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A018239 Primorial primes: primes of the form 1 + product of first k primes, for some k. 24
2, 3, 7, 31, 211, 2311, 200560490131, 1719620105458406433483340568317543019584575635895742560438771105058321655238562613083979651479555788009994557822024565226932906295208262756822275663694111 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Prime numbers that are the sum of two primorial numbers. - Juri-Stepan Gerasimov, Nov 08 2010

REFERENCES

F. Iacobescu, Smarandache Partition Type and Other Sequences, Bull. Pure Appl. Sciences, Vol. 16E, No. 2 (1997), pp. 237-240.

LINKS

Table of n, a(n) for n=1..8.

M. Fleuren, Factors and primes of Smarandache sequences.

M. Fleuren, Smarandache Prime Product Sequence.

R. Mestrovic, Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 BC--2012) and another new proof, arXiv preprint arXiv:1202.3670, 2012 - From N. J. A. Sloane, Jun 13 2012

F. Smarandache, Sequences of Numbers Involved in Unsolved Problems.

R. Ondrejka, Primorial-Plus-One Primes

FORMULA

a(n) = 1 + A002110(A014545(n)), where A002110(k) is the product of the first k primes. - M. F. Hasler, Jun 23 2019

EXAMPLE

From M. F. Hasler, Jun 23 2019:

a(1) = 2 = 1 + product of the first 0 primes (i.e., the empty product = 1).

a(2) = 3 = 1 + 2 = 1 + product of the first prime (= 2).

a(3) = 7 = 1 + 2*3 = 1 + product of the first two primes.

a(4) = 31 = 1 + 2*3*5 = 1 + product of the first three primes.

a(5) = 211 = 1 + 2*3*5*7 = 1 + product of the first four primes.

a(6) = 2311 = 1 + 2*3*5*7*11 = 1 + product of the first five primes.

Then the product of the first 6, 7, ..., 9 or 10 primes does not yield a primorial prime, the next one is:

a(7) = 200560490131 = 1 + 2*3*5*7*11*13*17*19*23*29*31 = 1 + product of the first eleven primes,

and so on. See A014545 = (0, 1, 2, 3, 4, 5, 11, 75, 171, 172, ...) for the k's that yield a term. (End)

MATHEMATICA

Select[FoldList[Times, 1, Prime[Range[200]]] + 1, PrimeQ]. (* Loreno Heer (helohe(AT)bluewin.ch), Jun 29 2004 *)

PROG

(PARI) P=1; print1(2); forprime(p=2, 1e6, if(isprime(1+P*=p), print1(", "P+1))) \\ Charles R Greathouse IV, Apr 28 2015

CROSSREFS

Primes in A006862 (primorials plus 1).

A005234 and A014545 (which are the main entries for this sequence) give more terms.

Cf. A002110.

Sequence in context: A241196 A073918 A096350 * A066279 A161471 A057677

Adjacent sequences:  A018236 A018237 A018238 * A018240 A018241 A018242

KEYWORD

nonn,nice

AUTHOR

Murray R. Bremner

EXTENSIONS

Edited by N. J. A. Sloane at the suggestion of Vladeta Jovovic, Jun 18 2007

Name edited by M. F. Hasler, Jun 23 2019

STATUS

approved

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Last modified October 19 04:40 EDT 2019. Contains 328211 sequences. (Running on oeis4.)