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%I #9 Sep 10 2022 12:20:54
%S 1,9,11,15,16,18,21,23,30,32,34,36,37,39,40,42,51,53,54,55,56,58,61,
%T 62,66,67,68,71,73,74,76,80,82,84,86,96,100,101,102,103,105,106,107,
%U 108,110,111,115,118,119,123,125,127,129,130,133,137,138,141,145,146,150
%N Integers n such that prime(n+1)-prime(n) is nonprime, squarefree. Except for the initial term of 1, the terms are k-semiprime for some k>=2.
%H Harvey P. Dale, <a href="/A113339/b113339.txt">Table of n, a(n) for n = 1..1000</a>
%F prime(n+1)-prime(n)=1 or p1*...*pk where p1, ..., pk are two or more distinct primes.
%e prime(69)-prime(68)=347-337=10=2*5.
%p L:=[]: for z to 1 do for k from 1 to 200 do p:=ithprime(k); q:=nextprime(p); x:=q-p; if not(isprime(x)) and issqrfree(x) then L:=[op(L),k] fi od od; L;
%t Flatten[Position[Flatten[Differences/@Partition[Prime[Range[200]],2,1]],_?(!PrimeQ[#] && SquareFreeQ[#]&)]]//Rest (* _Harvey P. Dale_, Sep 10 2022 *)
%Y Cf. A000040, A000469, A001358.
%K easy,nonn
%O 1,2
%A _Walter Kehowski_, Jan 08 2006