A068405 Numbers k such that (k+1) is squarefree and composite and such that there is one more distinct prime factor in k than in (k+1).

84, 90, 105, 110, 114, 132, 140, 154, 165, 182, 186, 204, 220, 234, 246, 252, 258, 264, 266, 273, 286, 290, 294, 300, 308, 318, 322, 340, 345, 354, 357, 364, 370, 380, 385, 402, 406, 410, 414, 426, 444, 450, 465, 468, 470, 480, 492, 504, 516, 518, 525, 532

Offset

1,1

Comments

Numbers k such that (1-isprime(k+1))*issquarefree(k+1)*omega(k) = omega(k+1)+1.

Contains 15*p if p is a prime > 5 such that (15*p + 1)/2 is prime. Thus Dickson's conjecture would imply that this sequence is infinite. - Robert Israel, May 24 2017

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Example

For example 322 = 2*7*23 has 3 distinct prime factors and 323=17*19 has 2 distinct prime factors, hence 322 is in the sequence.

Maple

Res:= NULL:
F:= []:
for n from 2 to 600 do
  Fp:= ifactors(n+1)[2];
  if nops(Fp) > 1 and max(seq(f[2], f=Fp)) = 1 and nops(F) = nops(Fp)+1 then Res:= Res, n fi;
  F:= Fp;
od:
Res; # Robert Israel, May 24 2017

Mathematica

Select[Range[500], CompositeQ[# + 1] && SquareFreeQ[# + 1] && PrimeNu[#] - PrimeNu[# + 1] == 1 &] (* Amiram Eldar, Jun 04 2022 *)

Crossrefs

Cf. A001221, A120944.

Sequence in context: A341304 A128873 A095607 * A349760 A045569 A219183

Adjacent sequences: A068402 A068403 A068404 * A068406 A068407 A068408

Keyword

easy,nonn

Author

Benoit Cloitre, Mar 09 2002

Status

approved