A064155 Primes whose product of digits equals the number of digits times the sum of digits.

2, 3, 5, 7, 167, 523, 617, 761, 1427, 2417, 2741, 4127, 4217, 4271, 4721, 126241, 126421, 146221, 212461, 216421, 221461, 224611, 226141, 241261, 242161, 246121, 261241, 262411, 264211, 421621, 426211, 621241, 642121, 642211, 1111457, 1111547, 1115417, 1117451

Offset: 1

Author

Felice Russo, Sep 14 2001

Keywords

• easy
• nonn
• base

Examples

			167 belongs to the sequence because 1*6*7 = 42 and 3*(1+6+7) = 42.
		

Links

• Michael S. Branicky, Table of n, a(n) for n = 1..10000 (terms 1..100 from Harry J. Smith)

Crossrefs

Primes in A064154.

Programs

Mathematica

Select[Prime@Range@1000000, Plus@@(i=IntegerDigits@#)*Length@i == Times@@i&] (*Hans Rudolf Widmer, Jun 13 2024*)

PARI

isok(k)={ if(isprime(k), my(d=digits(k)); vecprod(d)==#d * vecsum(d), 0) } \\ Harry J. Smith, Sep 09 2009

Python

from math import prod
from sympy import isprime
from sympy.utilities.iterables import multiset_permutations as mp
from itertools import count, islice, combinations_with_replacement as mc
def c(s):
    d = list(map(int, s))
    return prod(d) == len(d)*sum(d)
def agen():
    yield from (2, 3, 5, 7)
    for d in count(2):
        okset = set()
        for cand in ("".join(m) for m in mc("987654321", d)):
            if c(cand):
                for p in mp(cand, d):
                    t = int("".join(p))
                    if isprime(t): okset.add(t)
        yield from sorted(okset)
print(list(islice(agen(), 38))) # Michael S. Branicky, Nov 30 2022

Extensions

Name edited by Andrew Howroyd, Dec 05 2024