A063114 -id:A063114 - cp's OEIS frontend

A063425 Unattainable numbers: integers not expressible as k + product of nonzero digits of k (A063114).

1, 3, 5, 7, 9, 13, 15, 17, 19, 21, 25, 27, 30, 31, 36, 37, 39, 40, 43, 48, 49, 51, 52, 53, 57, 59, 61, 63, 64, 69, 71, 72, 73, 76, 79, 82, 83, 84, 87, 90, 91, 93, 96, 97, 103, 105, 113, 115, 117, 119, 121, 127, 131, 136, 137, 139, 148, 149, 151, 153, 157, 159, 163, 164

Offset: 1

Robert G. Wilson v, Aug 09 2001

Amiram Eldar, Table of n, a(n) for n = 1..10000
Paul A. Loomis, An Interesting Family of Iterated Sequences.
Paul A. Loomis, An Introduction to Digit Product Sequences, J. Rec. Math., 32 (2003-2004), 147-151.
Paul 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

Cf. A063114, A096347, A063425, A096922, A096923, A096924, A096925, A096926, A096927, A096928, A096929, A096930, A096931.

f[n_] := Block[{s = Sort[ IntegerDigits[n]]}, While[ s[[1]] == 0, s = Drop[s, 1]]; n + Times @@ s]; t = Table[0, {200}]; Do[ a = f[n]; If[a < 200, t[[a]]++ ], {n, 200}]; Select[ Range[ 200], t[[ # ]] == 0 &] (* Robert G. Wilson v, Jul 16 2004 *)

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

Original entry on oeis.org

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

Paul A. Loomis, Aug 08 2001

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

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

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

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

a063108_list = iterate a063114 1  -- Reinhard Zumkeller, Jan 15 2012

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

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 *)

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

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.

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

A096922 Numbers n for which there is a unique k such that n = k + (product of nonzero digits of k).

Original entry on oeis.org

2, 4, 6, 8, 10, 11, 20, 23, 24, 28, 29, 32, 33, 34, 35, 41, 42, 45, 46, 47, 54, 56, 58, 60, 65, 67, 68, 70, 75, 77, 78, 81, 85, 89, 92, 94, 95, 99, 100, 101, 106, 107, 108, 109, 111, 124, 125, 128, 129, 130, 132, 133, 135, 140, 141, 143, 145, 146, 147, 152, 154, 156, 158

Offset: 1

Klaus Brockhaus, Jul 15 2004

			21 is the unique k such that k + (product of nonzero digits of k) = 23, hence 23 is a term.

%H P. A. Loomis, An Interesting Family of Iterated Sequences
%H P. A. Loomis, An Introduction to Digit Product Sequences, J. Rec. Math., 32 (2003-2004), 147-151.
%H P. A. Loomis, An Introduction to Digit Product Sequences, J. Rec. Math., 32 (2003-2004), 147-151. [Annotated archived copy]
%H Index entries for Colombian or self numbers and related sequences

Cf. A063114, A096347, A063425, A096922, A096923, A096924, A096925, A096926, A096927, A096928, A096929, A096930, A096931.

f[n_] := Block[{s = Sort[ IntegerDigits[n]]}, While[ s[[1]] == 0, s = Drop[s, 1]]; n + Times @@ s]; t = Table[0, {200}]; Do[ a = f[n]; If[a < 200, t[[a]]++ ], {n, 200}]; Select[ Range[ 200], t[[ # ]] == 1 &] (* Robert G. Wilson v, Jul 16 2004 *)

addpnd(n)=local(k,s,d);k=n;s=1;while(k>0,d=divrem(k,10);k=d[1];s=s*max(1,d[2]));n+s
{c=1;z=160;v=vector(z);for(n=1,z+1,k=addpnd(n);if(k<=z,v[k]=v[k]+1));for(j=1,length(v),if(v[j]==c,print1(j,",")))}

A230099 a(n) = n + (product of digits of n).

Original entry on oeis.org

0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 20, 23, 26, 29, 32, 35, 38, 41, 44, 47, 30, 34, 38, 42, 46, 50, 54, 58, 62, 66, 40, 45, 50, 55, 60, 65, 70, 75, 80, 85, 50, 56, 62, 68, 74, 80, 86, 92, 98, 104, 60, 67, 74, 81, 88, 95, 102, 109, 116, 123, 70, 78, 86, 94, 102, 110, 118, 126

Offset: 0

N. J. A. Sloane, Oct 12 2013

A230099, A063114, A098736, A230101 are analogs of A092391 and A062028.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Index entries for Colombian or self numbers and related sequences

Cf. A007954, A011540, A092391, A062028, A004207, A230093, A003052, A176995.

