%I #92 Feb 14 2024 19:48:30
%S 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,
%T 50,52,53,55,62,64,67,68,73,74,77,80,82,85,88,89,94,95,97,98,104,110,
%U 112,113,115,118,119,124,127,130,133,134,137,139,140,145,152,154,155
%N Numbers k such that 2*k + 3 is a prime.
%C 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
%C 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
%C 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
%C This sequence is a subsequence of A047949. - _Jason Kimberley_, Aug 30 2012
%H Harry J. Smith, <a href="/A067076/b067076.txt">Table of n, a(n) for n = 1..1000</a>
%H Mutsumi Suzuki, <a href="http://mathforum.org/te/exchange/hosted/suzuki/Vincent.html">Vincenzo Librandi's method for sequential primes</a> (Librandi's description in Italian).
%F a(n) = A006254(n) - 2 = A086801(n+1)/2. [Corrected by _M. F. Hasler_, Feb 14 2024]
%F a(n) = A089253(n) - 4. - _Giovanni Teofilatto_, Dec 14 2003
%F Conjecture: a(n) = A008507(n) + n - 1 = A005097(n) - 1 = A102781(n+1) - 1. - _R. J. Mathar_, Jul 07 2009
%F a(n) = A179893(n) - A000040(n). - _Odimar Fabeny_, Aug 24 2010
%p select(t -> isprime(2*t+3), [$0..1000]); # _Robert Israel_, Feb 19 2015
%t (Prime[Range[100]+1]-3)/2 (* _Vladimir Joseph Stephan Orlovsky_, Sep 08 2008, modified by _G. C. Greubel_, May 21 2019 *)
%t Select[Range[0,200],PrimeQ[2#+3]&] (* _Harvey P. Dale_, Jun 10 2014 *)
%o (PARI) [k | k<-[0..99], isprime(2*k+3)] \\ for illustration
%o (PARI) A067076(n) = (prime(n+1)-3)/2 \\ _M. F. Hasler_, Feb 14 2024
%o (Magma)[n: n in [0..200]| IsPrime(2*n+3)]; // _Vincenzo Librandi_, Feb 23 2012
%o (Sage) [n for n in (0..200) if is_prime(2*n+3) ] # _G. C. Greubel_, May 21 2019
%o (GAP) Filtered([0..200], k-> IsPrime(2*k+3) ) # _G. C. Greubel_, May 21 2019
%Y Cf. A086801, A047949.
%Y 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
%Y 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).
%K nonn,easy
%O 1,3
%A _David Williams_, Aug 17 2002
%E Offset changed from 0 to 1 in 2008: some formulas here and elsewhere may need to be corrected.