A096923 -id:A096923 - cp's OEIS frontend

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

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,",")))}

A096347 Least number with n preimages (or immediate predecessors) under f(n) = n + (product of nonzero digits of n).

Original entry on oeis.org

1, 2, 12, 102, 116, 1098, 2072, 1014, 101134, 11014, 1011098, 1003525, 41021255, 210110985, 403130555, 481104655, 4401225555, 4811125555, 86413249555, 39011218055

Offset: 0

Jason Earls, Jun 29 2004

First occurrence of k in A096972.

a(20) > 10^11. [From Donovan Johnson, Nov 22 2009]

			a(3)=102 because 102 is the least number with three direct predecessors, 66: 66+6*6 = 102, 74: 74+7*4 = 102, 101: 101+1*1 = 102.

P. A. Loomis, An Introduction to Digit Product Sequences (see link).

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. A096972, A063108, A063425, A096922, A096923, A096924, A096925, A096926, A096927, A096928, A096929, A096930, A096931.

a(8) to a(11) from Klaus Brockhaus, Jul 07 2004
a(12) from Robert G. Wilson v, Jul 15 2004
a(13)-a(19) from Donovan Johnson, Nov 22 2009

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

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

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

Original entry on oeis.org

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 *)

A096972 Number of preimages of n (or immediate predecessors) under map f(k) = k + (product of nonzero digits of k).

Original entry on oeis.org

0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 2, 0, 2, 0, 2, 0, 2, 0, 1, 0, 2, 1, 1, 0, 2, 0, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 2, 0, 0, 1, 1, 0, 2, 1, 1, 1, 0, 0, 2, 0, 0, 0, 1, 2, 1, 0, 1, 0, 1, 0, 2, 0, 0, 1, 2, 1, 1, 0, 1, 0, 0, 0, 2, 1, 0, 1, 1, 0, 2, 1, 0, 0, 0, 1, 2, 0, 2, 1, 0, 0, 1, 0, 1, 1, 0, 0, 2, 1, 1, 1, 3, 0, 2, 0

Offset: 1

Robert G. Wilson v, Jul 16 2004

Index entries for Colombian or self numbers and related sequences

Cf. A063108, A096347, A063425, A096922, A096923, A096924, A096925, A096926, A096927, A096928, A096929, A096930, A096931, A230106 (a variant).

f[n_] := Block[{s = Sort[ IntegerDigits[ n]]}, While[ s[[1]] == 0, s = Drop[s, 1]]; n + Times @@ s]; t = Table[0, {111}]; Do[ t[[f [n]]]++, {n, 111}]; Table[ t[[n]], {n, 105}]

A095992 a(1) = 30; for n > 1, a(n+1) = a(n) + {product of nonzero digits of a(n)}.

Original entry on oeis.org

30, 33, 42, 50, 55, 80, 88, 152, 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, 4506, 4626, 4914, 5058, 5258, 5658

Offset: 1

Julien Piquet (julipiquet(AT)yahoo.fr), Jul 18 2004

From a puzzle; explanation found by Pierre Roger.

Frank Rubin, Number Series

Cf. A063108, A096347, A096972, A063108, A063425, A096922, A096923, A096924, A096925, A096926, A096927, A096928, A096929, A096930, A096931, A096973, A096987.

a[1] = 30; a[n_] := a[n] = Block[{s = Sort[ IntegerDigits[a[n - 1]]]}, While[ s[[1]] == 0, s = Drop[s, 1]]; a[n - 1] + Times @@ s]; Table[ a[n], {n, 50}]
nxt[n_] := n+Times@@Select[IntegerDigits[n], #>0&]; NestList[nxt,30,50] (* Harvey P. Dale, Jan 08 2011 *)

The proposer suggests that this web site may contain other sequences also.

Edited and extended by Robert G. Wilson v and Klaus Brockhaus, Jul 20 2004

cp's OEIS Frontend

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

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A096347 Least number with n preimages (or immediate predecessors) under f(n) = n + (product of nonzero 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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A096924 Numbers n for which there are exactly three k such that n = k + (product of nonzero digits of k).

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

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A096972 Number of preimages of n (or immediate predecessors) under map f(k) = k + (product of nonzero digits of k).

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A095992 a(1) = 30; for n > 1, a(n+1) = a(n) + {product of nonzero digits of a(n)}.

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