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A194281 Number of cycles under iteration of sum of cubes of digits in base b. 2
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 (list; graph; refs; listen; history; text; internal format)
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 A172168 A321095

Adjacent sequences:  A194278 A194279 A194280 * A194282 A194283 A194284

KEYWORD

nonn,base

AUTHOR

Martin Renner, Aug 22 2011

STATUS

approved

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Last modified August 12 11:10 EDT 2020. Contains 336438 sequences. (Running on oeis4.)