A046500 -id:A046500 - cp's OEIS frontend

A046501 Primes with multiplicative persistence value 1.

11, 13, 17, 19, 23, 31, 41, 61, 71, 101, 103, 107, 109, 113, 131, 151, 181, 191, 211, 241, 307, 311, 313, 331, 401, 409, 421, 503, 509, 601, 607, 701, 709, 809, 811, 907, 911, 1009, 1013, 1019, 1021, 1031, 1033, 1039, 1049, 1051, 1061, 1063, 1069, 1087

Offset: 1

Patrick De Geest, Sep 15 1998

The numbers < 10 have persistence 0. - T. D. Noe, Nov 23 2011

Also: Primes having either at least one digit "0", or any number of digits "1" and product of digits > 1 less than 10 (i.e., among {2, ..., 9, 2*2, 2*3, 2*4, 3*3, 2*2*2}). Terms without a digit "0" and such that deleting some digits "1" never yields an earlier term could be called "primitive". There are only finitely many such elements. If the terms < 10 are ignored, the primitive elements are 11, ..., 71, 151, 181, 211, 241, 313, 421, 811, 911, ... - M. F. Hasler, Sep 25 2012

			181 -> 1*8*1 = 8; one digit in one step.

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

Intersection of A000040 and A046510.

Cf. A046500.

Select[Prime[Range[179]], IntegerLength[Times @@ IntegerDigits[#]] <= 1 &] (* Jayanta Basu, Jun 26 2013 *)

is_A046501(n)={isprime(n) || return; my(P=n%10); while(P & n\=10, (P*=n%10)>9 & return);1}  \\ M. F. Hasler, Sep 25 2012

from math import prod
from sympy import isprime
def ok(n): return n > 9 and prod(map(int, str(n))) < 10 and isprime(n)
print([k for k in range(1088) if ok(k)]) # Michael S. Branicky, Mar 14 2022

Numbers < 10 removed, as they have a multiplicative persistence of 0, by Daniel Mondot, Mar 14 2022

A046502 Primes with multiplicative persistence value 2.

Original entry on oeis.org

29, 37, 43, 53, 67, 73, 83, 127, 137, 163, 167, 173, 199, 223, 233, 251, 257, 269, 271, 281, 317, 349, 431, 439, 457, 461, 521, 523, 541, 547, 563, 569, 587, 599, 613, 617, 631, 641, 653, 659, 761, 821, 827, 853, 857, 859, 919, 991, 1129, 1153, 1163, 1217

Offset: 1

Patrick De Geest, Sep 15 1998

			29 -> 2 * 9 -> [ 18 ] -> 1 * 8 -> [ 8 ] -> one digit in two steps.

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

Intersection of A000040 and A046511.

Cf. A046500.

A046503 Primes with multiplicative persistence value 3.

Original entry on oeis.org

47, 59, 79, 89, 97, 139, 149, 157, 179, 193, 197, 227, 229, 239, 263, 283, 293, 337, 347, 353, 359, 367, 373, 383, 389, 419, 433, 443, 449, 463, 479, 487, 491, 499, 571, 577, 593, 619, 643, 661, 673, 683, 691, 719, 733, 743, 751, 757, 797, 823, 829, 839

Offset: 1

Patrick De Geest, Sep 15 1998

			47 -> 4 * 7 -> [ 28 ] -> 2 * 8 -> [ 16 ] -> 1 * 6 -> [ 6 ] -> 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. A046500, A046512.

filter:= proc(n) local L;
  if not isprime(n) then return false fi;
  L:= convert(convert(n,base,10),`*`); if L < 10 then return false fi;
  L:= convert(convert(L,base,10),`*`); if L < 10 then return false fi;
  L:= convert(convert(L,base,10),`*`); evalb(L < 10)
end proc:
select(filter, [seq(i,i=11..1000,2)]); # Robert Israel, Jun 05 2018

pr3Q[n_] := Length[NestWhileList[Times @@ IntegerDigits[#] &, n, # > 9 &]] == 4; Select[Prime[Range[147]], pr3Q] (* Jayanta Basu, Jun 26 2013 *)

A046504 Primes with multiplicative persistence value 4.

Original entry on oeis.org

277, 379, 397, 467, 557, 647, 677, 727, 739, 773, 787, 877, 887, 937, 1277, 1483, 1489, 1553, 1699, 1747, 1777, 1783, 1787, 1867, 1873, 1877, 1973, 1999, 2267, 2347, 2377, 2399, 2437, 2467, 2473, 2477, 2647, 2683, 2687, 2689, 2699, 2797, 2837, 2861

Offset: 1

Patrick De Geest, Sep 15 1998

			277 -> [ 98 ][ 72 ][ 14 ][ 4 ] -> one digit in four steps.

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

Intersection of A000040 and A046513.

Cf. A046500.

pmp4Q[n_]:=Length[NestWhileList[Times@@IntegerDigits[#]&,n,#>9&]]==5; Select[Prime[Range[500]],pmp4Q] (* Harvey P. Dale, Oct 10 2016 *)

A046505 Primes with multiplicative persistence value 5.

