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%I #29 Sep 08 2022 08:45:46
%S 12721,16921,19441,24481,49681,61561,104161,229321,255361,259681,
%T 266401,291721,298201,311041,331921,419401,423481,436801,446881,
%U 471241,525241,532801,539401,581521,600601,663601,704161,709921,783721,867001,904801
%N Primes p such that (p+1)/2, (p+2)/3 and (p+3)/4 are also primes.
%C Are all terms == 1 (mod 10)?
%C Subsequence of A005383, of A091180 and of A036570. - _R. J. Mathar_, Aug 01 2009
%C 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
%C All terms are of the form 120k+1: a(n)=120*A163625(n)+1. - _Zak Seidov_, Aug 01 2009
%C 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
%C {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
%H Vincenzo Librandi and Chai Wah Wu, <a href="/A163573/b163573.txt">Table of n, a(n) for n = 1..10001</a> (First 1000 terms from Vincenzo Librandi)
%t 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
%o (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
%o (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
%o (Python)
%o from sympy import prime, isprime
%o 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
%Y Cf. A005383, A091180, A036570, A050498, A163623, A163624, A163625, A278583, A278585, A339883.
%K nonn,easy
%O 1,1
%A _Vladimir Joseph Stephan Orlovsky_, Jul 31 2009
%E Slightly edited by _R. J. Mathar_, Aug 01 2009
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