A061351 Between p and the next prime either there are no numbers or there is a single squarefree number.

2, 5, 29, 41, 101, 137, 281, 461, 569, 617, 641, 821, 857, 1229, 1289, 1301, 1481, 1697, 1721, 1877, 2081, 2129, 2237, 2309, 2381, 2657, 2729, 2801, 3389, 3461, 3557, 3917, 3929, 4001, 4217, 4241, 4421, 4637, 4721, 5009, 5441, 5477, 5501, 5657, 6089

Offset: 0

Author

Labos Elemer, Jun 07 2001

Keywords

• nonn

Comments

Apart from the initial 2, the lesser of twin primes {p, p+2} such that the middle term p+1 is squarefree: intersection[{A014574(n)},{A005117(n)}].

Examples

			Between 29 and 31 the only composite is 30, a squarefree number. If next(p)-p>2, a nonsquarefree integer always arises between them.
		

Links

• Harry J. Smith, Table of n, a(n) for n=0,...,1000

Crossrefs

Cf. A005117, A013929, A000040, A001359, A014574, A061398, A061399.

Programs

Mathematica

Join[{2},Select[Partition[Prime[Range[800]],2,1],#[[2]]-#[[1]]==2&&SquareFreeQ[#[[1]]+1]&][[;;,1]]] (* Harvey P. Dale, Jan 26 2025 *)

PARI

{ n=0; p=3; f="b061351.txt"; write(f, "0 2"); forprime (q=5, 355723, if (q-p == 2, if (issquarefree(p+1), write(f, n++, " ", p))); p=q ) } \\ Harry J. Smith, Jul 21 2009