Cf. also A063114, A098736, A230101, A230103, A230104, A230105, A062028, A230102, A063108.

a230099 n = a007954 n + n  -- Reinhard Zumkeller, Oct 13 2013

with transforms; [seq(n+digprod(n), n=0..200)];

a(n) = if (n, n + vecprod(digits(n)), 0); \\ Michel Marcus, Dec 18 2018

from math import prod
def a(n): return n + prod(map(int, str(n)))
print([a(n) for n in range(78)]) # Michael S. Branicky, Jan 09 2023

a(n) = n iff n contains a digit 0 (A011540). - Bernard Schott, Jul 31 2023

A096931 Numbers n for which there are exactly ten k such that n = k + (product of nonzero digits of k).

Original entry on oeis.org

1011098, 2102125, 2411305, 2711105, 4012055, 4042055, 4086725, 4101455, 4105555, 4132755, 4310145, 6021254, 6621256, 8012765, 8013495, 8111255, 8202555, 9012405, 9302165, 10011116, 10111014, 10113255, 11011098, 12102125

Offset: 1

Klaus Brockhaus, Jul 15 2004

			965738, 978842, 988058, 991658, 1009397, 1010874, 1010936, 1010972, 1011058 and 1011082 are the only ten k such that k + (product of nonzero digits of k) = 1011098, hence 1011098 is a term.

Cf. A063114, A096347, A096922, A096923, A096924, A096925, A096926, A096927, A096928, A096929, A096930.

f[n_] := Block[{s = Sort[ IntegerDigits[n]]}, While[ s[[1]] == 0, s = Drop[s, 1]]; n + Times @@ s]; t = Table[0, {12500000}]; Do[ a = f[n]; If[a < 12500000, t[[a]]++ ], {n, 12500000}]; Do[ If[ t[[n]] == 10, Print[n]], {n, 12500000}] (* Robert G. Wilson v, Jul 16 2004 *)

{c=10;z=3000000;v=vector(z);for(n=1,z+1,k=addpnd(n);if(k<=z,v[k]=v[k]+1));for(j=1,length(v),if(v[j]==c,print1(j,",")))} \\for function addpnd see A096922

More terms from Robert G. Wilson v, Jul 16 2004

A096926 Numbers n for which there are exactly five k such that n = k + (product of nonzero digits of k).

Original entry on oeis.org

1098, 1126, 1180, 1272, 1474, 1546, 1564, 2014, 2125, 2418, 3180, 3230, 3442, 5222, 5358, 5640, 6245, 7185, 7666, 8155, 8173, 8412, 9214, 9229, 9450, 9518, 10074, 10102, 10110, 10134, 10212, 10228, 10355, 10445, 10455, 10474, 10546, 10827

Offset: 1

Klaus Brockhaus, Jul 15 2004

			937, 982, 1077, 1118 and 1122 are the only five k such that k + (product of nonzero digits of k) = 1126, hence 1126 is a term.

Cf. A063114, A096347, A096922 - A096925, A096927 - A096931.

{c=5;z=11000;v=vector(z);for(n=1,z+1,k=addpnd(n);if(k<=z,v[k]=v[k]+1));for(j=1,length(v),if(v[j]==c,print1(j,",")))} \\for function addpnd see A096922

A096924 Numbers n for which there are exactly three k such that n = k + (product of nonzero digits of k).

Original entry on oeis.org

102, 110, 118, 126, 134, 150, 180, 202, 216, 225, 234, 260, 272, 312, 338, 366, 404, 414, 420, 455, 456, 512, 534, 542, 564, 576, 586, 635, 645, 712, 734, 750, 786, 808, 818, 827, 837, 840, 894, 920, 939, 970, 980, 1018, 1020, 1034, 1042, 1072, 1074, 1075

Offset: 1

Klaus Brockhaus, Jul 15 2004

			76, 109 and 114 are the only three k such that k + (product of nonzero digits of k) = 118, hence 118 is a term.

Cf. A063114, A096347, A096922, A096923, A096925 - A096931.

{c=3;z=1100;v=vector(z);for(n=1,z+1,k=addpnd(n);if(k<=z,v[k]=v[k]+1));for(j=1,length(v),if(v[j]==c,print1(j,",")))} \\for function addpnd see A096922

A096925 Numbers n for which there are exactly four k such that n = k + (product of nonzero digits of k).

