A014120 -id:A014120 - cp's OEIS frontend

A046500 Smallest prime with multiplicative persistence n.

2, 11, 29, 47, 277, 769, 8867, 186889, 2678789, 26899889, 3778888999, 277777788888989

Offset: 0

Patrick De Geest, Sep 15 1998

The persistence of a number is the number of times you need to multiply the digits together before reaching a single digit.

			47 -> 28 -> 16 -> 6 has persistence 3.

Shyam Sunder Gupta, Digital Root Wonders, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 1, 1-28.
Carlos Rivera, Puzzle 22. Primes & Persistence, The Prime Puzzles & Problems Connection.
N. J. A. Sloane, The persistence of a number, J. Recreational Math., 6 (1973), 97-98.
Eric Weisstein's World of Mathematics, Multiplicative Persistence.

Cf. A003001, A014120.

a[n_]:=Length[NestWhileList[Times@@IntegerDigits[#]&,n,#>9&]]-1; t={}; i=1; Do[While[a[p=Prime[i]]!=n,i++]; AppendTo[t,p],{n,0,9}]; t (* Jayanta Basu, Jun 02 2013 *)

a(10)-a(11) found by Jud McCranie

A046510 Numbers with multiplicative persistence value 1.

Original entry on oeis.org

10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 30, 31, 32, 33, 40, 41, 42, 50, 51, 60, 61, 70, 71, 80, 81, 90, 91, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 130, 131, 132, 133

Offset: 1

Patrick De Geest, Sep 15 1998

Numbers 0 to 9 have a multiplication persistence of 0, not 1. - Daniel Mondot, Mar 12 2022

			24 -> 2 * 4 = [ 8 ] -> one digit in one step.

Daniel Mondot, Table of n, a(n) for n = 1..10000
N. J. A. Sloane, The persistence of a number, J. Recreational Math., 6 (1973), 97-98.
Eric Weisstein's World of Mathematics, Multiplicative Persistence

Cf. A003001, A014120, A046501.

Numbers with multiplicative persistence m: this sequence (m=1), A046511 (m=2), A046512 (m=3), A046513 (m=4), A046514 (m=5), A046515 (m=6), A046516 (m=7), A046517 (m=8), A046518 (m=9), A352531 (m=10), A352532 (m=11).

Select[Range[10, 121], IntegerLength[Times @@ IntegerDigits[#]] <= 1 &] (* Jayanta Basu, Jun 26 2013 *)

isok(n) = my(d=digits(n)); (#d > 1) && (#digits(prod(k=1, #d, d[k])) <= 1); \\ Michel Marcus, Apr 12 2018 and Mar 13 2022

from math import prod
def ok(n): return n > 9 and prod(map(int, str(n))) < 10
print([k for k in range(134) if ok(k)]) # Michael S. Branicky, Mar 13 2022

Incorrect terms 0 to 9 removed by Daniel Mondot, Mar 12 2022

A046511 Numbers with multiplicative persistence value 2.

Original entry on oeis.org

25, 26, 27, 28, 29, 34, 35, 36, 37, 38, 43, 44, 45, 46, 48, 52, 53, 54, 56, 58, 62, 63, 64, 65, 67, 72, 73, 76, 82, 83, 84, 85, 92, 99, 125, 126, 127, 128, 129, 134, 135, 136, 137, 138, 143, 144, 145, 146, 148, 152, 153, 154, 156, 158, 162, 163, 164, 165, 167, 172

Offset: 1

Patrick De Geest, Sep 15 1998

			129 -> [ 18 ] -> [ 8 ] = one digit in two steps.

Daniel Mondot, Table of n, a(n) for n = 1..10000 (terms 1..3000 from Vincenzo Librandi)
Eric Weisstein's World of Mathematics, Multiplicative Persistence

Cf. A003001, A014120, A046502.

mpQ[n_]:=Length[NestWhileList[Times@@IntegerDigits[#]&,n,#>9&]]==3; Select[Range[200], mpQ] (* Harvey P. Dale, Apr 12 2014 *)

from math import prod
def pd(n): return prod(map(int, str(n)))
def ok(n): return n > 9 and (p := pd(n)) > 9 and pd(p) < 10
print([k for k in range(173) if ok(k)]) # Michael S. Branicky, Jan 16 2022

A046518 Numbers with multiplicative persistence value 9.

