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A246716
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Positive numbers that are not the product of (exactly) two distinct primes.
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3
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1, 2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 16, 17, 18, 19, 20, 23, 24, 25, 27, 28, 29, 30, 31, 32, 36, 37, 40, 41, 42, 43, 44, 45, 47, 48, 49, 50, 52, 53, 54, 56, 59, 60, 61, 63, 64, 66, 67, 68, 70, 71, 72, 73, 75, 76, 78, 79, 80, 81, 83, 84, 88, 89, 90, 92, 96, 97, 98, 99, 100
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OFFSET
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1,2
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COMMENTS
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Complement of A006881, then inheriting the "opposite" of the properties of A006881.
a(n+1) - a(n) <= 4 (gap upper bound) - (that is because among four consecutive integers there is always a multiple of 4, then there is a number which is not the product of two distinct primes). E.g., a(26)-a(25) = a(62)-a(61) = 4. Is it true that for any k <= 4 there are infinitely many numbers n such that a(n+1) - a(n) = k?
If r = A006881(n+1) - A006881(n) - 1 > 1, it indicates that there are r terms of (a(j)) starting with j = A006881(n) - n + 1 which are consecutive integers. E.g., A006881(8) - A006881(7) - 1 = 6, so there are 6 consecutive terms in (a(j)), starting with j = A006881(7) - 7 + 1 = 20.
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LINKS
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EXAMPLE
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7 is in the sequence because 7 is prime, so it has only one prime divisor.
8 and 9 are in the sequence because neither of them has two distinct prime divisors.
30 is in the sequence because it is the product of three primes.
On the other hand, 35 is not in the sequence because it is the product of two distinct primes.
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MAPLE
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filter:= n -> map(t -> t[2], ifactors(n)[2]) <> [1, 1]:
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MATHEMATICA
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Select[Range[125], Not[PrimeOmega[#] == PrimeNu[#] == 2] &] (* Alonso del Arte, Nov 03 2014 *)
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PROG
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(PARI) isok(n) = (omega(n)!=2) || (bigomega(n) != 2); \\ Michel Marcus, Nov 01 2014
(Magma) [n: n in [1..100] | #PrimeDivisors(n) ne 2 or &*[t[2]: t in Factorization(n)] ne 1]; // Bruno Berselli, Nov 12 2014
(Sage)
R = []
for i in (1..n) :
d = prime_divisors(i)
if len(d) != 2 or d[0]*d[1] != i : R.append(i)
return R
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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