A063114 -id:A063114 - cp's OEIS frontend

A096929 Numbers n for which there are exactly eight k such that n = k + (product of nonzero digits of k).

101134, 101180, 101642, 108305, 204205, 216425, 220145, 220725, 231014, 271855, 283055, 291705, 300180, 301205, 302125, 303555, 330776, 405555, 442055, 442395, 464255, 492055, 604425, 621136, 691865, 702145, 711486, 723205, 733585, 784985

Offset: 1

Klaus Brockhaus, Jul 15 2004

			88846, 97354, 98254, 99514, 100954, 101078, 101086 and 101131 are the only eight k such that k + (product of nonzero digits of k) = 101134, hence 101134 is a term.

Cf. A063114, A096347, A096922 - A096928, A096930, A096931.

{c=8;z=800000;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

A096923 Numbers n for which there are exactly two k such that n = k + (product of nonzero digits of k).

Original entry on oeis.org

12, 14, 16, 18, 22, 26, 38, 44, 50, 55, 62, 66, 74, 80, 86, 88, 98, 104, 112, 114, 120, 122, 123, 138, 142, 144, 155, 160, 162, 166, 170, 174, 186, 188, 198, 209, 210, 212, 218, 224, 230, 237, 240, 250, 258, 261, 265, 285, 286, 294, 303, 308, 314, 316, 326, 327

Offset: 1

Klaus Brockhaus, Jul 15 2004

			18 and 22 are the only two k such that k + (product of nonzero digits of k) = 26, hence 26 is a term.

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

knzd[n_]:=n+Times@@Select[IntegerDigits[n],#!=0&]; Sort[Transpose[ Select[ Tally[ Array[ knzd,400]],Last[#]==2&]][[1]]] (* Harvey P. Dale, Nov 05 2013 *)

{c=2;z=330;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

A096930 Numbers n for which there are exactly nine k such that n = k + (product of nonzero digits of k).

Original entry on oeis.org

11014, 100774, 111014, 412055, 510142, 511146, 633296, 931395, 983025, 1008305, 1011125, 1031414, 1100774, 1101642, 1108305, 1111014, 1412055, 1510142, 1511146, 1633296, 1931395, 1983025, 2011125, 2011305, 2012725, 2110145

Offset: 1

Klaus Brockhaus, Jul 15 2004

			88486, 96454, 99073, 99154, 99316, 100594, 100654, 100718 and 100732 are the only nine k such that k + (product of nonzero digits of k) = 100774, hence 100774 is a term.

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

Select[Tally[Table[n+Times@@DeleteCases[IntegerDigits[n],0],{n, 2111000}]],#[[2]]==9&][[All,1]]//Sort (* Harvey P. Dale, Sep 15 2019 *)

{c=9;z=2120000;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

A063543 a(n) = n - product of the nonzero digits of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 18, 19, 18, 17, 16, 15, 14, 13, 12, 11, 27, 28, 26, 24, 22, 20, 18, 16, 14, 12, 36, 37, 34, 31, 28, 25, 22, 19, 16, 13, 45, 46, 42, 38, 34, 30, 26, 22, 18, 14, 54, 55, 50, 45, 40, 35, 30, 25, 20, 15, 63, 64, 58

Offset: 1

N. J. A. Sloane, Aug 14 2001

The graph somewhat resembles wisteria flowers.

			a(20) = 20 - 2 = 18.

Harry J. Smith, Table of n, a(n) for n = 1..2000
N. J. A. Sloane and Brady Haran, Amazing Graphs II (including Star Wars), Numberphile video (2019).

Cf. A051801, A063114.

[n - &*[a: k in [1..#Intseq(n)] | a ne 0 where a is Intseq(n)[k]]: n in [1..100]]; // Marius A. Burtea, Sep 16 2019

a:= n-> n-mul(i, i=subs(0=1, convert(n, base, 10))):
seq(a(n), n=1..80);  # Alois P. Heinz, Aug 18 2019

Table[n - Times@@DeleteCases[IntegerDigits[n], 0], {n, 70}] (* Alonso del Arte, Dec 15 2013 *)

a(n) = my(d=select(x->(x!=0), digits(n))); n - vecprod(d); \\ Michel Marcus, Jan 13 2020

def a(n):
    digits = map(int, str(n))
    product = 1
    for d in digits:
        if d != 0:
            product *= d
    return n - product
[a(n) for n in range(20)]
# Elisabeth Zemack, Sep 16 2019; corrected by Fabio Somenzi, Jan 13 2020

a(n) = n - A051801(n).

More terms from Larry Reeves (larryr(AT)acm.org), Aug 14 2001

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

A230106 Number of m such that m + (product of nonzero digits of m) equals n.

Original entry on oeis.org

0, 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

Offset: 0

N. J. A. Sloane, Oct 13 2013

Number of times n appears in A063114.

Index entries for Colombian or self numbers and related sequences

Cf. A063108, A063114, A230099, A230104, A063425, A096922.

Variant of A096972.