Original entry on oeis.org

26888999, 26889899, 26889989, 26889998, 26898899, 26898989, 26898998, 26899889, 26899898, 26899988, 26988899, 26988989, 26988998, 26989889, 26989898, 26989988, 26998889, 26998898, 26998988, 26999888, 28688999

Offset: 1

Patrick De Geest, Sep 15 1998

			26888999 -> [ 4478976 ][ 338688 ][ 27648 ][ 2688 ][ 768 ][ 336 ][ 54 ][ 20 ][ 0 ] -> one digit in nine steps.

Daniel Mondot, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Robert Israel, terms 1001..7448 from David A. Corneth)
Eric Weisstein's World of Mathematics, Multiplicative Persistence

Cf. A003001, A014120, A046509.

mp:= proc(n) option remember;
    if n <= 9 then return 0 fi;
    1+procname(convert(convert(n,base,10),`*`))
end proc:
filter:= proc(n)
  evalb(mp(convert(convert(n,base,10),`*`))=8)
end proc:
select(filter, [$26111111..29999999]); # Robert Israel, Feb 12 2019

Offset corrected by Robert Israel, Feb 12 2019

A046512 Numbers with multiplicative persistence value 3.

Original entry on oeis.org

39, 47, 49, 55, 57, 59, 66, 68, 69, 74, 75, 78, 79, 86, 87, 88, 89, 93, 94, 95, 96, 97, 98, 139, 147, 149, 155, 157, 159, 166, 168, 169, 174, 175, 178, 179, 186, 187, 188, 189, 193, 194, 195, 196, 197, 198, 227, 229, 236, 238, 239, 246, 247, 248, 249, 263, 264

Offset: 1

Patrick De Geest, Sep 15 1998

			239 -> [ 54 ][ 20 ][ 0 ] -> one digit in three steps.

Robert Israel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Multiplicative Persistence

Cf. A003001, A014120, A046503.

mp:= proc(n) option remember;
    if n <= 9 then return 0 fi;
    1+procname(convert(convert(n,base,10),`*`))
end proc:
select(mp=3, [$1..1000]); # Robert Israel, Feb 12 2019

mp3Q[n_]:=Length[NestWhileList[Times@@IntegerDigits[#]&,n,#>9&]]==4; Select[Range[300],mp3Q] (* Harvey P. Dale, May 23 2015 *)

Offset corrected by Robert Israel, Feb 12 2019

A046513 Numbers with multiplicative persistence value 4.

Original entry on oeis.org

77, 177, 268, 277, 286, 348, 355, 377, 378, 379, 384, 387, 397, 438, 446, 464, 467, 476, 477, 483, 489, 498, 535, 553, 557, 575, 628, 644, 647, 668, 674, 677, 678, 682, 686, 687, 699, 717, 727, 737, 738, 739, 746, 747, 755, 764, 767, 768, 771, 772, 773

Offset: 1

Patrick De Geest, Sep 15 1998

			737 -> [ 147 ][ 28 ][ 16 ][ 6 ] -> one digit in four steps.

Robert Israel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Multiplicative Persistence

Cf. A003001, A014120, A046504.

mp:= proc(n) option remember;
    if n <= 9 then return 0 fi;
    1+procname(convert(convert(n,base,10),`*`))
end proc:
select(mp=4, [$1..1000]); # Robert Israel, Feb 12 2019

pr4Q[n_] := Length[NestWhileList[Times @@ IntegerDigits[#] &, n, # > 9 &]] == 5; Select[Range[773], pr4Q] (* Jayanta Basu, Jun 26 2013 *)

Offset corrected by Robert Israel, Feb 12 2019

A046514 Numbers with multiplicative persistence value 5.

Original entry on oeis.org

679, 688, 697, 769, 796, 868, 886, 967, 976, 1679, 1688, 1697, 1769, 1796, 1868, 1886, 1967, 1976, 2379, 2388, 2397, 2468, 2486, 2648, 2684, 2688, 2739, 2777, 2793, 2838, 2846, 2864, 2868, 2883, 2886, 2937, 2973, 3279, 3288, 3297, 3367, 3376, 3448

Offset: 1

Patrick De Geest, Sep 15 1998

			2777 -> [ 686 ][ 288 ][ 128 ][ 16 ][ 6 ] -> one digit in five steps.

Robert Israel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Multiplicative Persistence

Cf. A003001, A014120, A046505.

mp:= proc(n) option remember;
    if n <= 9 then return 0 fi;
    1+procname(convert(convert(n,base,10),`*`))
end proc:
select(mp=5, [$1..4000]); # Robert Israel, Feb 12 2019

Offset corrected by Robert Israel, Feb 12 2019

A046515 Numbers with multiplicative persistence value 6.

