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A368785 Least of three consecutive primes p, q, r such that p + q, p + r, q + r and p + q + r all have the same number of prime divisors, counted with multiplicity.

%I A368785 #13 May 25 2025 19:48:03
%S A368785 1559,4073,5237,5987,12119,14633,24697,29881,29947,30113,32003,41903,
%T A368785 45863,60169,64817,67601,69151,71263,73783,77713,78929,79633,86629,
%U A368785 88547,91493,95483,96181,108037,109859,110459,111667,125471,132833,133283,140419,142049,160001,165133,170579,171803,171827,171947
%N A368785 Least of three consecutive primes p, q, r such that p + q, p + r, q + r and p + q + r all have the same number of prime divisors, counted with multiplicity.
%C A368785 The number of prime divisors is at least 3, because p + q is even and not twice a prime.
%H A368785 Robert Israel, <a href="/A368785/b368785.txt">Table of n, a(n) for n = 1..10000</a>
%e A368785 a(2) = 4073 is a term because 4073, 4079, 4091 are consecutive primes with
%e A368785 4073 + 4079 = 8152 = 2^3 * 1019,
%e A368785 4073 + 4091 = 8164 = 2^2 * 13 * 157,
%e A368785 4079 + 4091 = 8170 = 2 * 5 * 19 * 43, and
%e A368785 4073 + 4079 + 4091 = 12243 = 3 * 7 * 11 * 53
%e A368785 all have 4 prime divisors, counted with multiplicity.
%p A368785 R:= NULL: count:= 0:
%p A368785 p:= 2: q:= 3: r:= 5: v:= numtheory:-bigomega(q+r);
%p A368785 while count < 100 do
%p A368785   p:= q; q:= r; r:= nextprime(r);
%p A368785   w:= numtheory:-bigomega(q+r);
%p A368785   if w = v and numtheory:-bigomega(p+r) = v and numtheory:-bigomega(p+q+r) = v then
%p A368785     R:= R,p; count:= count+1;
%p A368785     fi;
%p A368785   v:= w;
%p A368785 od:
%p A368785 R;
%t A368785 Select[Partition[Prime[Range[16000]],3,1],Length[Union[PrimeOmega[Total/@Subsets[#,{2,3}]]]]==1&][[;;,1]] (* _Harvey P. Dale_, May 25 2025 *)
%Y A368785 Cf. A001222, A368786.
%K A368785 nonn
%O A368785 1,1
%A A368785 _Zak Seidov_ and _Robert Israel_, Jan 05 2024