# Maple code for A063114, A230106, A063425, A096922
with(LinearAlgebra):
read transforms; # to get digprod0
M:=1000;
lis1:=Array(0..M);
lis2:=Array(0..M);
ctmax:=4;
for i from 0 to ctmax do ct[i]:=Array(0..M); od:
for n from 0 to M do
m:=n+digprod0(n);
lis1[n]:=m;
if (m <= M) then lis2[m]:=lis2[m]+1; fi;
od:
t1:=[seq(lis1[i],i=0..M)]; # A063114
t2:=[seq(lis2[i],i=0..M)]; # A230106
COMPl(t1); # A063425
for i from 1 to M do h:=lis2[i];
if h <= ctmax then ct[h]:=[op(ct[h]),i]; fi; od:
len:=nops(ct[0]); [seq(ct[0][i],i=1..len)]; # A063425 again
len:=nops(ct[1]); [seq(ct[1][i],i=1..len)]; # A096922

a(1) corrected by Zak Seidov, Oct 24 2013

A233692 The smallest prime that produces a set of n primes such that every prime after the first one is equal to the previous plus the product of its nonzero digits.

Original entry on oeis.org

2, 23, 23, 239, 239, 239, 3413, 14249, 524219, 4167379, 324550981, 2589767209, 346333812907

Offset: 1

Carlos Rivera, Dec 14 2013

This sequence was produced as solution to the problem 1270 of Claudio Meller's website (see link).

			For n=3, initial prime=23, set {23, 29, 47} because 23 -> 23+2*3=29 -> 29+2*9=47.
For n=2 to 9, the sets are
      23,     29;
      23,     29,     47;
     239,    293,    347,    431;
     239,    293,    347,    431,    443;
     239,    293,    347,    431,    443,    491;
    3413,   3449,   3881,   4073,   4157,   4297,   4801;
   14249,  14537,  14957,  16217,  16301,  16319,  16481,  16673;
  524219, 524939, 534659, 550859, 559859, 640859, 649499, 719483, 725531.

Claudio Meller, 1270 - Usando el producto digital II, Números y algos mas (in Spanish).

Cf. A051801, A063114, A233783.

checkp(p, n) = {ok = isprime(p); for (i=1, n, print1(p, ", "); digs = digits(p); np = p + prod(i=1, #digs, if (d=digs[i], d, 1)); p = np;if (i != n, ok = ok && isprime(p));); ok;} \\ Michel Marcus, Dec 15 2013

a(13) from Giovanni Resta, Dec 15 2013

A243140 Numbers n such that n appears in the sequence beginning with the digit-product of n and extended by adding successive digit-products.

Original entry on oeis.org

22, 26, 38, 55, 62, 88, 95, 102, 104, 108, 116, 122, 126, 138, 162, 174, 202, 206, 218, 234, 258, 410, 414, 430, 442, 474, 586, 826, 922, 1318, 1342, 1366, 1474, 1586, 1826, 1922, 1958, 2318, 2366, 2582, 2742, 3174, 3258, 3498, 4362

Offset: 1

Anthony Sand, May 30 2014

Numbers n>9 with following property: form a sequence b(i) whose initial term is digit-product(n). Later terms are given by the rule that b(i) = b(i-1) + digit-product(b(i-1)) and n itself appears in the sequence.

The function digit-product(n) multiplies all nonzero digits of n (A051801). For example, digit-product(1230) = 1 * 2 * 3 = 6. The resultant sequence appears in A063114, n + product of nonzero digits of n.

			The digit-product sequence for 22 begins with digit-product(22)= 4, 4 + 4 = 8, 8 + 8 = 16, 16 + 6 = 22. Since this procedure returns to the initial number 22, it belongs here.
The digit-product sequence for 102 begins with 2, 2 + 2 = 4, 4 + 4 = 8, 8 + 8 = 16, 16 + 6 = 22, 22 + 4 = 26, 26 + 12 = 38, 38 + 24 = 62, 62 + 12 = 74, 74 + 28 = 102. Since this procedure returns to the initial number 102, it belongs here.

Charles R Greathouse IV, Table of n, a(n) for n = 1..1000

Cf. A007629, A243021, A051801, A063114.

dp(n)=my(v=select(k->k>1,digits(n))); prod(i=1,#v,v[i])
is(n)=my(t=dp(n)); until(t>=n, t+=dp(t)); t==n \\ Charles R Greathouse IV, Jun 05 2014

b(i) = b(i-1) + digit-product(b(i-1)).

A096355 Number of unattainables <= 10^n, where unattainables are A063425.

Original entry on oeis.org

5, 44, 429, 4069, 39433, 388459, 3855173, 38374875, 382644491, 3819130611

Offset: 1

Jason Earls, Jun 30 2004

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. A063114.

a(9)-a(10) from Donovan Johnson, Jul 20 2010

cp's OEIS Frontend

A096929 Numbers n for which there are exactly eight k such that n = k + (product of nonzero digits of k).

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A096923 Numbers n for which there are exactly two k such that n = k + (product of nonzero digits of k).

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A096930 Numbers n for which there are exactly nine k such that n = k + (product of nonzero digits of k).

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A063543 a(n) = n - product of the nonzero digits of n.

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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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A230106 Number of m such that m + (product of nonzero digits of m) equals n.

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A233692 The smallest prime that produces a set of n primes such that every prime after the first one is equal to the previous plus the product of its nonzero digits.

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