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A067611 Numbers of the form 6xy +- x +- y, where x, y are positive integers. 15

%I #127 Sep 08 2022 08:45:05

%S 4,6,8,9,11,13,14,15,16,19,20,21,22,24,26,27,28,29,31,34,35,36,37,39,

%T 41,42,43,44,46,48,49,50,51,53,54,55,56,57,59,60,61,62,63,64,65,66,67,

%U 68,69,71,73,74,75,76,78,79,80,81,82,83,84,85,86,88,89,90,91,92,93,94

%N Numbers of the form 6xy +- x +- y, where x, y are positive integers.

%C Equivalently, numbers n such that either 6n-1 or 6n+1 is composite (or both are).

%C Numbers k such that 36*k^2 - 1 is not a product of twin primes. - _Artur Jasinski_, Dec 12 2007

%C Apart from initial zero, union of A046953 and A046954. - _Reinhard Zumkeller_, Jul 13 2014

%C From _Bob Selcoe_, Nov 18 2014: (Start)

%C Complementary sequence to A002822.

%C For all k >= 1, a(n) are the only positive numbers congruent to the following residue classes:

%C f == k (mod 6k+-1);

%C g == (5k-1) (mod 6k-1);

%C h == (5k+1) (mod 6k+1).

%C All numbers in classes g and h will be in this sequence; for class f, the quotient must be >= 1.

%C 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)).

%C (End)

%C From _Jason Kimberley_, Oct 14 2015: (Start)

%C Numbers n such that A001222(A136017(n)) > 2.

%C The disjoint union of A060461, A121763, and A121765.

%C (End)

%C From _Ralf Steiner_, Aug 08 2018 (Start)

%C 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).

%C 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)

%C 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

%H Reinhard Zumkeller, <a href="/A067611/b067611.txt">Table of n, a(n) for n = 1..10000</a>

%H F. Balestrieri, <a href="https://arxiv.org/abs/1106.6050">An Equivalent Problem To The Twin Prime Conjecture</a>, arXiv:1106.6050 [math.GM], 2011.

%e 4 = 6ab - a - b with a = 1, b = 1.

%e 6 = 6ab + a - b or 6ab - a + b with a = 1, b = 1.

%e 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.

%p filter:= n -> not isprime(6*n+1) or not isprime(6*n-1):

%p select(filter, [$1..1000]); # _Robert Israel_, Nov 18 2014

%t Select[Range[100], !PrimeQ[6# - 1] || !PrimeQ[6# + 1] &]

%t Select[Range[100],AnyTrue[6#+{1,-1},CompositeQ]&] (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale_, Oct 05 2019 *)

%o (Haskell)

%o a067611 n = a067611_list !! (n-1)

%o a067611_list = map (`div` 6) $

%o filter (\x -> a010051' (x-1) == 0 || a010051' (x+1) == 0) [6,12..]

%o -- _Reinhard Zumkeller_, Jul 13 2014

%o (Magma) [n: n in [1..100] | not IsPrime(6*n-1) or not IsPrime(6*n+1)]; // _Vincenzo Librandi_, Nov 19 2014

%o (PARI) for(n=1, 1e2, if(!isprime(6*n+1) || !isprime(6*n-1), print1(n", "))) \\ _Altug Alkan_, Nov 10 2015

%o (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

%o (GAP) Filtered([1..120], k-> not IsPrime(6*k-1) or not IsPrime(6*k+1)) # _G. C. Greubel_, Feb 21 2019

%Y Cf. A002822, A010051, A037074, A046953, A046954, A060461, A070043, A070799, A121763, A121765, A136017, A136050, A071538 (pi_2).

%Y Cf. A323674 (numbers 6xy +- x +- y including repetitions). - _Sally Myers Moite_, Jan 27 2019

%K nonn

%O 1,1

%A _Jon Perry_, Feb 01 2002

%E Edited by _Robert G. Wilson v_, Feb 05 2002

%E Edited by _Dean Hickerson_, May 07 2002

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Last modified March 29 05:48 EDT 2024. Contains 371265 sequences. (Running on oeis4.)