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Primes which are the sum of two consecutive composite numbers.
23

%I #15 Jan 12 2019 16:06:42

%S 17,19,29,31,41,43,53,67,71,79,89,97,101,103,109,113,127,131,137,139,

%T 149,151,163,173,181,191,197,199,211,223,229,233,239,241,251,257,269,

%U 271,281,283,293,307,311,317,331,337,349,353,367,373,379,389,401,409

%N Primes which are the sum of two consecutive composite numbers.

%C For the smaller of the consecutive composite pair (p-+1)/2, see A096784

%C This sequence also contains exactly those odd primes p where neither p-1 nor p+1 is the product of exactly 2 (not necessarily distinct) primes. - _Leroy Quet_, Sep 09 2008

%C 5 together with the prime numbers A060254=(5,17,19,29,31,41,43,53,..)=primes which are the sum of two consecutive nonprime numbers. - _Juri-Stepan Gerasimov_, Aug 30 2009

%C Conjecture: a(n) ~ n log n. - _Charles R Greathouse IV_, Apr 29 2015

%H Charles R Greathouse IV, <a href="/A060254/b060254.txt">Table of n, a(n) for n = 1..10000</a>

%e The prime 19 is an entry since it is the sum of 9=3^2 and 10=2*5.

%t 2Select[ Range[210], PrimeQ[ # ] == PrimeQ[ # + 1] == False && PrimeQ[2# + 1] == True &] + 1

%t Select[Total/@Partition[Select[Range[300],CompositeQ],2,1],PrimeQ] (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale_, Jan 12 2019 *)

%o (PARI) is(n)=!isprime(n\2) && !isprime(n\2+1) && isprime(n) \\ _Charles R Greathouse IV_, Apr 29 2015

%Y Cf. A096783, A096784, A096785, A096786, A096787, A096788, A096677.

%K nonn

%O 1,1

%A _Robert G. Wilson v_, Mar 22 2001