A199593 Numbers n such that 3n-2, 3n-1 and 3n are all composite.

9, 12, 17, 19, 22, 26, 29, 31, 32, 39, 40, 41, 42, 45, 48, 49, 52, 54, 57, 59, 62, 63, 68, 69, 70, 72, 73, 74, 79, 82, 83, 85, 87, 89, 92, 96, 97, 99, 100, 101, 102, 107, 108, 109, 110, 112, 114, 115, 119, 121, 122, 124, 126, 129, 131, 132, 135, 136, 138, 139, 142, 143, 146, 149, 151, 152, 157, 158, 159, 161, 162, 165, 166, 169, 171, 172, 173, 176, 177, 178

Offset

1,1

Comments

From Antti Karttunen, Apr 17 2015: (Start)

Other, equivalent definitions:

Numbers n such that A007310(n) is composite, from which it follows that the function c(1) = 0, c(n) = 1-A075743(n-1) is the characteristic function of this sequence.

Numbers n such that A084967(n) has at least three prime factors (when counted with bigomega, A001222).

Numbers n such that A249823(n) is composite.

(End)

There are n - pi(3n) + 1 terms in this sequence up to n; with an efficient algorithm for pi(x) this allows isolated large values to be computed. Using David Baugh and Kim Walisch's calculation that pi(10^27) = 16352460426841680446427399 one can see that a(316980872906491652886905934) = 333333333333333333333333333 (since 999999999999999999999999997 is composite). - Charles R Greathouse IV, Sep 13 2016

References

Ernest V. Miliauskas, letter to N. J. A. Sloane, Dec 21 1985.

Links

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Bogart B. Strauss, Formula Explanation, pp. 1, 2, 3.

Formula

((1+(-1)^k)((-1)^n)(2n+3)+2k(6n+9+(-1)^n)+((-1)^k)+(12n^2)+36n+29)/4 n,k are all natural numbers and zero. - Bogart B. Strauss, Jul 10 2013

a(n) = n + 3n/log n + o(n/log n). - Charles R Greathouse IV, Oct 27 2013, corrected Aug 07 2016

Maple

remove(t -> isprime(3*t-1 - (t mod 2)), {$2..2000}); # Robert Israel, Apr 17 2015

Mathematica

Select[Range[200], Union[PrimeQ[{3# - 2, 3# - 1, 3#}]] == {False} &] (* Alonso del Arte, Jul 06 2013 *)

Prog

(PARI) is(n)=!isprime(bitor(3*n-2, 1)) && n>1 \\ Charles R Greathouse IV, Oct 27 2013
(Scheme, after Greathouse's PARI-program above, requiring also Antti Karttunen's IntSeq-library)
(define A199593 (MATCHING-POS 1 2 (lambda (n) (not (prime? (A003986bi (+ n n n -2) 1)))))) ;; A003986bi implements binary inclusive or (A003986).
;; Antti Karttunen, Apr 17 2015
(Magma) [n: n in [1..200] | not IsPrime(3*n) and not IsPrime(3*n-1) and not IsPrime(3*n-2)]; // Vincenzo Librandi, Apr 18 2015
(Python)
from sympy import isprime
def ok(n): return n > 0 and not any(isprime(3*n-i) for i in [2, 1, 0])
print([k for k in range(179) if ok(k)]) # Michael S. Branicky, Apr 16 2022

Crossrefs

Cf. A053726, A199595, A001222, A007310, A075743, A084967, A249823.

Sequence in context: A027571 A342757 A154631 * A356842 A174525 A141552

Adjacent sequences: A199590 A199591 A199592 * A199594 A199595 A199596

Keyword

nonn,easy

Author

N. J. A. Sloane, Nov 08 2011

Status

approved