14, 6, 21, 7, 10, 17, 18, 9, 30, 8, 10, 24, 12, 4, 28, 4, 19, 12, 11, 6, 36

Offset

2,6

Comments

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

Links

Table of n, a(n) for n=2..85.

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) = 2 + 2 = 4.

Maple

S:=proc(n, p, b) local Q, k, N, z; Q:=[n]; 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:
a:=proc(b) local Z, i, A, Q, B, C; A:=[]: for i from 1 to 2*b^3 do Q:=S(convert(i, base, b), 3, b); A:={op(A), Q[nops(Q)]}; od: Z:={}: for i from 1 while nops(A)>0 do B:=S(A[1], 3, b); C:=[seq(B[i], i=1..nops(B)-1)]: if nops(C)<>1 then Z:={op(Z), C}: fi: A:=A minus {op(B)}; od: return(nops(Z)); end:
# Martin Renner, Aug 24 2011

Prog

(Sage)
def A194281(n):
    cycle_mins = set()
    seen = {}
    for i in (1..2*n**3):
        if i not in seen:
            path = []
            while not i in path and not i in seen:
                path.append(i)
                i = sum(d**3 for d in i.digits(base=n))
            if i not in seen:
                m = min(path[path.index(i):])
                if sf(m) != m: cycle_mins.add(m)
            else: m = seen[i]
            for p in path: seen[p] = m
    return len(cycle_mins) # D. S. McNeil, Aug 24 2011

Crossrefs

Cf. A193594, A194025.

Sequence in context: A197590 A154190 A019981 * A117038 A365935 A172168

Adjacent sequences: A194278 A194279 A194280 * A194282 A194283 A194284

Keyword

nonn,base

Author

Martin Renner, Aug 22 2011

Status

approved