A031347 -id:A031347 - cp's OEIS frontend

A034051 Numbers with multiplicative digital root value 4.

4, 14, 22, 27, 39, 41, 72, 89, 93, 98, 114, 122, 127, 139, 141, 172, 189, 193, 198, 212, 217, 221, 249, 266, 271, 277, 294, 319, 333, 338, 346, 364, 379, 383, 391, 397, 411, 429, 436, 463, 492, 626, 634, 643, 662, 677, 712, 721, 727, 739, 767, 772, 776, 793

Offset: 1

Patrick De Geest, Sep 15 1998

Robert Israel, Table of n, a(n) for n = 1..5000
Eric Weisstein's World of Mathematics, Multiplicative Digital Root
Index entries for 10-automatic sequences.

Cf. A031347.

Cf. A034048, A002275, A034049, A034050, A034051, A034052, A034053, A034054, A034055, A034056 (numbers having multiplicative digital roots 0-9).

mdr:= proc(n) option remember;
if n < 10 then return(n) fi;
procname(convert(convert(n,base,10),`*`))
end proc:
select(mdr=4, [$1..10^5]); # Robert Israel, Oct 19 2015

Select[Range@ 800, FixedPoint[Times @@ IntegerDigits@ # &, #] == 4 &] (* Michael De Vlieger, Oct 19 2015 *)

t(n) = {while(n>9, n=prod(i=1, #n=digits(n), n[i])); n};
for(n=0, 1e4, if(t(n) == 4, print1(n", "))); \\ Altug Alkan, Oct 19 2015

Incorrect formula removed by Martin Renner, Oct 19 2015

A034049 Numbers with multiplicative digital root value 2.

Original entry on oeis.org

2, 12, 21, 26, 34, 37, 43, 62, 73, 112, 121, 126, 134, 137, 143, 162, 173, 211, 216, 223, 232, 261, 278, 279, 287, 297, 299, 314, 317, 322, 341, 367, 369, 371, 376, 389, 396, 398, 413, 431, 447, 469, 474, 496, 612, 621, 637, 639, 649, 666, 673, 693, 694, 713

Offset: 1

Patrick De Geest, Sep 15 1998

Robert Israel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Multiplicative Digital Root
Index entries for 10-automatic sequences.

Cf. A031347.

Cf. A034048, A002275, A034049, A034050, A034051, A034052, A034053, A034054, A034055, A034056 (numbers having multiplicative digital roots 0-9).

A031347:= proc(n) local x;
  x:= n;
  while x > 10 do
    x:= convert(convert(x,base,10),`*`)
  od;
  x
end proc:
select(A031347=2, [$1..1000]); # Robert Israel, Jan 23 2023

t = {}; n = 0; While[Length[t] < 100, n++; s = n; While[s > 10, s = Times @@ IntegerDigits[s]]; If[s == 2, AppendTo[t, n]]]; t (* T. D. Noe, Nov 15 2011 *)

t(n) = {while(n>9, n=prod(i=1, #n=digits(n), n[i])); n};
for(n=0, 1e3, if(t(n) == 2, print1(n", "))); \\ Altug Alkan, Oct 19 2015

Incorrect formula removed by Martin Renner, Oct 19 2015

A034053 Numbers with multiplicative digital root value 6.

Original entry on oeis.org

6, 16, 23, 28, 32, 44, 47, 48, 61, 68, 74, 82, 84, 86, 116, 123, 128, 132, 144, 147, 148, 161, 168, 174, 182, 184, 186, 213, 218, 224, 227, 228, 231, 238, 242, 244, 246, 264, 267, 272, 276, 281, 282, 283, 288, 289, 298, 312, 321, 328, 344, 347, 368, 374, 377

Offset: 1

Patrick De Geest, Sep 15 1998

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Multiplicative Digital Root
Index entries for 10-automatic sequences.

Cf. A031347.

Cf. A034048, A002275, A034049, A034050, A034051, A034052, A034053, A034054, A034055, A034056 (numbers having multiplicative digital roots 0-9).

mdr6Q[n_]:=NestWhile[Times@@IntegerDigits[#]&,n,#>9&]==6; Select[Range[400],mdr6Q] (* Harvey P. Dale, Jul 14 2024 *)

A034055 Numbers with multiplicative digital root value 8.

