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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.
2

%I #9 Jan 12 2024 22:47:22

%S 1559,4073,5237,5987,12119,14633,24697,29881,29947,30113,32003,41903,

%T 45863,60169,64817,67601,69151,71263,73783,77713,78929,79633,86629,

%U 88547,91493,95483,96181,108037,109859,110459,111667,125471,132833,133283,140419,142049,160001,165133,170579,171803,171827,171947

%N 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 The number of prime divisors is at least 3, because p + q is even and not twice a prime.

%H Robert Israel, <a href="/A368785/b368785.txt">Table of n, a(n) for n = 1..10000</a>

%e a(2) = 4073 is a term because 4073, 4079, 4091 are consecutive primes with

%e 4073 + 4079 = 8152 = 2^3 * 1019,

%e 4073 + 4091 = 8164 = 2^2 * 13 * 157,

%e 4079 + 4091 = 8170 = 2 * 5 * 19 * 43, and

%e 4073 + 4079 + 4091 = 12243 = 3 * 7 * 11 * 53

%e all have 4 prime divisors, counted with multiplicity.

%p R:= NULL: count:= 0:

%p p:= 2: q:= 3: r:= 5: v:= numtheory:-bigomega(q+r);

%p while count < 100 do

%p p:= q; q:= r; r:= nextprime(r);

%p w:= numtheory:-bigomega(q+r);

%p if w = v and numtheory:-bigomega(p+r) = v and numtheory:-bigomega(p+q+r) = v then

%p R:= R,p; count:= count+1;

%p fi;

%p v:= w;

%p od:

%p R;

%Y Cf. A001222, A368786.

%K nonn

%O 1,1

%A _Zak Seidov_ and _Robert Israel_, Jan 05 2024