A109133 Numbers k such that (sum of digits)*(number of digits) + 1 is prime.

1, 2, 4, 6, 10, 11, 12, 14, 15, 17, 18, 20, 21, 23, 24, 26, 27, 29, 30, 32, 33, 35, 36, 38, 41, 42, 44, 45, 47, 50, 51, 53, 54, 56, 59, 60, 62, 63, 65, 68, 69, 71, 72, 74, 77, 78, 80, 81, 83, 86, 87, 90, 92, 95, 96, 99, 101, 103, 105, 109, 110, 112, 114, 118, 121, 123, 127

Offset: 1

Author

Jason Earls, Aug 17 2005

Keywords

• base
• easy
• nonn

Comments

By Dirichlet's theorem on primes in arithmetic progressions, for any positive integer k this sequence has infinitely many terms of the form k*10^m. - Robert Israel, Dec 19 2021

Examples

			1234 is a term because 4*(1+2+3+4)+1 = 41.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A110805.

Programs

Maple

filter:= proc(n) local L;
  L:= convert(n,base,10);
  isprime(convert(L,`+`)*nops(L)+1)
end proc:
select(filter, [$1..200]); # Robert Israel, Dec 19 2021

Mathematica

Select[Range[130],PrimeQ[Total[IntegerDigits[#]]IntegerLength[ #]+ 1]&] (* Harvey P. Dale, Jul 12 2011 *)

Python

from sympy import isprime
def ok(n): s = str(n); return isprime(sum(map(int, s))*len(s) + 1)
print([k for k in range(128) if ok(k)]) # Michael S. Branicky, Dec 19 2021