A063108 a(1) = 1; thereafter a(n+1) = a(n) + product of nonzero digits of a(n).

1, 2, 4, 8, 16, 22, 26, 38, 62, 74, 102, 104, 108, 116, 122, 126, 138, 162, 174, 202, 206, 218, 234, 258, 338, 410, 414, 430, 442, 474, 586, 826, 922, 958, 1318, 1342, 1366, 1474, 1586, 1826, 1922, 1958, 2318, 2366, 2582, 2742, 2854, 3174, 3258, 3498, 4362

Offset

1,2

Comments

Conjecture: no matter what the starting term is, the sequence eventually joins this one. This should be true in any base - base 2, for example, is trivial.

A063114 iterated, beginning with 1. - Reinhard Zumkeller, Jan 15 2012

Links

T. D. Noe, Table of n, a(n) for n = 1..10000

P. A. Loomis, An Interesting Family of Iterated Sequences

P. A. Loomis, An Introduction to Digit Product Sequences, J. Rec. Math., 32 (2003-2004), 147-151.

P. A. Loomis, An Introduction to Digit Product Sequences, J. Rec. Math., 32 (2003-2004), 147-151. [Annotated archived copy]

Index entries for Colombian or self numbers and related sequences

Formula

A crude heuristic analysis suggests that a(n) grows roughly like (8/9 * (1-y))^(1/(1-y)) * n^(1/1-y) where y = log_10(4.5), i.e., that a(n) ~ 0.033591*n^2.8836.

Example

a(2) = 1 + 1 = 2; a(3) = 4; a(6) = 16 + 1*6 = 22; a(22) = 206 + 2*6 = 218.

Maple

with transforms;
f:=proc(n) option remember; if n=1 then 1
else f(n-1)+digprod(f(n-1)); fi; end;
[seq(f(n), n=1..20)];
# N. J. A. Sloane, Oct 12 2013

Mathematica

f[ n_Integer ] := Block[{s = Sort[ IntegerDigits[ n ]]}, While[ s[[ 1 ]] == 0, s = Drop[ s, 1 ]]; n + Times @@ s]; NestList[ f, 1, 65 ]
nxt[n_]:=n+Times@@Select[IntegerDigits[n], #>0&]; NestList[nxt, 1, 50] (* Harvey P. Dale, Oct 10 2012 *)

Prog

(PARI) ProdNzD(x)= { p=1; while (x>9, d=x-10*(x\10); if (d, p*=d); x\=10); return(p*x) } { for (n=1, 10000, if (n>1, a+=ProdNzD(a), a=1); write("b063108.txt", n, " ", a) ) } \\ Harry J. Smith, Aug 18 2009
(Haskell)
a063108_list = iterate a063114 1  -- Reinhard Zumkeller, Jan 15 2012

Crossrefs

Cf. A063112, A063113, A063114, A097050, A051801, A096355, A230102, A232485, A232486, A232487, A232488.

Sequence in context: A254062 A341817 A230102 * A161140 A257350 A257165

Adjacent sequences: A063105 A063106 A063107 * A063109 A063110 A063111

Keyword

base,easy,nonn,nice,look

Author

Paul A. Loomis, Aug 08 2001

Extensions

More terms from Robert G. Wilson v, Aug 09 2001

Status

approved