A100564 Normal sequence of primes with a(1) = 3.

3, 5, 17, 23, 29, 53, 83, 89, 113, 149, 173, 197, 257, 263, 269, 293, 317, 353, 359, 383, 389, 419, 449, 467, 479, 503, 509, 557, 563, 569, 593, 617, 653, 659, 677, 683, 773, 797, 809, 827, 857, 863, 887, 947, 977, 983, 1049, 1097, 1109, 1217, 1223, 1229, 1283

Offset: 1

Author

Franz Vrabec, Nov 28 2004

Keywords

• nonn

Comments

A sequence {a(1), a(2), a(3), ... } is called a "normal sequence of primes" if a(1) is prime and if for every n > 1 a(n) is the smallest prime greater than a(n-1) such that the primes a(1), a(2), ..., a(n-1) are not divisors of a(n)-1.

The existence of the primes a(n) is guaranteed by Dirichlet's theorem on primes in arithmetic progressions.

Erdős proved that the number of terms in this sequence which do not exceed x is ~ (1 + o(1)) x/(logx loglogx), and that the sum of their reciprocals diverges. - Amiram Eldar, May 15 2017

The sum of reciprocals diverges slowly: the sum exceeds 1 only after adding 159989 terms: 1/3 + 1/5 + ... + 1/11321273 = 1.0000000628... - Amiram Eldar, May 28 2017

The product a(1)*a(2)*...*a(n) gives a cyclic number (A003277) with n factors. For the smallest cyclic number with n prime factors, see A264907. - Jeppe Stig Nielsen, May 22 2021

Examples

			a(2) = 5 because a(1) = 3 is not a divisor of 4 = 5 - 1.
a(3) = 17 because a(1) = 3 is a divisor of 6 and 12 (so 7 and 13 are not possible for a(3)); a(2) = 5 is a divisor of 10 (so 11 is not possible for a(3)), but a(1) = 3 and a(2) = 5 both not divisors of 16 = 17 - 1.
		

References

• S. W. Golomb, Problems in the Distribution of the Prime Numbers, Ph.D. dissertation, Dept. of Mathematics, Harvard University, May 1956. See page 8.

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..10000
• Paul Erdős, On a problem of G. [sic] Golomb, Journal of the Australian Mathematical Society, Vol 2, Issue 1 (1961), pp. 1-8.
• Solomon W. Golomb, Sets of primes with intermediate density, Mathematica Scandinavica, Vol. 3 (1956), pp. 264-274.
• H. G. Meijer, Sets of Primes with Intermediate Density, Mathematica Scandinavica, Vol. 34 (1974), pp. 37-43.
• Erick Wong, Computations on Normal Families of Primes.

Programs

Maple

a:= proc(n) option remember; local p;
      if n=1 then 3
    else p:= a(n-1);
         do p:= nextprime(p);
            if {} = numtheory[factorset](p-1) intersect
               {seq(a(i), i=1..n-1)} then return p fi
         od
      fi
    end:
seq(a(n), n=1..70);  # Alois P. Heinz, Feb 05 2017

Mathematica

a[1] = 3; a[n_] := a[n] = Block[{k = PrimePi[a[n - 1]] + 1, t = Table[a[i], {i, n - 1}]}, While[ Union[ Mod[ Prime[k] - 1, t]][[1]] == 0, k++ ]; Prime[k]]; Table[ a[n], {n, 53}] (* Robert G. Wilson v, Dec 04 2004 *)

Extensions

More terms from Robert G. Wilson v, Dec 04 2004