A319630 Positive numbers that are not divisible by two consecutive prime numbers.

1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 16, 17, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 31, 32, 33, 34, 37, 38, 39, 40, 41, 43, 44, 46, 47, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 61, 62, 63, 64, 65, 67, 68, 69, 71, 73, 74, 76, 79, 80, 81, 82, 83, 85, 86, 87

Offset: 1

Author

Rémy Sigrist, Sep 25 2018

Keywords

• nonn

Comments

This sequence is the complement of A104210.

Equivalently, this sequence corresponds to the positive numbers k such that:

- A300820(k) <= 1,

- A087207(k) is a Fibbinary number (A003714).

For any n > 0 and k >= 0, a(n)^k belongs to the sequence.

The numbers of terms not exceeding 10^k, for k=1,2,..., are 9, 78, 758, 7544, 75368, 753586, 7535728, 75356719, 753566574, ... Apparently, the asymptotic density of this sequence is 0.75356... - Amiram Eldar, Apr 10 2021

Numbers not divisible by any term of A006094. - Antti Karttunen, Jul 29 2022

Examples

			The number 10 is only divisible by 2 and 5, hence 10 appears in the sequence.
The number 42 is divisible by 2 and 3, hence 42 does not appear in the sequence.
		

Links

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

Crossrefs

Cf. A003714, A006094, A087207, A104210, A300820, A356171 (odd terms only).

Positions of 1's in A322361 and in A356173 (characteristic function).

Programs

Maple

N:= 1000: # for terms <= N
R:= {}:
p:= 2:
do
  q:= p; p:= nextprime(p);
  if p*q > N then break fi;
  R:= R union {seq(i,i=p*q..N,p*q)}
od:
sort(convert({$1..N} minus R,list)); # Robert Israel, Apr 13 2020

Mathematica

q[n_] := SequenceCount[FactorInteger[n][[;; , 1]], {p1_, p2_} /; p2 == NextPrime[p1]] ==  0; Select[Range[100], q] (* Amiram Eldar, Apr 10 2021 *)

PARI

is(n) = my (f=factor(n)); for (i=1, #f~-1, if (nextprime(f[i,1]+1)==f[i+1,1], return (0))); return (1)

Formula

A300820(a(n)) <= 1.