Original entry on oeis.org

8, 18, 24, 29, 36, 38, 42, 46, 49, 63, 64, 66, 67, 76, 77, 79, 81, 83, 88, 92, 94, 97, 99, 118, 124, 129, 136, 138, 142, 146, 149, 163, 164, 166, 167, 176, 177, 179, 181, 183, 188, 192, 194, 197, 199, 214, 219, 222, 226, 229, 233, 234, 236, 237, 241, 243, 248

Offset: 1

Patrick De Geest, Sep 15 1998

Eric Weisstein's World of Mathematics, Multiplicative Digital Root
Index entries for 10-automatic sequences.

Cf. A031347.

Cf. A034048, A002275, A034049, A034050, A034051, A034052, A034053, A034054, A034055, A034056 (numbers having multiplicative digital roots 0-9).

dr8Q[n_]:=NestWhile[Times@@IntegerDigits[#]&,n,#>9&]==8; Select[ Range[ 250],dr8Q] (* Harvey P. Dale, Jun 10 2011 *)

A034054 Numbers with multiplicative digital root value 7.

Original entry on oeis.org

7, 17, 71, 117, 171, 711, 1117, 1171, 1711, 7111, 11117, 11171, 11711, 17111, 71111, 111117, 111171, 111711, 117111, 171111, 711111, 1111117, 1111171, 1111711, 1117111, 1171111, 1711111, 7111111, 11111117, 11111171, 11111711, 11117111

Offset: 1

Patrick De Geest, Sep 15 1998

Numbers with one 7, and zero or more 1s. - Daniel Forgues, Oct 09 2011

Eric Weisstein's World of Mathematics, Multiplicative Digital Root
Index entries for 10-automatic sequences.

Cf. A031347.

Cf. A034048, A002275, A034049, A034050, A034051, A034052, A034053, A034054, A034055, A034056 (numbers having multiplicative digital roots 0-9).

Sort[Flatten[Table[FromDigits/@Permutations[Join[{7},PadRight[{},n,1]]],{n,0,10}]]] (* Harvey P. Dale, Jul 20 2015 *)

t(k)=while(k>9, k=prod(i=1, #k=digits(k), k[i])); k
for(n=1, 1e8, if(t(n) == 7, print1(n, ", "))); \\ Altug Alkan, Oct 22 2015

There are n(n+1)/2 elements up to 10^n, so a(n) is about 10^sqrt(2n).

A034050 Numbers with multiplicative digital root value 3.

Original entry on oeis.org

3, 13, 31, 113, 131, 311, 1113, 1131, 1311, 3111, 11113, 11131, 11311, 13111, 31111, 111113, 111131, 111311, 113111, 131111, 311111, 1111113, 1111131, 1111311, 1113111, 1131111, 1311111, 3111111, 11111113, 11111131, 11111311, 11113111

Offset: 1

Patrick De Geest, Sep 15 1998

Numbers with one 3, and zero or more 1s. - Daniel Forgues, Oct 09 2011

Robert Israel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Multiplicative Digital Root
Index entries for 10-automatic sequences.

Cf. A031347.

Cf. A034048, A002275, A034049, A034050, A034051, A034052, A034053, A034054, A034055, A034056 (numbers having multiplicative digital roots 0-9).

seq(seq((10^m-1)/9 + 2*10^j,j=0..m-1),m=1..10); # Robert Israel, Sep 27 2016

Sort[Flatten[Table[FromDigits/@Permutations[Join[{3},PadRight[{},n,1]]],{n,0,8}]]] (* Harvey P. Dale, Jul 16 2012 *)

# through 8-digit terms
print([int("1"*(d-i)+"3"+"1"*i) for d in range(8) for i in range(d+1)]) # Michael S. Branicky, Mar 13 2021

There are n(n+1)/2 members up to 10^n, so a(n) is about 10^sqrt(2n).

a(m*(m-1)/2+j+1) = (10^m-1)/9 + 2*10^j for 0 <= j < m. - Robert Israel, Sep 27 2016

A034056 Numbers with multiplicative digital root value 9.

