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

%I A109133 #16 Dec 20 2021 19:00:45
%S A109133 1,2,4,6,10,11,12,14,15,17,18,20,21,23,24,26,27,29,30,32,33,35,36,38,
%T A109133 41,42,44,45,47,50,51,53,54,56,59,60,62,63,65,68,69,71,72,74,77,78,80,
%U A109133 81,83,86,87,90,92,95,96,99,101,103,105,109,110,112,114,118,121,123,127
%N A109133 Numbers k such that (sum of digits)*(number of digits) + 1 is prime.
%C A109133 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
%H A109133 Robert Israel, <a href="/A109133/b109133.txt">Table of n, a(n) for n = 1..10000</a>
%e A109133 1234 is a term because 4*(1+2+3+4)+1 = 41.
%p A109133 filter:= proc(n) local L;
%p A109133   L:= convert(n,base,10);
%p A109133   isprime(convert(L,`+`)*nops(L)+1)
%p A109133 end proc:
%p A109133 select(filter, [$1..200]); # _Robert Israel_, Dec 19 2021
%t A109133 Select[Range[130],PrimeQ[Total[IntegerDigits[#]]IntegerLength[ #]+ 1]&] (* _Harvey P. Dale_, Jul 12 2011 *)
%o A109133 (Python)
%o A109133 from sympy import isprime
%o A109133 def ok(n): s = str(n); return isprime(sum(map(int, s))*len(s) + 1)
%o A109133 print([k for k in range(128) if ok(k)]) # _Michael S. Branicky_, Dec 19 2021
%Y A109133 Cf. A110805.
%K A109133 base,easy,nonn
%O A109133 1,2
%A A109133 _Jason Earls_, Aug 17 2005