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A067611
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Numbers of the form 6xy +- x +- y, where x, y are positive integers.
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15
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4, 6, 8, 9, 11, 13, 14, 15, 16, 19, 20, 21, 22, 24, 26, 27, 28, 29, 31, 34, 35, 36, 37, 39, 41, 42, 43, 44, 46, 48, 49, 50, 51, 53, 54, 55, 56, 57, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 71, 73, 74, 75, 76, 78, 79, 80, 81, 82, 83, 84, 85, 86, 88, 89, 90, 91, 92, 93, 94
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OFFSET
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1,1
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COMMENTS
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Equivalently, numbers n such that either 6n-1 or 6n+1 is composite (or both are).
Numbers k such that 36*k^2 - 1 is not a product of twin primes. - Artur Jasinski, Dec 12 2007
For all k >= 1, a(n) are the only positive numbers congruent to the following residue classes:
f == k (mod 6k+-1);
g == (5k-1) (mod 6k-1);
h == (5k+1) (mod 6k+1).
All numbers in classes g and h will be in this sequence; for class f, the quotient must be >= 1.
When determining which numbers are contained in this sequence, it is only necessary to evaluate f, g and h when the moduli are prime and the dividends are >= 2*k*(3*k - 1) (i.e., A033579(k)).
(End)
(End)
Conjecture 1: With u(k) = floor(k(k + 1)/4) one has A071538(a(u(k))*6) = a(u(k)) - u(k) + 1, for k >= 2 (u > 1).
Conjecture 2: In the interval [T(k-1)+1, T(k)], with T(k) = A000217(k), k >= 2, there exists at least one number that is not a member of the present sequence. (End)
Also: numbers of the form n*p +- round(p/6) with some positive integer n and prime p >= 5. [Proof available on demand.] - M. F. Hasler, Jun 25 2019
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LINKS
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EXAMPLE
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4 = 6ab - a - b with a = 1, b = 1.
6 = 6ab + a - b or 6ab - a + b with a = 1, b = 1.
5 cannot be obtained by any values of a and b in 6ab - a - b, 6ab - a + b, 6ab + a - b or 6ab + a + b.
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MAPLE
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filter:= n -> not isprime(6*n+1) or not isprime(6*n-1):
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MATHEMATICA
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Select[Range[100], !PrimeQ[6# - 1] || !PrimeQ[6# + 1] &]
Select[Range[100], AnyTrue[6#+{1, -1}, CompositeQ]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Oct 05 2019 *)
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PROG
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(Haskell)
a067611 n = a067611_list !! (n-1)
a067611_list = map (`div` 6) $
filter (\x -> a010051' (x-1) == 0 || a010051' (x+1) == 0) [6, 12..]
(Magma) [n: n in [1..100] | not IsPrime(6*n-1) or not IsPrime(6*n+1)]; // Vincenzo Librandi, Nov 19 2014
(PARI) for(n=1, 1e2, if(!isprime(6*n+1) || !isprime(6*n-1), print1(n", "))) \\ Altug Alkan, Nov 10 2015
(Sage) [n for n in (1..120) if not is_prime(6*n-1) or not is_prime(6*n+1)] # G. C. Greubel, Feb 21 2019
(GAP) Filtered([1..120], k-> not IsPrime(6*k-1) or not IsPrime(6*k+1)) # G. C. Greubel, Feb 21 2019
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CROSSREFS
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Cf. A002822, A010051, A037074, A046953, A046954, A060461, A070043, A070799, A121763, A121765, A136017, A136050, A071538 (pi_2).
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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