A395103 - OEIS

%I #41 May 28 2026 22:46:12

%S 22,25,33,35,55,57,77,115,133,177,213,217,219,222,235,237,247,253,255,

%T 259,267,273,275,289,295,319,322,323,325,329,332,333,341,357,361,371,

%U 377,413,415,417,437,473,511,517,519,529,533,535,537,543,553,555,559,573,575,579,583,589,597

%N Numbers with exactly k prime factors, counting repeats, with k>1, that are formed by concatenating k primes.

%C Conjecture: The sequence has infinitely many terms.

%C The conjecture is true if A117360 is infinite. - _Yehune Jeong_, May 24 2026

%e 237 has 2 prime factors (3, 79) and is the concatenation of 2 primes (23, 7).

%e 332 has 3 prime factors (2, 2, 83) and is the concatenation of 3 primes (3, 3, 2).

%e 3255 has 4 prime factors (3, 5, 7, 31) and is the concatenation of 4 primes (3, 2, 5, 5).

%p filter:= n -> not isprime(n) and g(n,NumberTheory:-Omega(n)):

%p g:= proc(n,t)

%p   local d,i,a;

%p   if t = 1 then return isprime(n) fi;

%p   d:= 1+ilog10(n);

%p   if d < t then return false fi;

%p   for i from 1 to d-t+1 do

%p     a:= n mod 10^i;

%p     if a > 10^(i-1) and isprime(a) and procname((n-a)/10^i,t-1) then return true fi;

%p   od;

%p   false

%p end proc:

%p select(filter, [$10..1000]); # _Robert Israel_, May 18 2026

%o (PARI) is(n) = {my(b = bigomega(n)); if(b <= 1, return(0)); isConcatenationOfbPrimes(n, b)}

%o isConcatenationOfbPrimes(n, b) = {if(b == 1, return(isprime(n))); my(qd = #digits(n), cn, res = 0, g = n%10); if(gcd(g, 10) > 1, if(g == 2 || g == 5, return(isConcatenationOfbPrimes(n\10, b-1))); return); for(i = 1, qd - b + 1, cn = n\10^i; if((isprime(n - 10^i*cn) && n - 10^i*cn >= 10^(i-1)), if(isConcatenationOfbPrimes(cn, b-1), res = 1; return(res)))); res} \\ _David A. Corneth_, May 19 2026

%Y Cf. A001222, A117360.

%K nonn,base

%O 1,1

%A _Philip Jameson_, May 10 2026