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A100564 Normal sequence of primes with a(1) = 3. 2
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 (list; graph; refs; listen; history; text; internal format)
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 the 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

LINKS

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Erick Wong, Computations on Normal Families of Primes.

Paul Erdős, On a problem of G. 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.

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

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Last modified April 20 06:19 EDT 2019. Contains 322294 sequences. (Running on oeis4.)