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A319630
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Positive numbers that are not divisible by two consecutive prime numbers.
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25
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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
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OFFSET
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1,2
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COMMENTS
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This sequence is the complement of A104210.
Equivalently, this sequence corresponds to the positive numbers k such that:
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
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LINKS
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FORMULA
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EXAMPLE
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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.
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MAPLE
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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
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MATHEMATICA
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q[n_] := SequenceCount[FactorInteger[n][[;; , 1]], {p1_, p2_} /; p2 == NextPrime[p1]] == 0; Select[Range[100], q] (* Amiram Eldar, Apr 10 2021 *)
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PROG
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(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)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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