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

Author

Patrick De Geest, Sep 15 1998

Keywords

• nonn
• base

Comments

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

Examples

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

Links

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

Crossrefs

Intersection of A000040 and A046510.

Cf. A046500.

Programs

Mathematica

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

PARI

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

Python

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

Extensions

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