A067076 Numbers k such that 2*k + 3 is a prime.

0, 1, 2, 4, 5, 7, 8, 10, 13, 14, 17, 19, 20, 22, 25, 28, 29, 32, 34, 35, 38, 40, 43, 47, 49, 50, 52, 53, 55, 62, 64, 67, 68, 73, 74, 77, 80, 82, 85, 88, 89, 94, 95, 97, 98, 104, 110, 112, 113, 115, 118, 119, 124, 127, 130, 133, 134, 137, 139, 140, 145, 152, 154, 155

Offset: 1

Author

David Williams, Aug 17 2002

Keywords

• nonn
• easy
• changed

Comments

The following sequences (allowing offset of first term) all appear to have the same parity: A034953, triangular numbers with prime indices; A054269, length of period of continued fraction for sqrt(p), p prime; A082749, difference between the sum of next prime(n) natural numbers and the sum of next n primes; A006254, numbers n such that 2n-1 is prime; A067076, 2n+3 is a prime. - Jeremy Gardiner, Sep 10 2004

n is in the sequence iff none of the numbers (n-3k)/(2k+1), 1 <= k <= (n-1)/5, is positive integer. - Vladimir Shevelev, May 31 2009

Zeta(s) = Sum_{n>=1} 1/n^s = 1/1 - 2^(-s) * Product_{p=prime=(2*A067076)+3} 1/(1 - (2*A067076+3)^(-s)). - Eric Desbiaux, Dec 15 2009

This sequence is a subsequence of A047949. - Jason Kimberley, Aug 30 2012

Links

• Harry J. Smith, Table of n, a(n) for n = 1..1000
• Mutsumi Suzuki, Vincenzo Librandi's method for sequential primes (Librandi's description in Italian).

Crossrefs

Cf. A086801, A047949.

Numbers n such that 2n+k is prime: A005097 (k=1), this seq(k=3), A089038 (k=5), A105760 (k=7), A155722 (k=9), A101448 (k=11), A153081 (k=13), A089559 (k=15), A173059 (k=17), A153143 (k=19). - Jason Kimberley, Sep 07 2012

Numbers n such that 2n-k is prime: A006254 (k=1), A098090 (k=3), A089253 (k=5), A089192 (k=7), A097069 (k=9), A097338 (k=11), A097363 (k=13), A097480 (k=15), A098605 (k=17), A097932 (k=19).

Programs

GAP

Filtered([0..200], k-> IsPrime(2*k+3) ); # G. C. Greubel, May 21 2019

Magma

[n: n in [0..200]| IsPrime(2*n+3)]; // Vincenzo Librandi, Feb 23 2012

Maple

select(t -> isprime(2*t+3), [$0..1000]); # Robert Israel, Feb 19 2015

Mathematica

(Prime[Range[100]+1]-3)/2  (* Vladimir Joseph Stephan Orlovsky, Sep 08 2008, modified by G. C. Greubel, May 21 2019 *)
Select[Range[0,200],PrimeQ[2#+3]&] (* Harvey P. Dale, Jun 10 2014 *)

PARI

[k | k<-[0..99], isprime(2*k+3)] \\ for illustration

PARI

A067076(n) = (prime(n+1)-3)/2 \\ M. F. Hasler, Feb 14 2024

Sage

[n for n in (0..200) if is_prime(2*n+3) ] # G. C. Greubel, May 21 2019

Formula

a(n) = A006254(n) - 2 = A086801(n+1)/2. [Corrected by M. F. Hasler, Feb 14 2024]

a(n) = A089253(n) - 4. - Giovanni Teofilatto, Dec 14 2003

Conjecture: a(n) = A008507(n) + n - 1 = A005097(n) - 1 = A102781(n+1) - 1. - R. J. Mathar, Jul 07 2009

a(n) = A179893(n) - A000040(n). - Odimar Fabeny, Aug 24 2010

Extensions

Offset changed from 0 to 1 in 2008: some formulas here and elsewhere may need to be corrected.