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

Author

Martin Renner, Aug 22 2011

Keywords

• nonn
• base

Comments

If b>=2 and n >= 2*b^3, then S(n,3,b)

Examples

			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.
		

Links

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

Crossrefs

Cf. A193594, A194025.

Programs

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

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

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