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%I #32 Jun 09 2023 14:59:38
%S 9,25,49,121,289,1849,5041,11881,32761,109561,196249,6466849,9369721,
%T 30283009,52374169,72880369,1000267129
%N Table read by rows: n-th row lists odd composites c such that Q + c is prime, where Q is the n-th primorial and c < prime(n+1)*prime(n+2).
%C (Stems from a post at mymathforum.com)
%C Outstanding questions include:
%C (1) Is every term a (prime) square?
%C (2) How many more terms are there?
%C The primorial associated with each term hopefully is self-evident. I had to use the "ispseudoprime" function at the higher end. After the first dozen members, c > 0.99 times the defining limit prime(n+1)*prime(n+2). I believe there is a lone entry per primorial.
%C From _Jon E. Schoenfield_, Sep 13 2017: (Start)
%C There is at most one entry per primorial, and every term is the square of a prime. Proof: By definition, since c is a composite, it is the product of at least two primes (not necessarily distinct), and Q (the n-th primorial) is the product of all primes up through prime(n). If c is divisible by any prime <= prime(n), then so is Q + c, so Q + c cannot be prime. The only way that c can be a composite < prime(n+1)*prime(n+2) without being divisible by any prime <= prime(n) is to have c = prime(n+1)^2.
%C Thus, this sequence simply consists of numbers c of the form prime(k+1)^2 such that primorial(k) + c is prime. (Although k=0 would yield primorial(k) + prime(k+1)^2 = primorial(0) + prime(0+1)^2 = 1 + 2^2 = 5, which is prime, c would be 4, hence not an odd composite.)
%C Terms of the sequence correspond to the following positive values of k (which are the ones for which primorial(k) + prime(k+1)^2 is prime): 1, 2, 3, 4, 6, 13, 19, 28, 41, 66, 85, 371, 437, 726, 924, 1063, ... (End)
%H <a href="http://mymathforum.com/number-theory/45472-primorial-prime-function-theory.html"> Jul 27 2014 forum post</a>
%e Considering primorial 2*3*5*7 = 210, we seek all the odd composites c < 11*13 such that (210 + c) is prime. We begin 210+9, 210+15, 210+21, ..., 210+141.
%e Only one prime results: 331 = 210 + 121. 121 is thus a term in the sequence. A similar evaluation for primorial 2310 produces no terms for the sequence.
%t Table[With[{Q = Product[Prime@ i, {i, n}], s = Prime[n + 1] Prime[n + 2]}, Select[Range[4, s - 1], And[CompositeQ@ #, PrimeQ[Q + #]] &]], {n, 100}] // Flatten (* _Michael De Vlieger_, Sep 11 2017 *)
%o (PARI) genit()=n=1;while(n<100,Q=prod(x=1,n,prime(x));r=prime(n+1);s=prime(n+2);Z=r*s;c=9;while(c<Z,if(isprime(Q+c)&& !isprime(c), print1(c,", ");); c=c+2 ); n+=1;);
%Y Cf. A002110 (primorials), A038773.
%K more,tabf,nonn
%O 1,1
%A _Bill McEachen_, Jul 29 2014
%E a(16)-a(17) from _Michael S. Branicky_, Jun 09 2023
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