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

%I A100564 #45 Aug 06 2024 09:39:17
%S A100564 3,5,17,23,29,53,83,89,113,149,173,197,257,263,269,293,317,353,359,
%T A100564 383,389,419,449,467,479,503,509,557,563,569,593,617,653,659,677,683,
%U A100564 773,797,809,827,857,863,887,947,977,983,1049,1097,1109,1217,1223,1229,1283
%N A100564 Normal sequence of primes with a(1) = 3.
%C A100564 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.
%C A100564 The existence of the primes a(n) is guaranteed by Dirichlet's theorem on primes in arithmetic progressions.
%C A100564 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
%C A100564 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
%C A100564 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
%D A100564 S. W. Golomb, Problems in the Distribution of the Prime Numbers, Ph.D. dissertation, Dept. of Mathematics, Harvard University, May 1956. See page 8.
%H A100564 Alois P. Heinz, <a href="/A100564/b100564.txt">Table of n, a(n) for n = 1..10000</a>
%H A100564 Paul Erdős, <a href="https://doi.org/10.1017/S144678870002632X">On a problem of G. [sic] Golomb</a>, Journal of the Australian Mathematical Society, Vol 2, Issue 1 (1961), pp. 1-8.
%H A100564 Solomon W. Golomb, <a href="https://eudml.org/doc/165595">Sets of primes with intermediate density</a>, Mathematica Scandinavica, Vol. 3 (1956), pp. 264-274.
%H A100564 H. G. Meijer, <a href="https://eudml.org/doc/166341">Sets of Primes with Intermediate Density</a>, Mathematica Scandinavica, Vol. 34 (1974), pp. 37-43.
%H A100564 Erick Wong, <a href="https://citeseerx.ist.psu.edu/pdf/d0e8aa41e7b707845360ace88a4cfa4666cc2cc0">Computations on Normal Families of Primes</a>.
%e A100564 a(2) = 5 because a(1) = 3 is not a divisor of 4 = 5 - 1.
%e A100564 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.
%p A100564 a:= proc(n) option remember; local p;
%p A100564       if n=1 then 3
%p A100564     else p:= a(n-1);
%p A100564          do p:= nextprime(p);
%p A100564             if {} = numtheory[factorset](p-1) intersect
%p A100564                {seq(a(i), i=1..n-1)} then return p fi
%p A100564          od
%p A100564       fi
%p A100564     end:
%p A100564 seq(a(n), n=1..70);  # _Alois P. Heinz_, Feb 05 2017
%t A100564 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 *)
%K A100564 nonn
%O A100564 1,1
%A A100564 _Franz Vrabec_, Nov 28 2004
%E A100564 More terms from _Robert G. Wilson v_, Dec 04 2004