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%I #49 Jan 13 2024 10:44:29
%S 2,3,7,31,211,2311,200560490131
%N Primorial primes: primes of the form 1 + product of first k primes, for some k.
%C Prime numbers that are the sum of two primorial numbers. - _Juri-Stepan Gerasimov_, Nov 08 2010
%D F. Iacobescu, Smarandache Partition Type and Other Sequences, Bull. Pure Appl. Sciences, Vol. 16E, No. 2 (1997), pp. 237-240.
%H Michel Marcus, <a href="/A018239/b018239.txt">Table of n, a(n) for n = 1..10</a>
%H M. Fleuren, <a href="http://www.gallup.unm.edu/~smarandache/michafleuren.htm">Factors and primes of Smarandache sequences</a>.
%H M. Fleuren, <a href="http://www.gallup.unm.edu/~smarandache/SmPrimProd.txt">Smarandache Prime Product Sequence</a>.
%H Ernest G. Hibbs, <a href="https://www.proquest.com/openview/4012f0286b785cd732c78eb0fc6fce80">Component Interactions of the Prime Numbers</a>, Ph. D. Thesis, Capitol Technology Univ. (2022), see p. 33.
%H R. Mestrovic, <a href="http://arxiv.org/abs/1202.3670">Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 BC--2012) and another new proof</a>, arXiv preprint arXiv:1202.3670 [math.HO], 2012-2023. - From N. J. A. Sloane, Jun 13 2012
%H F. Smarandache, <a href="http://www.gallup.unm.edu/~smarandache/Sequences-book.pdf">Sequences of Numbers Involved in Unsolved Problems</a>.
%H R. Ondrejka, <a href="https://t5k.org/lists/top_ten/topten.pdf">Primorial-Plus-One Primes</a>
%F a(n) = 1 + A002110(A014545(n)), where A002110(k) is the product of the first k primes. - _M. F. Hasler_, Jun 23 2019
%e From _M. F. Hasler_, Jun 23 2019:
%e a(1) = 2 = 1 + product of the first 0 primes (i.e., the empty product = 1).
%e a(2) = 3 = 1 + 2 = 1 + product of the first prime (= 2).
%e a(3) = 7 = 1 + 2*3 = 1 + product of the first two primes.
%e a(4) = 31 = 1 + 2*3*5 = 1 + product of the first three primes.
%e a(5) = 211 = 1 + 2*3*5*7 = 1 + product of the first four primes.
%e a(6) = 2311 = 1 + 2*3*5*7*11 = 1 + product of the first five primes.
%e Then the product of the first 6, 7, ..., 9 or 10 primes does not yield a primorial prime, the next one is:
%e a(7) = 200560490131 = 1 + 2*3*5*7*11*13*17*19*23*29*31 = 1 + product of the first eleven primes,
%e and so on. See A014545 = (0, 1, 2, 3, 4, 5, 11, 75, 171, 172, ...) for the k's that yield a term. (End)
%t Select[FoldList[Times, 1, Prime[Range[200]]] + 1, PrimeQ] (* Loreno Heer (helohe(AT)bluewin.ch), Jun 29 2004 *)
%o (PARI) P=1;print1(2); forprime(p=2,1e6, if(isprime(1+P*=p), print1(", "P+1))) \\ _Charles R Greathouse IV_, Apr 28 2015
%Y Primes in A006862 (primorials plus 1).
%Y A005234 and A014545 (which are the main entries for this sequence) give more terms.
%Y Cf. A002110.
%K nonn,nice
%O 1,1
%A _Murray R. Bremner_
%E Edited by _N. J. A. Sloane_ at the suggestion of _Vladeta Jovovic_, Jun 18 2007
%E Name edited by _M. F. Hasler_, Jun 23 2019
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