cp's OEIS Frontend

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

%I A067611 #127 Sep 08 2022 08:45:05
%S A067611 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 A067611 41,42,43,44,46,48,49,50,51,53,54,55,56,57,59,60,61,62,63,64,65,66,67,
%U A067611 68,69,71,73,74,75,76,78,79,80,81,82,83,84,85,86,88,89,90,91,92,93,94
%N A067611 Numbers of the form 6xy +- x +- y, where x, y are positive integers.
%C A067611 Equivalently, numbers n such that either 6n-1 or 6n+1 is composite (or both are).
%C A067611 Numbers k such that 36*k^2 - 1 is not a product of twin primes. - _Artur Jasinski_, Dec 12 2007
%C A067611 Apart from initial zero, union of A046953 and A046954. - _Reinhard Zumkeller_, Jul 13 2014
%C A067611 From _Bob Selcoe_, Nov 18 2014: (Start)
%C A067611 Complementary sequence to A002822.
%C A067611 For all k >= 1, a(n) are the only positive numbers congruent to the following residue classes:
%C A067611 f == k (mod 6k+-1);
%C A067611 g == (5k-1) (mod 6k-1);
%C A067611 h == (5k+1) (mod 6k+1).
%C A067611 All numbers in classes g and h will be in this sequence; for class f, the quotient must be >= 1.
%C A067611 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 A067611 (End)
%C A067611 From _Jason Kimberley_, Oct 14 2015: (Start)
%C A067611 Numbers n such that A001222(A136017(n)) > 2.
%C A067611 The disjoint union of A060461, A121763, and A121765.
%C A067611 (End)
%C A067611 From _Ralf Steiner_, Aug 08 2018 (Start)
%C A067611 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 A067611 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 A067611 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 A067611 Reinhard Zumkeller, <a href="/A067611/b067611.txt">Table of n, a(n) for n = 1..10000</a>
%H A067611 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 A067611 4 = 6ab - a - b with a = 1, b = 1.
%e A067611 6 = 6ab + a - b or 6ab - a + b with a = 1, b = 1.
%e A067611 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 A067611 filter:= n -> not isprime(6*n+1) or not isprime(6*n-1):
%p A067611 select(filter, [$1..1000]); # _Robert Israel_, Nov 18 2014
%t A067611 Select[Range[100], !PrimeQ[6# - 1] || !PrimeQ[6# + 1] &]
%t A067611 Select[Range[100],AnyTrue[6#+{1,-1},CompositeQ]&] (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale_, Oct 05 2019 *)
%o A067611 (Haskell)
%o A067611 a067611 n = a067611_list !! (n-1)
%o A067611 a067611_list = map (`div` 6) $
%o A067611    filter (\x -> a010051' (x-1) == 0 || a010051' (x+1) == 0) [6,12..]
%o A067611 -- _Reinhard Zumkeller_, Jul 13 2014
%o A067611 (Magma) [n: n in [1..100] | not IsPrime(6*n-1) or not IsPrime(6*n+1)]; // _Vincenzo Librandi_, Nov 19 2014
%o A067611 (PARI) for(n=1, 1e2, if(!isprime(6*n+1) || !isprime(6*n-1), print1(n", "))) \\ _Altug Alkan_, Nov 10 2015
%o A067611 (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 A067611 (GAP) Filtered([1..120], k-> not IsPrime(6*k-1) or not IsPrime(6*k+1)) # _G. C. Greubel_, Feb 21 2019
%Y A067611 Cf. A002822, A010051, A037074, A046953, A046954, A060461, A070043, A070799, A121763, A121765, A136017, A136050, A071538 (pi_2).
%Y A067611 Cf. A323674 (numbers 6xy +- x +- y including repetitions). - _Sally Myers Moite_, Jan 27 2019
%K A067611 nonn
%O A067611 1,1
%A A067611 _Jon Perry_, Feb 01 2002
%E A067611 Edited by _Robert G. Wilson v_, Feb 05 2002
%E A067611 Edited by _Dean Hickerson_, May 07 2002