A008507 -id:A008507 - cp's OEIS frontend

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

David Williams, Aug 17 2002

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

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).

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).

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

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

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

(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 *)

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

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

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

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

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

A129768 Number of odd nonprime numbers less than the n-th prime.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 3, 3, 4, 6, 6, 8, 9, 9, 10, 12, 14, 14, 16, 17, 17, 19, 20, 22, 25, 26, 26, 27, 27, 28, 34, 35, 37, 37, 41, 41, 43, 45, 46, 48, 50, 50, 54, 54, 55, 55, 60, 65, 66, 66, 67, 69, 69, 73, 75, 77, 79, 79, 81, 82, 82, 86, 92, 93, 93, 94, 100, 102, 106, 106, 107, 109

Offset: 1

Giovanni Teofilatto, May 16 2007

nonn

Also, a(n)=sum_{k=n-th even number..n-th odd prime together with 1}-k*(-1)^k. - Juri-Stepan Gerasimov, Jul 30 2009

A129768 := proc(n) local res,p,i ; res := 0 ; p := ithprime(n) ; for i from 1 to p-1 by 2 do if not isprime(i) then res := res+1 ; fi ; od ; RETURN(res) ; end: for n from 1 to 80 do printf("%d, ",A129768(n)) ; od ; # R. J. Mathar, May 17 2007

f[n_] := Ceiling[ Prime@n / 2] - n + 1; Array[f, 72] (* Robert G. Wilson v *)
nn=401;With[{np=Select[Range[1,nn,2],!PrimeQ[#]&]},Table[Count[np,?(#<n&)],{n,Prime[Range[PrimePi[nn]]]}]] (* _Harvey P. Dale, Mar 01 2015 *)

a(n)=A008507(n-1)+1. - R. J. Mathar, Jul 06 2009

a(n)=sum_{k=A005843(n)..A006005(n)}-k*(-1)^k. - Juri-Stepan Gerasimov, Jul 30 2009

Edited, corrected and extended by Robert G. Wilson v and R. J. Mathar, May 16 2007

cp's OEIS Frontend

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

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