A265684 - OEIS

%I #20 Jun 29 2019 02:24:48

%S 645,7957,11305,15841,25761,35333,126217,194221,212421,332949,464185,

%T 635401,656601,741751,934021,1193221,1357441,1459927,1620385,1690501,

%U 1969417,2704801,3911197,4154161,4209661,5095177,5284333,5351537,5758273,6189121,6212361,7820201,8134561,8209657

%N Sarrus numbers (A001567) that are the average of two consecutive primes.

%C Inspired by A265669.

%C Motivation was the form of differences between consecutive primes that generate this sequence. It seems that 12*k appears in differences most of the time. For the first 175 term of this sequence, the relevant proportion is 161/175.

%C Differences between corresponding consecutive primes are 4, 12, 12, 36, 4, 12, 12, 36, 4, 4, 24, 24, 4, 60, 24, 24, 24, 12, 12, 36, 12, 24, 12, 24, 36, 12, 12, 12, 12, 24, 4, 60, 24, 48, 36, 12, 24, 36, 24, 20, 12, 84, 36, 12, 24, 24, 12, 24, 36, 12, 12, 36, ...

%H Amiram Eldar, <a href="/A265684/b265684.txt">Table of n, a(n) for n = 1..10000</a>

%e 645 is a term because it is a Sarrus number and the average of the consecutive primes 643 and 647.

%e 7957 is a term because it is a Sarrus number and the average of the consecutive primes 7951 and 7963.

%t Select[Range[200000], CompositeQ[#] && PowerMod[2, (# - 1), #] == 1 && NextPrime[#] - # == # - NextPrime[#, -1] &] (* _Amiram Eldar_, Jun 28 2019 *)

%o (PARI) is(n)={Mod(2, n)^n==2 && !isprime(n)}

%o forcomposite(n=2, 1e7, if(is(n) && (nextprime(n)-n)==(n-precprime(n)), print1(n, ", ")))

%Y Intersection of A001567 and A024675.

%Y Cf. A265669.

%K nonn

%O 1,1

%A _Altug Alkan_, Dec 13 2015