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

1,1

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

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.

Example

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.

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 *)

Crossrefs

Sequence in context: A069687 A079017 A211440 * A231232 A154608 A024862

Adjacent sequences: A100561 A100562 A100563 * A100565 A100566 A100567

Keyword

nonn

Author

Franz Vrabec, Nov 28 2004

Extensions

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

Status

approved