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A245694
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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).
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1
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9, 25, 49, 121, 289, 1849, 5041, 11881, 32761, 109561, 196249, 6466849, 9369721, 30283009, 52374169, 72880369, 1000267129
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OFFSET
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1,1
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COMMENTS
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(Stems from a post at mymathforum.com)
Outstanding questions include:
(1) Is every term a (prime) square?
(2) How many more terms are there?
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.
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.
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.)
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)
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LINKS
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EXAMPLE
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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.
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.
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MATHEMATICA
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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 *)
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PROG
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(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; );
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CROSSREFS
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KEYWORD
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more,tabf,nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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