A106002 -id:A106002 - cp's OEIS frontend

A105399 Largest prime <= numbers of the form 6k+3 (duplicates removed).

3, 7, 13, 19, 23, 31, 37, 43, 47, 53, 61, 67, 73, 79, 83, 89, 97, 103, 109, 113, 127, 131, 139, 151, 157, 163, 167, 173, 181, 193, 199, 211, 223, 229, 233, 241, 251, 257, 263, 271, 277, 283, 293, 307, 313, 317, 331, 337, 349, 353, 359, 367, 373, 379, 383, 389

Offset: 1

Giovanni Teofilatto, May 01 2005

Apart from the initial 3, the same as A049591. [Proof from T. Khovanova, Jan 23 2008: True for primes up to 5 by inspection. Higher primes must be of the form 6k+1 or 6k+5 since 6k+2 and 6k+4 are divisible by 2 and 6k+3 is divisible by 3. So searching the prime p backwards from the composite, odd 6k+3 in steps of 2 implies that p+2, skipped during that scan, is composite. So p is not in A001359 but in A049591.] - R. J. Mathar, Jan 28 2008

			7 is in the sequence because 7 is the largest prime < 9=6*1+3.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A106002.

Cf. A049591.

pp[n_] := Block[{k = n},While[ ! PrimeQ[k], k-- ];k];Union[Table[pp[6n + 3], {n, 0, 65}]] (* Ray Chandler, Oct 17 2006 *)
Union[If[PrimeQ[#],#,NextPrime[#,-1]]&/@(6*Range[0,70]+3)] (* Harvey P. Dale, Aug 20 2021 *)

Edited, corrected and extended by Ray Chandler, Oct 17 2006

A105470 a(n)=1 if there is number of the form 6k+3 with prime(n) <= 6k+3 <= prime(n+1), otherwise 0.

Original entry on oeis.org

1, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1

Offset: 1

Giovanni Teofilatto, May 02 2005

Except for the first pair of primes and for twin primes there is always at least one number of the form 6n+3 between two successive primes.

			a(3)=0 because between prime(3) and prime(4) there are no numbers of the form 6k+3;
a(4)=1 because between prime(4) and prime(5) there is one number of the form 6k+3: 9.

Cf. A100810, A106002.

f[n_] := Count[Table[Mod[k, 6], {k, Prime[n], Prime[n + 1]}], 3];Table[If[f[n] == 0, 0, 1], {n, 120}] (* Ray Chandler, Oct 17 2006 *)
Join[{1,1},If[Last[#]-First[#]==2,0,1]&/@Partition[Prime[Range[ 3,200]],2,1]] (* Harvey P. Dale, Nov 27 2013 *)

Edited by Ray Chandler, Oct 17 2006

cp's OEIS Frontend

A105399 Largest prime <= numbers of the form 6k+3 (duplicates removed).

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