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A163573 Primes p such that (p+1)/2, (p+2)/3 and (p+3)/4 are also primes. 12
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 all terms == 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
Sequence in context: A205939 A278585 A288355 * A236882 A252325 A252322
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
Slightly edited by R. J. Mathar, Aug 01 2009
STATUS
approved

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Last modified May 30 17:56 EDT 2024. Contains 372971 sequences. (Running on oeis4.)