1,1

A sequence of primes generated recursively as follows: a(n+1)=q(n)*a(n)+1, where q=q(n) is the smallest (even) number such that a(n+1)=q*a(n)+1 is prime and the initial value a(1)=2. q(n)=[a(n+1)-1]/a(n) is the satellite "almost-quotient-sequence".

It has been established in the Murthy reference that for every prime p there exists at least one prime of the form k*p +1. Hence the sequence is infinite. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Mar 02 2002

Amarnath Murthy, On the divisors of Smarandache Unary Sequence. Smarandache Notions Journal, Vol. 11, No. 1-2-3, Spring 2000.

a(n+1)=a(n)*q(n)+1, q(n)=Min{q|qa(n)+1 is prime}

a(5) = 47 and a(6) = 283 = 6*47 +1 is the smallest such prime.

The initial values are safe primes: (2), 5, 11, 23, 47, ... To obtain qa(i)+1 primes q>2 multiplier arises and such a q always exists in arithmetic progression of difference a(i). E.g. {1699*2k+1} gives the first prime when 2k=12. So a(7)=1699 is followed by 1699*12+1=20389=a(8). The emergent "quotient-sequence" is {2, 2, 2, 2, 6, 6, 12, 12, 30, 28, 2, 12, 60, 16, 48, 54, 72, 28, 180, 102, 4, 12, 106, 50, 18}. A059411 is an infinite sequence of primes increasing at least with exponential speed.

i := 0:a[0] := 2:while(i<40) do k := 2:while(not isprime(k*a[i]+1)) do k := k+1; end do; i := i+1; a[i] := k*a[i-1]+1; end do:q := seq(a[i], i=0..39);

Cf. A061092.

Sequence in context: A084403 A055011 A007505 this_sequence A126017 A034468 A130668

Adjacent sequences: A059408 A059409 A059410 this_sequence A059412 A059413 A059414

nonn

Labos E. (labos(AT)ana.sote.hu), Jan 30 2001