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

171869, 5609369, 7653119, 11177069, 11320709, 11479319, 12325619, 13530119, 15426419, 15558119, 17136619, 17541869, 17851919, 18809369, 18843119, 19593869, 19634369, 19938119, 20806619, 21600419, 22470953, 23637839, 23796869, 23999369, 24006119, 24275819, 25798739, 25879001, 25965869, 26278169

Offset: 1

Author

Zak Seidov and Robert Israel, Sep 25 2024

Keywords

• nonn

Comments

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.

Examples

			a(3) = 7653119 is a term because 7653119 is prime,
7653121 = 7 * 61 * 17923 has 3 prime factors,
7653123 = 3^4 * 94483 has 5 prime factors, and
7653125 = 5^5 * 31 * 79 has 7 prime factors, counted with multiplicity.
		

Links

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

Crossrefs

Cf. A001222, A255092.

Programs

Maple

with(priqueue):
R:= NULL: count:= 0:
initialize(pq):
insert([-5^7,[5$7]],pq):
for iter from 1 while count < 100 do
  t:= extract(pq);
  v:= -t[1]; w:= t[2];
  if isprime(v-6) and numtheory:-bigomega(v-4) = 3 and numtheory:-bigomega(v-2) = 5 then
    R:= R,v-6; count:= count+1;
  fi;
  p:= nextprime(w[-1]);
  for i from 7 to 1 by -1 while w[i] = w[7] do
    insert([t[1]*(p/w[7])^(8-i),[op(w[1..i-1]),p$(8-i)]],pq);
  od;
od:
R;