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A369898 Numbers k such that k and k + 1 each have 9 prime divisors, counted with multiplicity.

%I A369898 #7 Feb 05 2024 09:27:10
%S A369898 203391,698624,1245375,1942784,2176064,2282175,2536191,2858624,
%T A369898 2953664,3282687,3560192,3655935,3914000,4068224,4135616,4205600,
%U A369898 4244967,4586624,4695488,4744575,4991679,5055615,5450624,5475519,5519744,6141824,6246800,6410096,6655040,6660224,6753375,6816879,6862400
%N A369898 Numbers k such that k and k + 1 each have 9 prime divisors, counted with multiplicity.
%C A369898 Numbers k such that k and k + 1 are in A046312.
%C A369898 If a and b are coprime terms of A046310, one of them even, then Dickson's conjecture implies there are infinitely many terms k where k/a and (k+1)/b are primes.
%H A369898 Robert Israel, <a href="/A369898/b369898.txt">Table of n, a(n) for n = 1..10000</a>
%e A369898 a(3) = 1245375 is a term because 1245375 = 3^5 * 5^3 * 41 and 1245376 = 2^6 * 11 * 29 * 61 each have 9 prime factors, counted with multiplicity.
%p A369898 with(priqueue):
%p A369898 R:= NULL:  count:= 0:
%p A369898 initialize(Q); r:= 0:
%p A369898 insert([-2^9, [2$9]], Q);
%p A369898 while count < 40 do
%p A369898   T:= extract(Q);
%p A369898   if -T[1] = r + 1 then
%p A369898     R:= R, r; count:= count+1;
%p A369898   fi;
%p A369898   r:= -T[1];
%p A369898   p:= T[2][-1];
%p A369898   q:= nextprime(p);
%p A369898   for i from 9 to 1 by -1 while T[2][i] = p do
%p A369898     insert([-r*(q/p)^(10-i), [op(T[2][1..i-1]), q$(10-i)]], Q);
%p A369898   od
%p A369898 od:
%p A369898 R;
%Y A369898 Cf. A001222, A046310, A046312, A115186, A369897.
%K A369898 nonn
%O A369898 1,1
%A A369898 _Robert Israel_, Feb 04 2024