A083771 Rearrangement of primes such that every partial product + 1 is a prime.

2, 3, 5, 7, 11, 19, 29, 13, 59, 37, 31, 47, 67, 53, 41, 97, 73, 113, 103, 43, 71, 233, 61, 151, 109, 101, 251, 107, 587, 79, 223, 167, 311, 239, 137, 139, 359, 181, 257, 337, 163, 173, 881, 563, 149, 409, 157, 179, 293, 127, 331, 191, 269, 317, 83, 277, 23, 821, 373, 271, 283, 461, 569, 853, 487, 433, 647, 953, 383, 199, 367, 1231, 397, 307, 457, 691, 523, 463, 1061, 281, 787, 421, 197, 857, 1103, 347, 631, 499, 991, 643, 769, 983, 607, 811, 449, 1223, 733, 1327, 683, 1021

Offset: 1

Author

Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), May 06 2003

Keywords

• nonn

Comments

Though initial terms match it is different from A039726, in that a smaller prime may appear later.

Some of the larger entries may only correspond to probable primes.

A158076 suggests that the numbers in this sequence can be generated quite easily/quickly. Perhaps this sequence is a fast method to generate large probable primes. [Dmitry Kamenetsky, Mar 12 2009]

Records: 2, 3, 5, 7, 11, 19, 29, 59, 67, 97, 113, 233, 251, 587, 881, 953, 1231, 1327, 1553, 1657, 2383, 3251, 3769, 6737, 6947, 7103, 7879, 8263, 10159, 11369, 22003, ..., . - Robert G. Wilson v, Jul 20 2017

Position of the n_th prime: 1, 2, 3, 4, 5, 8, 472, 6, 57, 7, 11, 10, 15, 20, 12, 14, 9, 23, 13, 21, 17, 30, 55, 478, 16, 26, 19, 28, 25, 18, 50, 345, 35, 36, 45, 24, ..., . - Robert G. Wilson v, Jul 20 2017

Examples

			The n-th term is the smallest prime that is not already in the sequence, such that one plus the product of the first n terms is prime. [_Dmitry Kamenetsky_, Mar 12 2009]
		

Links

• Robert G. Wilson v, Table of n, a(n) for n = 1..700 (first 100 terms from Amarnath Murthy and Meenakshi Srikanth)

Crossrefs

Cf. A005235, A039726, A083772.

Cf. number of primality tests required for each term in this sequence is in A158076. [Dmitry Kamenetsky, Mar 12 2009]

Programs

Mathematica

f[s_List] := Block[{p = Times @@ s, q = 2}, While[ MemberQ[s, q] || !PrimeQ[p*q + 1], q = NextPrime@ q]; Append[s, q]]; Nest[f, {2}, 63] (* Robert G. Wilson v, Jul 20 2017 *)

PARI

{ terms=100; a=A083772=vector(terms); a[1]=2; tmp=1; A083772[1]=3; for(k=2,terms, tmp=tmp*a[k-1]; p=1; while(1, until(isprime(p), p=p+2); for(m=1,k-1, if(p==a[m], break, if(m==k-1, if(isprime(tmp*p+1), a[k]=p; A083772[k]=tmp*p+1; print1(a[k],","); break(2))))))); a }

Extensions

More terms from Rick L. Shepherd, Mar 18 2004