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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 (list; graph; refs; listen; history; text; internal format)
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*q-3 for q in (prime(i) for i in range(1, 10000)) if isprime(4*q-3) and isprime(2*q-1) and (not (4*q-1) % 3) and isprime((4*q-1)//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

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Last modified April 19 16:18 EDT 2019. Contains 322282 sequences. (Running on oeis4.)