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A251609 Least k such that prime(k) + prime(k+1) contains n distinct prime divisors.

%I A251609 #17 Jan 20 2025 05:35:32
%S A251609 1,3,6,27,276,1755,24865,646029,7946521,195711271,4129119136,
%T A251609 198635909763,6351380968517,322641218722443,11068897188590241,
%U A251609 501741852481602261,24367382928343066431,1292304206793356882286
%N A251609 Least k such that prime(k) + prime(k+1) contains n distinct prime divisors.
%F A251609 a(n) = A000720(A230518(n)). - _Amiram Eldar_, Feb 17 2019
%e A251609 a(1) = 1 because prime(1) + prime(2) = 2 + 3 = 5, which is a prime power and so has one distinct prime divisor; the other prime indices yielding a prime power are 2, 18, 564,...(A071352) since prime(2) + prime(3) = 3 + 5 = 2^3, prime(18) + prime(19) = 61 + 67 = 2^7, prime(564) + prime(565)= 4093 + 4099 = 2^13,...
%p A251609 N:= 10^6: # to use primes <= N
%p A251609 Primes:= select(isprime, [2,seq(2*i+1,i=1..(N-1)/2)]):
%p A251609 for i from 1 to nops(Primes)-1 do
%p A251609   f:= nops(numtheory:-factorset(Primes[i]+Primes[i+1]));
%p A251609   if not assigned(A[f]) then A[f]:= i fi
%p A251609 od:
%p A251609 seq(A[j],j=1..max(indices(A))); # _Robert Israel_, Dec 05 2014
%t A251609 lst={};Do[k=1;While[Length[FactorInteger[Prime[k]+Prime[k+1]]]!=n,k++];AppendTo[lst,k],{n,1,5}];lst
%Y A251609 Cf. A071215, A001221, A071352, A230518, A251600.
%K A251609 nonn,more
%O A251609 1,2
%A A251609 _Michel Lagneau_, Dec 05 2014
%E A251609 a(10)-a(18) from _Amiram Eldar_, Feb 17 2019