A064155 Let p = x1x2x3...xk be a prime in base 10 with k digits. The sequence gives the primes p such that x1*x2*x3*...xk = k*(x1 + x2 + x3 + ... + xk).

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

1,1

Links

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

Example

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

Mathematica

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

Prog

(PARI) ProdD(x)= { local(p); p=1; while (x>9 && p>0, p*=(x-10*(x\10)); x\=10); return(p*x) }
SumD(x)= { local(s); s=0; while (x>9, s+=x-10*(x\10); x\=10); return(s + x) }
digitsIn(x)= { local(d); if (x==0, return(1)); d=1 + log(x)\log(10); if (10^d == x, d++, if (10^(d-1) > x, d--)); return(d) }
{ n=r=0; for (m=1, 10^9, r=nextprime(r+1); if ((p=ProdD(r)) && p == digitsIn(r)*SumD(r), write("b064155.txt", n++, " ", r); if (n==100, break)) ) } \\ 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

Crossrefs

Sequence in context: A281021 A230046 A098463 * A230047 A230042 A230045

Adjacent sequences: A064152 A064153 A064154 * A064156 A064157 A064158

Keyword

easy,nonn,base

Author

Felice Russo, Sep 14 2001

Status

approved