A113339 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.

1, 9, 11, 15, 16, 18, 21, 23, 30, 32, 34, 36, 37, 39, 40, 42, 51, 53, 54, 55, 56, 58, 61, 62, 66, 67, 68, 71, 73, 74, 76, 80, 82, 84, 86, 96, 100, 101, 102, 103, 105, 106, 107, 108, 110, 111, 115, 118, 119, 123, 125, 127, 129, 130, 133, 137, 138, 141, 145, 146, 150

Offset: 1

Author

Walter Kehowski, Jan 08 2006

Keywords

• easy
• nonn

Examples

			prime(69)-prime(68)=347-337=10=2*5.
		

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A000040, A000469, A001358.

Programs

Maple

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;

Mathematica

Flatten[Position[Flatten[Differences/@Partition[Prime[Range[200]],2,1]],?(!PrimeQ[#] && SquareFreeQ[#]&)]]//Rest (* _Harvey P. Dale, Sep 10 2022 *)

Formula

prime(n+1)-prime(n)=1 or p1*...*pk where p1, ..., pk are two or more distinct primes.