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A163573 Primes p such that (p+1)/2, (p+2)/3 and (p+3)/4 are also primes. 10
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

{6*a(n))_{n >= 1), is a subsequence of A050498. Proof: with p = a(n) the arithmetic progression with four terms of difference 6 and constant value of Euler's phi, namely 2*(p-1), is  6*(p, 2*(p+1)/2, 3*(p + 2)/3, 4*(p+3)/4). Use phi(n, prime) = phi(n)*(prime-1) if gcd(n, prime) = 1. Here n = 6, 12, 18, 24 and prime > 3 for p >= a(1). Thanks to Hugo Pfoertner for a link to the present sequence in connection with A339883. - Wolfdieter Lang, Jan 11 2021

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. A005383, A091180, A036570, A050498, A163623, A163624, A163625, A278583, A278585, A339883.

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 February 28 06:33 EST 2021. Contains 341695 sequences. (Running on oeis4.)