Original entry on oeis.org

9, 19, 33, 91, 119, 133, 191, 313, 331, 911, 1119, 1133, 1191, 1313, 1331, 1911, 3113, 3131, 3311, 9111, 11119, 11133, 11191, 11313, 11331, 11911, 13113, 13131, 13311, 19111, 31113, 31131, 31311, 33111, 91111, 111119, 111133, 111191, 111313, 111331

Offset: 1

Patrick De Geest, Sep 15 1998

Numbers with one 9 or two 3s, and zero or more 1s. - Daniel Forgues, Oct 09 2011

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Multiplicative Digital Root
Index entries for 10-automatic sequences.

Cf. A031347.

Cf. A034048, A002275, A034049, A034050, A034051, A034052, A034053, A034054, A034055, A034056 (numbers having multiplicative digital roots 0-9).

Module[{nn=6,ne,te},ne=Union[FromDigits/@Flatten[Permutations/@Table[PadRight[{9},n,1],{n,nn}],1]];te=Rest[Union[FromDigits/@ Flatten[ Permutations/@Table[PadRight[{3,3},n,1],{n,nn}],1]]];Join[ne,te]]//Sort (* Harvey P. Dale, Apr 14 2025 *)

There are n(n+1)(n+2)/6 elements up to 10^n, so a(n) is about 10^sqrt(6n).

A028843 Numbers whose iterated product of digits is a prime.

Original entry on oeis.org

2, 3, 5, 7, 12, 13, 15, 17, 21, 26, 31, 34, 35, 37, 43, 51, 53, 57, 62, 71, 73, 75, 112, 113, 115, 117, 121, 126, 131, 134, 135, 137, 143, 151, 153, 157, 162, 171, 173, 175, 211, 216, 223, 232, 261, 278, 279, 287, 297, 299, 311, 314, 315, 317, 322, 341, 351, 355

Offset: 1

N. J. A. Sloane

			For 53, the product of digits is 5 * 3 = 15, iterated to 1 * 5 = 5, which is a prime, so 53 is in the sequence.
For 54, the product of digits is 5 * 4 = 20, iterated to 2 * 0 = 0, which is not prime, so 54 is not in the sequence.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A003001, A028835, A028842, A031347, A333955.

iterDigitProd[n_] := NestWhile[Times@@IntegerDigits[#] &, n, # > 9 &]; Select[Range[355], PrimeQ[iterDigitProd[#]] &] (* Jayanta Basu, Jun 02 2013 *)

def iterDigitProd(n: Int): Int = n.toString.length match {
  case 1 => n
  case  => iterDigitProd(n.toString.toCharArray.map( - 48).scanRight(1)( * ).head)
}
(1 to 400).filter(n => List(2, 3, 5, 7).contains(iterDigitProd(n))) // Alonso del Arte, Apr 11 2020

More terms from Patrick De Geest, Jun 15 1999

Corrected by Franklin T. Adams-Watters, Jan 17 2007

A350180 Numbers of multiplicative persistence 1 which are themselves the product of digits of a number.

Original entry on oeis.org

10, 12, 14, 15, 16, 18, 20, 21, 24, 30, 32, 40, 42, 50, 60, 70, 80, 81, 90, 100, 105, 108, 112, 120, 140, 150, 160, 180, 200, 210, 240, 250, 270, 280, 300, 320, 350, 360, 400, 405, 420, 450, 480, 490, 500, 504, 540, 560, 600, 630, 640, 700, 720, 750, 800

Offset: 1

Daniel Mondot, Dec 18 2021

The multiplicative persistence of a number mp(n) is the number of times the product of digits function p(n) must be applied to reach a single digit, i.e., A031346(n).
The product of digits function partitions all numbers into equivalence classes. There is a one-to-one correspondence between values in this sequence and equivalence classes of numbers with multiplicative persistence 2.
There are infinitely many numbers with mp of 1 to 11, but the classes of numbers (p(n)) are postulated to be finite for subsequent sequences A350181..., but not for this sequence (where mp(p(n)) = 1). That is because there are infinitely many numbers that include both an even digit (2, 4, 6 or 8), a 5 and no 0. For these numbers n, p(n) will include a zero and p(p(n)) will be 0.
Equivalently: This sequence contains all numbers A007954(k) such that A031346(k) = 2, and they are the numbers k in A002473 such that A031346(k) = 1.
Or, they factor into powers of 2, 3, 5 and 7 exclusively and p(n) goes to a single digit in 1 step.

