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A376503 Primes p such that p + 2, p + 4 and p + 6 are products of 3, 5 and 7 primes respectively (counted with multiplicity).

%I A376503 #8 Jun 02 2025 15:28:13
%S A376503 171869,5609369,7653119,11177069,11320709,11479319,12325619,13530119,
%T A376503 15426419,15558119,17136619,17541869,17851919,18809369,18843119,
%U A376503 19593869,19634369,19938119,20806619,21600419,22470953,23637839,23796869,23999369,24006119,24275819,25798739,25879001,25965869,26278169
%N A376503 Primes p such that p + 2, p + 4 and p + 6 are products  of 3, 5 and 7 primes respectively (counted with multiplicity).
%C A376503 Since 2 and 3 are not terms, the least possible prime factor of p + 6 is 5.  This is why, at least initially, it seems most terms end in 9.
%H A376503 Robert Israel, <a href="/A376503/b376503.txt">Table of n, a(n) for n = 1..10000</a>
%e A376503 a(3) = 7653119 is a term because 7653119 is prime,
%e A376503 7653121 = 7 * 61 * 17923 has 3 prime factors,
%e A376503 7653123 = 3^4 * 94483 has 5 prime factors, and
%e A376503 7653125 = 5^5 * 31 * 79 has 7 prime factors, counted with multiplicity.
%p A376503 with(priqueue):
%p A376503 R:= NULL: count:= 0:
%p A376503 initialize(pq):
%p A376503 insert([-5^7,[5$7]],pq):
%p A376503 for iter from 1 while count < 100 do
%p A376503   t:= extract(pq);
%p A376503   v:= -t[1]; w:= t[2];
%p A376503   if isprime(v-6) and numtheory:-bigomega(v-4) = 3 and numtheory:-bigomega(v-2) = 5 then
%p A376503     R:= R,v-6; count:= count+1;
%p A376503   fi;
%p A376503   p:= nextprime(w[-1]);
%p A376503   for i from 7 to 1 by -1 while w[i] = w[7] do
%p A376503     insert([t[1]*(p/w[7])^(8-i),[op(w[1..i-1]),p$(8-i)]],pq);
%p A376503   od;
%p A376503 od:
%p A376503 R;
%Y A376503 Cf. A001222, A255092.
%K A376503 nonn
%O A376503 1,1
%A A376503 _Zak Seidov_ and _Robert Israel_, Sep 25 2024