Original entry on oeis.org

6788, 6878, 6887, 7688, 7868, 7886, 8678, 8687, 8768, 8786, 8867, 8876, 16788, 16878, 16887, 17688, 17868, 17886, 18678, 18687, 18768, 18786, 18867, 18876, 23788, 23878, 23887, 24678, 24687, 24768, 24786, 24867, 24876, 26478, 26487

Offset: 1

Patrick De Geest, Sep 15 1998

			6788 -> [ 2688 ][ 768 ][ 336 ][ 54 ][ 20 ][ 0 ] -> one digit in six steps.

Robert Israel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Multiplicative Persistence

Cf. A003001, A014120, A046506.

mp:= proc(n) option remember;
    if n <= 9 then return 0 fi;
    1+procname(convert(convert(n,base,10),`*`))
end proc:
select(mp=6, [$1..30000]); # Robert Israel, Feb 12 2019

Offset corrected by Robert Israel, Feb 12 2019

A046516 Numbers with multiplicative persistence value 7.

Original entry on oeis.org

68889, 68898, 68988, 69888, 86889, 86898, 86988, 88689, 88698, 88869, 88896, 88968, 88986, 89688, 89868, 89886, 96888, 98688, 98868, 98886, 168889, 168898, 168988, 169888, 186889, 186898, 186988, 188689, 188698, 188869, 188896, 188968

Offset: 1

Patrick De Geest, Sep 15 1998

From Daniel Mondot, Mar 26 2022: (Start)
The product of the digits of each term is 27648, 47628, 64827, 84672, 134217728, 914838624, 1792336896, 3699376128, 48814981614, 134481277728, 147483721728 or 1438916737499136 (sequence A350185).
The first 62 terms produce 27648.
The first term that produces 47628 is a(63).
The first term that produces 64827 is a(233).
The first term that produces 84672 is a(235).
The first term that produces 134217728 is a(1753110).
The first term that produces 914838624 is a(17835449).
The first term that produces 1792336896 is a(18235677).
The first term that produces 3699376128 is a(23853261).
The first term that produces 48814981614 is a(66441891).
The first term that produces 134481277728 is a(452601087).
The first term that produces 147483721728 is a(425636434).
The first term that produces 1438916737499136 is somewhere after a(500*10^6). (End)

			68889 -> [ 27648 ][ 2688 ][ 768 ][ 336 ][ 54 ][ 20 ][ 0 ] -> one digit in seven steps.

Robert Israel, Table of n, a(n) for n = 1..10000
Daniel Mondot, Multiplicative Persistence Tree
Eric Weisstein's World of Mathematics, Multiplicative Persistence

Cf. A003001, A014120, A046507, A350185.

mp:= proc(n) option remember;
    if n <= 9 then return 0 fi;
    1+procname(convert(convert(n,base,10),`*`))
end proc:
select(mp=7, [$1..200000]); # Robert Israel, Feb 12 2019

A046517 Numbers with multiplicative persistence value 8.

Original entry on oeis.org

2677889, 2677898, 2677988, 2678789, 2678798, 2678879, 2678897, 2678978, 2678987, 2679788, 2679878, 2679887, 2687789, 2687798, 2687879, 2687897, 2687978, 2687987, 2688779, 2688797, 2688977, 2689778, 2689787, 2689877

Offset: 1

Patrick De Geest, Sep 15 1998

			2677889 -> [ 338688 ][ 27648 ][ 2688 ][ 768 ][ 336 ][ 54 ][ 20 ][ 0 ] -> one digit in eight steps.

Daniel Mondot, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Robert Israel)
Eric Weisstein's World of Mathematics, Multiplicative Persistence

Cf. A003001, A014120, A046508.

mp:= proc(n) option remember;
    if n <= 9 then return 0 fi;
    1+procname(convert(convert(n,base,10),`*`))
end proc:
select(mp=8, [$1..4000000]); # Robert Israel, Feb 12 2019

okQ[n_]:=Length[NestWhileList[Times@@IntegerDigits[#]&, n,IntegerLength[ #]>1&]]==9; Select[Range[2700000],okQ]  (* Harvey P. Dale, Jan 29 2011 *)

Offset corrected by Robert Israel, Feb 12 2019

cp's OEIS Frontend

A046500 Smallest prime with multiplicative persistence n.

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A046510 Numbers with multiplicative persistence value 1.

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A046511 Numbers with multiplicative persistence value 2.

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A046518 Numbers with multiplicative persistence value 9.

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A046512 Numbers with multiplicative persistence value 3.

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A046513 Numbers with multiplicative persistence value 4.

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A046514 Numbers with multiplicative persistence value 5.

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A046515 Numbers with multiplicative persistence value 6.

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