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

1,2

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

Links

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

Formula

A300820(a(n)) <= 1.

Example

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.

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 *)

Prog

(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)

Crossrefs

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

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

Sequence in context: A153381 A307750 A356734 * A324849 A091010 A306528

Adjacent sequences: A319627 A319628 A319629 * A319631 A319632 A319633

Keyword

nonn

Author

Rémy Sigrist, Sep 25 2018

Status

approved