A369897 Numbers k such that k and k + 1 each have 10 prime divisors, counted with multiplicity.

3290624, 4122495, 4402431, 5675264, 6608384, 6890624, 7914752, 8614592, 9454400, 11553920, 12613887, 13466816, 14493248, 14853375, 15473024, 16719615, 17494784, 18272384, 18309375, 22784895, 24890624, 25200800, 25869375, 25957503, 26903744, 26921727, 27510272, 28350080, 29761424, 31802624

Offset: 1

Author

Zak Seidov and Robert Israel, Feb 04 2024

Keywords

• nonn

Comments

Numbers k such that k and k + 1 are in A046314.

If a and b are coprime terms of A046312, one of them even, then Dickson's conjecture implies there are infinitely many terms k where k/a and (k+1)/b are primes.

Examples

			a(5) = 6608384 is a term because 6608384 = 2^9 * 12907 and 6608385 = 3^6 * 5 * 7^2 * 37 each have 10 prime divisors, counted with multiplicity.
		

Links

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

Crossrefs

Cf. A001222, A046312, A046314, A115186.

Programs

Maple

with(priqueue):
R:= NULL:  count:= 0:
initialize(Q); r:= 0:
insert([-2^10, [2$10]],Q);
while count < 30 do
  T:= extract(Q);
  if -T[1] = r + 1 then
    R:= R, r; count:= count+1;
  fi;
  r:= -T[1];
  p:= T[2][-1];
  q:= nextprime(p);
  for i from 10 to 1 by -1 while T[2][i] = p do
    insert([-r*(q/p)^(11-i), [op(T[2][1..i-1]),q$(11-i)]],Q);
  od
od:
R;