A273913 Consider the sequence b(k) with initial values b(1) = 1 and b(2) = n and satisfying b(k) = b(k-1) + Pd(b(k-2)), where Pd(x) is the product of the digits of x. Then b(k) eventually becomes constant, and this constant is a(n).

1902, 1902, 730, 230, 550, 420, 502, 1902, 2150, 1074, 1074, 1074, 1902, 1902, 8170, 730, 550, 730, 600, 230, 80, 230, 470, 550, 1074, 4045, 4990, 180, 230, 106, 90, 4990, 1062, 102, 902, 1230, 730, 108, 1406, 1017, 1410, 630, 2038, 505, 230, 1810, 150, 2306, 630

Offset

1,1

Comments

Maximum value in the first 10^5 terms is a(6874) = 209875, from b(128) on.

First n's whose last repetitive number of the sequence b(k) is a multiple: 1, 2, 5, 6, 34, 42, 135, 195, 460, 893, 2370, 4230, 7165, 237945.

Links

Paolo P. Lava, Table of n, a(n) for n = 1..10000

Example

b(1) = 1, b(2) = 7. Then:
b(3) = 7 + Pd(1) = 7+1 = 8; b(4) = 8 + Pd(7) = 8+7 = 15;
b(5) = 15 + Pd(8) = 15+8 = 23; b(6) = 23 + Pd(15) = 23+5 = 28;
b(7) = 28 + Pd(23) = 28+6 = 34; b(8) = 34 + Pd(28) = 34+16 = 50;
…
b(19) = 270 + Pd(214) = 270+8 = 278; b(20) = 278 + Pd(270) = 278+0 = 278;
b(21) = 278 + Pd(278) = 278+112 = 390; b(22) = 390 + Pd(278) = 390+112 = 502;
b(23) = 502 + Pd(502) = 502+0 = 502; therefore a(7) = 502.

Maple

with(numtheory); T:=proc(w) local x, y, z; x:=w; y:=1;
for z from 1 to ilog10(x)+1 do y:=y*(x mod 10); x:=trunc(x/10); od; y; end:
P:=proc(q) local a1, a2, a3, n; for n from 1 to q do a1:=1; a2:=n; a3:=T(a1)+a2;
while not (a1=a2 and a2=a3) do a1:=a2; a2:=a3; a3:=T(a1)+a2; od;  print(a1);
od; end: P(10^7);

Mathematica

a[n_] := Block[{b=0, c=1, d=n, p}, While[! (b == c == d), b=c; p = Times @@ IntegerDigits@ c; c = d; d += p]; d]; Array[a, 50] (* Giovanni Resta, Jun 20 2016 *)

Prog

(PARI) pd(n) = my(d=digits(n)); prod(k=1, #d, d[k]);
a(n) = {ba = 1; bb = n; bc = bb + pd(ba); while (!((ba ==bb) && (bc == bb)), newb = bb + pd(ba); ba = bb; bb = bc; bc = bb + pd(ba); ); bc; } \\ Michel Marcus, Jun 20 2016

Crossrefs

Cf. A007954, A063108.

Sequence in context: A078862 A190652 A177137 * A190651 A237787 A190653

Adjacent sequences: A273910 A273911 A273912 * A273914 A273915 A273916

Keyword

nonn,base,easy

Author

Paolo P. Lava, Jun 17 2016

Status

approved