

A163573


Primes p such that (p+1)/2, (p+2)/3 and (p+3)/4 are also primes.


9



12721, 16921, 19441, 24481, 49681, 61561, 104161, 229321, 255361, 259681, 266401, 291721, 298201, 311041, 331921, 419401, 423481, 436801, 446881, 471241, 525241, 532801, 539401, 581521, 600601, 663601, 704161, 709921, 783721, 867001, 904801
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OFFSET

1,1


COMMENTS

Are they all == 1 (mod 10) ?
Subsequence of A005383, of A091180 and of A036570.  R. J. Mathar, Aug 01 2009
Since (p+2)/3 and (p+3)/4 must be integer, the Chinese remainder theorem shows that all terms are ==1 (mod 12).  R. J. Mathar, Aug 01 2009
All terms are of the form 120k+1: a(n)=120*A163625(n)+1.  Zak Seidov, Aug 01 2009
Each term is congruent to 1 mod 120, so the last digits are always '1': For all four values to be integers it must be that p = 1 (mod 12). As p is prime, it must be that p = 1, 13, 37, 49, 61, 73, 97, or 109 (mod 120). In all but the first case either (p+3)/4 is even or one of the three expressions gives a value divisible by 5 (or both, and possibly the same expression).  Rick L. Shepherd, Aug 01 2009


LINKS

Vincenzo Librandi and Chai Wah Wu, Table of n, a(n) for n = 1..10001 (First 1000 terms from Vincenzo Librandi)


MATHEMATICA

lst={}; Do[p=Prime[n]; If[PrimeQ[(p+1)/2]&&PrimeQ[(p+2)/3]&&PrimeQ[(p+3)/ 4], AppendTo[lst, p]], {n, 2*9!}]; lst


PROG

(MAGMA) [p: p in PrimesInInterval(6, 1200000)  IsPrime((p+1) div 2) and IsPrime((p+2) div 3) and IsPrime((p+3) div 4)]; // Vincenzo Librandi, Apr 09 2013
(PARI) is(n)=n%120==1 && isprime(n) && isprime(n\2+1) && isprime(n\3+1) && isprime(n\4+1) \\ Charles R Greathouse IV, Nov 30 2016
(Python)
from sympy import prime, isprime
A163573_list = [4*q3 for q in (prime(i) for i in range(1, 10000)) if isprime(4*q3) and isprime(2*q1) and (not (4*q1) % 3) and isprime((4*q1)//3)] # Chai Wah Wu, Nov 30 2016


CROSSREFS

Cf. A163623, A163624, A163625, A036570, A278583, A278585.
Sequence in context: A205939 A278585 A288355 * A236882 A252325 A252322
Adjacent sequences: A163570 A163571 A163572 * A163574 A163575 A163576


KEYWORD

nonn,easy


AUTHOR

Vladimir Joseph Stephan Orlovsky, Jul 31 2009


EXTENSIONS

Slightly edited by R. J. Mathar, Aug 01 2009


STATUS

approved



