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A386228 Primes that are the sum of prime factors (with multiplicity) of a triprime which is the concatenation of three consecutive primes.

%I A386228 #7 Jul 20 2025 15:33:00
%S A386228 8539,11813,19181,27827,45013,52859,64621,64969,81077,103583,105373,
%T A386228 127493,228203,264791,297397,318161,324491,439753,466247,480299,
%U A386228 491353,496631,561091,613559,638431,678943,779981,822631,827537,906673,908893,1039477,1046029,1079927,1090577,1176871,1220327
%N A386228 Primes that are the sum of prime factors (with multiplicity) of a triprime which is the concatenation of three consecutive primes.
%C A386228 Numbers that are the sum of prime factors (with multiplicity) of at least one member of A385968.
%H A386228 Robert Israel, <a href="/A386228/b386228.txt">Table of n, a(n) for n = 1..5098</a>
%e A386228 a(3) = 19181 is a term because 487491499 = A385968(4) is the concatenation of consecutive primes 487, 491, 499 and 487491499 = 11 * 2689 * 16481 with 11 + 2689 + 16481 = 19181 prime.
%e A386228 The only term < 3 * 10^9 that arises in more than one way is
%e A386228 a(756) = 149573911 = 53281 + 121110841 + 28409789
%e A386228                  = 143597911 + 524453 + 5451547
%e A386228 where 53281 * 121110841 * 28409789 = 183325718332591833269 = A385968(3382)
%e A386228 and 143597911 * 524453 * 5451547 = 410557941055894105601 = A385968(6601).
%p A386228 tcat:= proc(a, b, c);
%p A386228   c + 10^(1+ilog10(c))*(b + 10^(1+ilog10(b))*a)
%p A386228 end proc:
%p A386228 xmax:= 10^15: Bmax:= 3*10^6:
%p A386228 B:= NULL: count:= 0:
%p A386228 q:= 2: r:= 3:
%p A386228 do
%p A386228   p:= q; q:= r; r:= nextprime(r);
%p A386228   x:= tcat(p, q, r);
%p A386228   if x > xmax then break fi;
%p A386228   F:= ifactors(x)[2];
%p A386228   if add(t[2], t=F) = 3 then
%p A386228      b:= add(t[1]*t[2], t=F);
%p A386228      if b <= Bmax and isprime(b) then
%p A386228        count:= count+1; B:= B, b;
%p A386228   fi fi;
%p A386228 od:
%p A386228 sort(convert({B},list));
%Y A386228 Cf. A385968.
%K A386228 nonn,base
%O A386228 1,1
%A A386228 _Will Gosnell_ and _Robert Israel_, Jul 15 2025