A063108 -id:A063108 - cp's OEIS frontend

A096287 Number of iterations of n -> n + (product of nonzero digits of n) needed for the trajectory of n to join the trajectory of A063108.

0, 0, 5, 0, 6, 4, 3, 0, 2, 5, 4, 3, 1, 2, 2, 0, 3, 1, 5, 1, 10, 0, 9, 2, 7, 0, 9, 4, 8, 8, 5, 1, 7, 4, 6, 2, 15, 0, 2, 4, 8, 6, 5, 3, 7, 3, 7, 4, 16, 5, 17, 1, 2, 1, 4, 16, 7, 14, 1, 2, 4, 0, 322, 3, 6, 1, 3, 1, 17, 2, 16, 16, 17, 0, 6, 2, 1, 15, 14, 3, 321, 14, 4, 1, 15, 15, 13, 2, 320, 12, 3, 6, 2, 16

Offset: 1

Jason Earls, Jun 23 2004

Loomis has verified that all n up to 1000000 eventually join the trajectory of A063108.

			a(3)=5 because the trajectory for 1 (Sequence A063108) starts
1->2->4->8->16->22->26->38->62->74...
and the sequence for 3 starts
3->6->12->14->18->26->38->62->74...
so the sequence beginning with 3 joins A063108 after 5 steps.

Paul Tek, 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]
Paul Tek, C program for this sequence
Index entries for Colombian or self numbers and related sequences

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A063114 a(n) = n + product of the nonzero digits of n.

Original entry on oeis.org

2, 4, 6, 8, 10, 12, 14, 16, 18, 11, 12, 14, 16, 18, 20, 22, 24, 26, 28, 22, 23, 26, 29, 32, 35, 38, 41, 44, 47, 33, 34, 38, 42, 46, 50, 54, 58, 62, 66, 44, 45, 50, 55, 60, 65, 70, 75, 80, 85, 55, 56, 62, 68, 74, 80, 86, 92, 98, 104, 66, 67, 74, 81, 88, 95, 102, 109, 116

Offset: 1

N. J. A. Sloane, Aug 08 2001

			a(59) = 59 + 5*9 = 104. a(66) = 66 + 6*6 = 102.

Harry J. Smith, Table of n, a(n) for n = 1..1000
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. A051801, A063108, A063112, A063113, A063543.

var stk: stack; end; for n := 1 to 80 do s := itoa(n); for j := 0 to length(s) -1 do k := atoi(s[j..j]); if k > 0 then stack_push(stk,k); end; end; write(n + product(stack2array(stk))," "); end;

a063114 n = n + a051801 n -- Reinhard Zumkeller, Jan 15 2012

Table[i+Times@@(IntegerDigits[i]/. 0->1), {i, 70}]

a(n) = n + vecprod(select(x->(x!=0), digits(n))) \\ Harry J. Smith, Aug 19 2009

a(n) = n + A051801(n). - Reinhard Zumkeller, Jan 15 2012

More terms from Robert G. Wilson v and Klaus Brockhaus, 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

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

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 *)

A063112 a(1) = 1; a(n+1) = a(n) + product of nonzero digits of a(n) when written in base 3. Display sequence in base 3.

Original entry on oeis.org

1, 2, 11, 12, 21, 100, 101, 102, 111, 112, 121, 200, 202, 220, 1001, 1002, 1011, 1012, 1021, 1100, 1101, 1102, 1111, 1112, 1121, 1200, 1202, 1220, 2001, 2010, 2012, 2100, 2102, 2120, 2201, 2212, 10011, 10012, 10021, 10100, 10101, 10102, 10111, 10112

Offset: 1

N. J. A. Sloane, Aug 08 2001

Harry J. Smith, Table of n, a(n) for n = 1..1000
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. A063108, A063113, A063114.

baseE(x, b)= { local(d, e, f); e=0; f=1; while (x>0, d=x-b*(x\b); x\=b; e+=d*f; f*=10); return(e) }
ProdNzD(x)= { local(d, p); p=1; while (x>9, d=x-10*(x\10); if (d, p*=d); x\=10); return(p*x) }
{ for (n=1, 1000, if (n>1, a=baseE(b+= ProdNzD(a), 3), a=1; b=1); write("b063112.txt", n, " ", a) ) } \\ Harry J. Smith, Aug 19 2009

More terms from Vladeta Jovovic, Aug 10 2001

A063113 a(1) = 1; a(n+1) = a(n) + product of nonzero digits of a(n) when written in base 3. But display sequence in base 10.