Original entry on oeis.org

769, 967, 1697, 2777, 3637, 3673, 4483, 6197, 6337, 6373, 6719, 6733, 6779, 6791, 6917, 6971, 6977, 7559, 7691, 7727, 8443, 8681, 8861, 9677, 9767, 12379, 12739, 12973, 13297, 13367, 13729, 13763, 14779, 14797, 14843, 17239, 17293, 17497

Offset: 1

Patrick De Geest, Sep 15 1998

			769 -> [ 378 ][ 168 ][ 48 ][ 32 ][ 6 ] -> one digit in five steps.

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

Intersection of A000040 and A046514.

Cf. A046500.

mp5Q[n_]:=Length[NestWhileList[Times@@IntegerDigits[#]&,n,#>9&]]==6; Select[ Prime[Range[2100]],mp5Q] (* Harvey P. Dale, Dec 30 2019 *)

A046506 Primes with multiplicative persistence value 6.

Original entry on oeis.org

8867, 23887, 27883, 28387, 28837, 32887, 34487, 34847, 38287, 38447, 43487, 44647, 46447, 47843, 48437, 48473, 49999, 72883, 74843, 78283, 78823, 82387, 82837, 84347, 84437, 87443, 88237, 88327, 94999, 118687, 123887, 126487, 128467

Offset: 1

Patrick De Geest, Sep 15 1998

			8867 -> [ 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. A046500, A046515.

filter:= proc(n) local L,i;
  L:= convert(convert(n,base,10),`*`); if L < 10 then return false fi;
  for i from 2 to 5 do
     L:= convert(convert(L,base,10),`*`); if L < 10 then return false fi
    od;
  L:= convert(convert(L,base,10),`*`); evalb(L < 10)
end proc:
count:= 0: Res:= NULL;
p:= 11:
while count < 100 do
  p:= nextprime(p);
  if filter(p) then count:= count+1; Res:= Res,p fi
od:
Res; # Robert Israel, Jun 06 2018

Offset corrected by Robert Israel, Jun 06 2018

A046507 Primes with multiplicative persistence value 7.

Original entry on oeis.org

186889, 188869, 246889, 248869, 264889, 269779, 279679, 279967, 284689, 288389, 288649, 288839, 296797, 297967, 376679, 376769, 377779, 379667, 379777, 426889, 444869, 467879, 467897, 468289, 468683, 469787, 469877, 478679, 478697, 478769

Offset: 1

Patrick De Geest, Sep 15 1998

			186889 -> [ 27648 ][ 2688 ][ 768 ][ 336 ][ 54 ][ 20 ][ 0 ] -> one digit in seven steps.

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

Cf. A046500, A046516.

A046508 Primes with multiplicative persistence value 8.

Original entry on oeis.org

2678789, 2687879, 2687897, 2688797, 2688977, 2689877, 2697887, 2768789, 2768897, 2769887, 2778689, 2786789, 2787689, 2788769, 2798687, 2798867, 2868779, 2868977, 2876789, 2876879, 2878679, 2878697, 2878769, 2879687

Offset: 1

Patrick De Geest, Sep 15 1998

			2678789 -> [ 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..9001 from Harvey P. Dale)
Eric Weisstein's World of Mathematics, Multiplicative Persistence

Intersection of A000040 and A046517.

Cf. A046500.

mp8Q[n_]:=Length[NestWhileList[Times@@IntegerDigits[#]&,n,#>9&]]==9; Select[ Prime[ Range[210000]],mp8Q] (* Harvey P. Dale, Mar 30 2019 *)

A046509 Primes with multiplicative persistence value 9.

Original entry on oeis.org

26899889, 28698899, 28699889, 28869989, 28889699, 28899869, 28989689, 29868989, 29888699, 29968889, 29988869, 38849989, 38894899, 38898949, 38899489, 38984989, 39849889, 43898989, 43988899, 43989889, 46668899

Offset: 1

Patrick De Geest, Sep 15 1998

			26899889 -> [ 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
Eric Weisstein's World of Mathematics, Multiplicative Persistence

Intersection of A000040 and A046518.

Cf. A046500.

A157837 Smallest emirp with multiplicative persistence n.

Original entry on oeis.org

13, 37, 79, 739, 769, 32887, 186889, 3748879, 92888699, 3778889899, 727777887889889

Offset: 1

Lekraj Beedassy, Mar 07 2009

Cf. A046500

Corrected a(11). Lekraj Beedassy, Mar 10 2009

a(8) and a(11) corrected by Ray Chandler, Mar 12 2009

cp's OEIS Frontend

A046501 Primes with multiplicative persistence value 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Python

Extensions

A046502 Primes with multiplicative persistence value 2.

Views

Author

Keywords

Examples

Links

Crossrefs

A046503 Primes with multiplicative persistence value 3.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

A046504 Primes with multiplicative persistence value 4.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

A046505 Primes with multiplicative persistence value 5.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

A046506 Primes with multiplicative persistence value 6.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Extensions

A046507 Primes with multiplicative persistence value 7.

Views

Author

Keywords

Examples

Links

Crossrefs

A046508 Primes with multiplicative persistence value 8.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

A046509 Primes with multiplicative persistence value 9.

Views

Author

Keywords

Examples