cp's OEIS Frontend

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

%I A319630 #19 Jul 29 2022 23:48:19
%S A319630 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,
%T A319630 32,33,34,37,38,39,40,41,43,44,46,47,49,50,51,52,53,55,56,57,58,59,61,
%U A319630 62,63,64,65,67,68,69,71,73,74,76,79,80,81,82,83,85,86,87
%N A319630 Positive numbers that are not divisible by two consecutive prime numbers.
%C A319630 This sequence is the complement of A104210.
%C A319630 Equivalently, this sequence corresponds to the positive numbers k such that:
%C A319630 - A300820(k) <= 1,
%C A319630 - A087207(k) is a Fibbinary number (A003714).
%C A319630 For any n > 0 and k >= 0, a(n)^k belongs to the sequence.
%C A319630 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
%C A319630 Numbers not divisible by any term of A006094. - _Antti Karttunen_, Jul 29 2022
%H A319630 Robert Israel, <a href="/A319630/b319630.txt">Table of n, a(n) for n = 1..10000</a>
%F A319630 A300820(a(n)) <= 1.
%e A319630 The number 10 is only divisible by 2 and 5, hence 10 appears in the sequence.
%e A319630 The number 42 is divisible by 2 and 3, hence 42 does not appear in the sequence.
%p A319630 N:= 1000: # for terms <= N
%p A319630 R:= {}:
%p A319630 p:= 2:
%p A319630 do
%p A319630   q:= p; p:= nextprime(p);
%p A319630   if p*q > N then break fi;
%p A319630   R:= R union {seq(i,i=p*q..N,p*q)}
%p A319630 od:
%p A319630 sort(convert({$1..N} minus R,list)); # _Robert Israel_, Apr 13 2020
%t A319630 q[n_] := SequenceCount[FactorInteger[n][[;; , 1]], {p1_, p2_} /; p2 == NextPrime[p1]] ==  0; Select[Range[100], q] (* _Amiram Eldar_, Apr 10 2021 *)
%o A319630 (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)
%Y A319630 Cf. A003714, A006094, A087207, A104210, A300820, A356171 (odd terms only).
%Y A319630 Positions of 1's in A322361 and in A356173 (characteristic function).
%K A319630 nonn
%O A319630 1,2
%A A319630 _Rémy Sigrist_, Sep 25 2018