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A319630
Positive numbers that are not divisible by two consecutive prime numbers.
25
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
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
KEYWORD
nonn
AUTHOR
Rémy Sigrist, Sep 25 2018
STATUS
approved