Original entry on oeis.org

116, 405, 430, 474, 530, 546, 642, 744, 774, 836, 854, 855, 930, 958, 1038, 1055, 1070, 1088, 1104, 1110, 1116, 1134, 1154, 1162, 1236, 1366, 1400, 1405, 1418, 1430, 1455, 1530, 1642, 1744, 1774, 1836, 1854, 1855, 1930, 1958, 2112, 2137, 2185, 2199, 2205

Offset: 1

Klaus Brockhaus, Jul 15 2004

			279, 345, 381 and 401 are the only four k such that k + (product of nonzero digits of k) = 405, hence 405 is a term.

Cf. A063114, A096347, A096922 - A096924, A096926 - A096931.

Take[Select[Tally[Table[k+Times@@(IntegerDigits[k]/.(0->1)),{k,100000}]],#[[2]] == 4&][[All,1]]//Sort,50] (* Harvey P. Dale, Oct 12 2022 *)

{c=4;z=2210;v=vector(z);for(n=1,z+1,k=addpnd(n);if(k<=z,v[k]=v[k]+1));for(j=1,length(v),if(v[j]==c,print1(j,",")))} \\for function addpnd see A096922

A096927 Numbers n for which there are exactly six k such that n = k + (product of nonzero digits of k).

Original entry on oeis.org

2072, 3525, 9170, 9190, 11098, 11116, 11474, 11564, 12072, 12125, 13525, 19170, 19190, 20165, 20228, 20445, 21125, 24305, 29395, 30488, 31105, 31255, 31305, 31825, 40339, 40344, 40455, 41255, 42355, 45555, 50745, 51175, 54742, 58300

Offset: 1

Klaus Brockhaus, Jul 15 2004

			1688, 1928, 1991, 2036, 2052 and 2060 are the only six k such that k + (product of nonzero digits of k) = 2072, hence 2072 is a term.

Cf. A063114, A096347, A096922 - A096926, A096928 - A096931.

With[{nn=60000},Select[Tally[Table[n+Times@@(IntegerDigits[n]/.(0->1)),{n,nn}]],#[[2]]==6&&#[[1]]<=nn&]][[All,1]]//Sort(* Harvey P. Dale, Aug 16 2018 *)

{c=6;z=60000;v=vector(z);for(n=1,z+1,k=addpnd(n);if(k<=z,v[k]=v[k]+1));for(j=1,length(v),if(v[j]==c,print1(j,",")))} \\for function addpnd see A096922

A096928 Numbers n for which there are exactly seven k such that n = k + (product of nonzero digits of k).

Original entry on oeis.org

1014, 8305, 18305, 26425, 30205, 30725, 31414, 39186, 41156, 51358, 71110, 71136, 72505, 74470, 80305, 82765, 90985, 100405, 100786, 100855, 101014, 101098, 101126, 102072, 110474, 112418, 118305, 126425, 130205, 130725, 131414, 139186

Offset: 1

Klaus Brockhaus, Jul 15 2004

			678, 854, 933, 942, 960, 1007 and 1012 are the only seven k such that k + (product of nonzero digits of k) = 1014, hence 1014 is a term.

Cf. A063114, A096347, A096922 - A096927, A096929 - A096931.

Select[Tally[Table[n+Times@@Select[IntegerDigits[n],#!=0&],{n,200000}]], #[[2]] == 7&][[All,1]]//Sort (* Harvey P. Dale, Apr 21 2018 *)

{c=7;z=140000;v=vector(z);for(n=1,z+1,k=addpnd(n);if(k<=z,v[k]=v[k]+1));for(j=1,length(v),if(v[j]==c,print1(j,",")))} \\for function addpnd see A096922

cp's OEIS Frontend

A063425 Unattainable numbers: integers not expressible as k + product of nonzero digits of k (A063114).

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

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A096922 Numbers n for which there is a unique k such that n = k + (product of nonzero digits of k).

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A230099 a(n) = n + (product of digits of n).

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A096931 Numbers n for which there are exactly ten k such that n = k + (product of nonzero digits of k).

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A096926 Numbers n for which there are exactly five k such that n = k + (product of nonzero digits of k).

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A096924 Numbers n for which there are exactly three k such that n = k + (product of nonzero digits of k).

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A096925 Numbers n for which there are exactly four k such that n = k + (product of nonzero digits of k).

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