Original entry on oeis.org

1, 2, 4, 5, 7, 9, 10, 11, 13, 14, 16, 18, 20, 24, 28, 29, 31, 32, 34, 36, 37, 38, 40, 41, 43, 45, 47, 51, 55, 57, 59, 63, 65, 69, 73, 77, 85, 86, 88, 90, 91, 92, 94, 95, 97, 99, 101, 105, 109, 110, 112, 113, 115, 117, 118, 119, 121, 122, 124, 126, 128, 132, 136, 138

Offset: 1

N. J. A. Sloane, Aug 08 2001

Harry J. Smith, Table of n, a(n) for n = 1..1000
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. A063108, A063112, A063114.

f[n_Integer] := (a = Sort[ IntegerDigits[n, 3]]; While[ a[[1]] == 0, a = Delete[a, 1]]; n + Apply[ Times, a] ); NestList[f, 1, 65]

baseE(x, b)= { local(d,e,f); e=0; f=1; while (x>0, d=x-b*(x\b); x\=b; e+=d*f; f*=10); return(e) }
ProdNzD(x)= { local(d,p); p=1; while (x>9, d=x-10*(x\10); if (d, p*=d); x\=10); return(p*x) }
{ for (n=1, 1000, if (n>1, a=baseE(b+= ProdNzD(a), 3), a=1; b=1); write("b063113.txt", n, " ", b) ) } \\ Harry J. Smith, Aug 19 2009

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

A230102 a(0)=1; thereafter a(n+1) = a(n) + (product of digits of a(n)).

Original entry on oeis.org

1, 2, 4, 8, 16, 22, 26, 38, 62, 74, 102, 102, 102, 102, 102, 102, 102, 102, 102, 102, 102, 102, 102, 102, 102, 102, 102, 102, 102, 102, 102, 102, 102, 102, 102, 102, 102, 102, 102, 102, 102, 102, 102, 102, 102, 102, 102, 102, 102, 102, 102

Offset: 0

N. J. A. Sloane, Oct 12 2013

Never gets above 102.

Cf. A007954, A051801, A063108, A230099, A232485.

a230102 n = a230102_list !! n
a230102_list = iterate a230099 1  -- Reinhard Zumkeller, Oct 13 2013

NestList[#+Times@@IntegerDigits[#]&,1,50] (* Harvey P. Dale, Jul 30 2023 *)

A232485 a(1) = 3; thereafter a(n+1) = a(n) + product of digits of a(n).

Original entry on oeis.org

3, 6, 12, 14, 18, 26, 38, 62, 74, 102, 102, 102, 102, 102, 102, 102, 102, 102, 102, 102, 102, 102, 102, 102, 102, 102, 102, 102, 102, 102, 102, 102, 102, 102, 102, 102, 102, 102, 102, 102, 102, 102, 102, 102, 102, 102, 102, 102, 102, 102, 102, 102, 102, 102, 102, 102, 102, 102, 102, 102, 102, 102, 102, 102, 102, 102

Offset: 1

N. J. A. Sloane, Nov 29 2013

GCHQ, The GCHQ Puzzle Book, Penguin, 2016. See page 36.

Cf. A063108, A230102, A232486.

NestList[#+Times@@IntegerDigits[#]&,3,70] (* Harvey P. Dale, Jul 02 2017 *)

cp's OEIS Frontend

A096287 Number of iterations of n -> n + (product of nonzero digits of n) needed for the trajectory of n to join the trajectory of A063108.

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A063114 a(n) = n + product of the nonzero digits of 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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A096347 Least number with n preimages (or immediate predecessors) under f(n) = n + (product of nonzero digits of n).

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

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A063112 a(1) = 1; a(n+1) = a(n) + product of nonzero digits of a(n) when written in base 3. Display sequence in base 3.

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A063113 a(1) = 1; a(n+1) = a(n) + product of nonzero digits of a(n) when written in base 3. But display sequence in base 10.

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