			10 is in this sequence because:
- 10 goes to a single digit in 1 step: p(10) = 0.
- 25, 52, 125, 152, 215, 512, 251, 521, 1125, 1152, 1215, 1512, 1251, 1521, 2115, 5112, 2511, 5211, etc. all lead to 10, i.e., p(25)=10, p(52)=10, etc.
Some of these (25, 125, 512, 1125, 1152, 1215, 1512) are in the next layer of classes, A350181, and the rest are not.
12 is in this sequence because:
- 12 goes to a single digit in 1 step: p(12) = 2.
- 12, 21, 112, 211, 121, 11112, 11211, etc. all lead to 12.
(12, 21 and 112 are in the next layer of classes, A350181, but the rest are not)
14 is in this sequence because:
- 14 goes to a single digit in 1 step: p(14) = 4.
- 27, 72, 127, 172, 217, 712, 271, 721, 12111711, etc. all lead to 14.
(27 and 72 are in the next layer of classes, A350181, the rest are not).

Daniel Mondot, Table of n, a(n) for n = 1..20000
Daniel Mondot, Multiplicative Persistence Tree
Eric Weisstein's World of Mathematics, Multiplicative Persistence

Intersection of A002473 and A046510
Cf. A003001 (smallest number with multiplicative persistence n), A031346 (multiplicative persistence), A031347 (multiplicative digital root), A046510 (all numbers with mp of 1).
Cf. A350181, A350182, A350183, A350184, A350185, A350186, A350187 (numbers with mp 2 to 10 that are themselves 7-smooth numbers).

mp(n)={my(k=0); while(n>=10, k++; n=vecprod(digits(n))); k}
isparent(n)={my(m=0); while(m<>n, m=n; n/=gcd(n,2*3*5*7)); n==1}
isok(n)={mp(n)==1 && isparent(n)} \\ Andrew Howroyd, Dec 20 2021

A263473 Total number of positive integers < 10^n with multiplicative digital root value 5.

Original entry on oeis.org

1, 7, 40, 172, 607, 2073, 7414, 26070, 84099, 243529, 636130, 1518166, 3354325, 6940831, 13579716, 25318372, 45270813, 78039555, 130259668, 211289368, 334074499, 516217405, 781284010, 1160386410, 1694081935, 2434633461, 3448679742, 4820368690, 6655010857

Offset: 1

Martin Renner, Oct 19 2015

Partial sums of A263479.

Hiroaki Yamanouchi, Table of n, a(n) for n = 1..50
Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).

Cf. A031347, A034052, A263479.

lim = 6; t = Select[Range[1, 10^lim - 1], FixedPoint[Times @@ IntegerDigits@ # &, #] == 5 &]; Count[t, n_ /; n <= 10^#] & /@ Range@ lim (* Michael De Vlieger, Oct 21 2015 *)

t(k) = {while(k>9, k=prod(i=1, #k=digits(k), k[i])); k}
a(n) = sum(i=1, 10^n - 1, if(t(i) == 5, 1, 0)); \\ Altug Alkan, Oct 19 2015

A263470(n) + A000027(n) + A263471(n) + A000217(n) + A263472(n) + a(n) + A263474(n) + A000217(n) + A263475(n) + A000292(n) = A002283(n).
From Chai Wah Wu, Apr 17 2024: (Start)
a(n) = 10*a(n-1) - 45*a(n-2) + 120*a(n-3) - 210*a(n-4) + 252*a(n-5) - 210*a(n-6) + 120*a(n-7) - 45*a(n-8) + 10*a(n-9) - a(n-10) for n > 10.
G.f.: x*(-235*x^7 + 205*x^6 + 161*x^5 + 57*x^4 - 33*x^3 + 15*x^2 - 3*x + 1)/(x - 1)^10. (End)

a(9)-a(29) from Hiroaki Yamanouchi, Oct 25 2015

cp's OEIS Frontend

A034051 Numbers with multiplicative digital root value 4.

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A034049 Numbers with multiplicative digital root value 2.

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A034053 Numbers with multiplicative digital root value 6.

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A034055 Numbers with multiplicative digital root value 8.

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A034054 Numbers with multiplicative digital root value 7.

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A034050 Numbers with multiplicative digital root value 3.

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A034056 Numbers with multiplicative digital root value 9.

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A028843 Numbers whose iterated product of digits is a prime.

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