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

%I A063108 #46 Mar 19 2025 20:31:26
%S A063108 1,2,4,8,16,22,26,38,62,74,102,104,108,116,122,126,138,162,174,202,
%T A063108 206,218,234,258,338,410,414,430,442,474,586,826,922,958,1318,1342,
%U A063108 1366,1474,1586,1826,1922,1958,2318,2366,2582,2742,2854,3174,3258,3498,4362
%N A063108 a(1) = 1; thereafter a(n+1) = a(n) + product of nonzero digits of a(n).
%C A063108 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.
%C A063108 A063114 iterated, beginning with 1. - _Reinhard Zumkeller_, Jan 15 2012
%H A063108 T. D. Noe, <a href="/A063108/b063108.txt">Table of n, a(n) for n = 1..10000</a>
%H A063108 P. A. Loomis, <a href="http://facstaff.bloomu.edu/ploomis/sequences.html">An Interesting Family of Iterated Sequences</a>
%H A063108 P. A. Loomis, <a href="http://facstaff.bloomu.edu/ploomis/itseq4.ps">An Introduction to Digit Product Sequences</a>, J. Rec. Math., 32 (2003-2004), 147-151.
%H A063108 P. A. Loomis, <a href="/A063108/a063108.pdf">An Introduction to Digit Product Sequences</a>, J. Rec. Math., 32 (2003-2004), 147-151. [Annotated archived copy]
%H A063108 <a href="/index/Coi#Colombian">Index entries for Colombian or self numbers and related sequences</a>
%F A063108 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.
%e A063108 a(2) = 1 + 1 = 2; a(3) = 4; a(6) = 16 + 1*6 = 22; a(22) = 206 + 2*6 = 218.
%p A063108 with transforms;
%p A063108 f:=proc(n) option remember; if n=1 then 1
%p A063108 else f(n-1)+digprod(f(n-1)); fi; end;
%p A063108 [seq(f(n),n=1..20)];
%p A063108 # _N. J. A. Sloane_, Oct 12 2013
%t A063108 f[ n_Integer ] := Block[{s = Sort[ IntegerDigits[ n ]]}, While[ s[[ 1 ]] == 0, s = Drop[ s, 1 ]]; n + Times @@ s]; NestList[ f, 1, 65 ]
%t A063108 nxt[n_]:=n+Times@@Select[IntegerDigits[n],#>0&]; NestList[nxt,1,50] (* _Harvey P. Dale_, Oct 10 2012 *)
%o A063108 (PARI) lista(n)={ my(a=vector(n)); a[1]=1; for(i=1, #a-1, a[i+1] = a[i] + vecprod(select(x->x, digits(a[i])))); a } \\ _Harry J. Smith_, Aug 18 2009
%o A063108 (Haskell)
%o A063108 a063108_list = iterate a063114 1  -- _Reinhard Zumkeller_, Jan 15 2012
%Y A063108 Cf. A063112, A063113, A063114, A097050, A051801, A096355, A230102, A232485, A232486, A232487, A232488.
%K A063108 base,easy,nonn,nice,look
%O A063108 1,2
%A A063108 _Paul A. Loomis_, Aug 08 2001
%E A063108 More terms from _Robert G. Wilson v_, Aug 09 2001