

A067611


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


15



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

1,1


COMMENTS

Equivalently, numbers n such that either 6n1 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
Apart from initial zero, union of A046953 and A046954.  Reinhard Zumkeller, Jul 13 2014
From Bob Selcoe, Nov 18 2014: (Start)
Complementary sequence to A002822.
For all k >= 1, a(n) are the only positive numbers congruent to the following residue classes:
f == k (mod 6k+1);
g == (5k1) (mod 6k1);
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)
From Jason Kimberley, Oct 14 2015: (Start)
Numbers n such that A001222(A136017(n)) > 2.
The disjoint union of A060461, A121763, and A121765.
(End)
From Ralf Steiner, Aug 08 2018 (Start)
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(k1)+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


LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
F. Balestrieri, An Equivalent Problem To The Twin Prime Conjecture, arXiv:1106.6050 [math.GM], 2011.


EXAMPLE

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.


MAPLE

filter:= n > not isprime(6*n+1) or not isprime(6*n1):
select(filter, [$1..1000]); # Robert Israel, Nov 18 2014


MATHEMATICA

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 *)


PROG

(Haskell)
a067611 n = a067611_list !! (n1)
a067611_list = map (`div` 6) $
filter (\x > a010051' (x1) == 0  a010051' (x+1) == 0) [6, 12..]
 Reinhard Zumkeller, Jul 13 2014
(MAGMA) [n: n in [1..100]  not IsPrime(6*n1) or not IsPrime(6*n+1)]; // Vincenzo Librandi, Nov 19 2014
(PARI) for(n=1, 1e2, if(!isprime(6*n+1)  !isprime(6*n1), print1(n", "))) \\ Altug Alkan, Nov 10 2015
(Sage) [n for n in (1..120) if not is_prime(6*n1) or not is_prime(6*n+1)] # G. C. Greubel, Feb 21 2019
(GAP) Filtered([1..120], k> not IsPrime(6*k1) or not IsPrime(6*k+1)) # G. C. Greubel, Feb 21 2019


CROSSREFS

Cf. A002822, A010051, A037074, A046953, A046954, A060461, A070043, A070799, A121763, A121765, A136017, A136050, A071538 (pi_2).
Cf. A323674 (numbers 6xy + x + y including repetitions).  Sally Myers Moite, Jan 27 2019
Sequence in context: A180638 A213320 A024887 * A190485 A105803 A202268
Adjacent sequences: A067608 A067609 A067610 * A067612 A067613 A067614


KEYWORD

nonn


AUTHOR

Jon Perry, Feb 01 2002


EXTENSIONS

Edited by Robert G. Wilson v, Feb 05 2002
Edited by Dean Hickerson, May 07 2002


STATUS

approved



