cp's OEIS Frontend

A230504 Smallest prime in r(k) = r(k-1) + gcd(k,r(k-1)) with r(1) = n.

%I A230504 #14 Mar 15 2023 11:45:59
%S A230504 2,2,3,19,5,19,7,11,11,17,11,17,13,17,17,23,17,23,19,23,23,29,23,29,
%T A230504 29,29,29,37,29,37,31,37,37,53,53,53,37,41,41,47,41,47,43,47,47,53,47,
%U A230504 53,53,53,53,59,53,59,59,59,59,67,59,67,61,67,67,79,79,79
%N A230504 Smallest prime in r(k) = r(k-1) + gcd(k,r(k-1)) with r(1) = n.
%C A230504 a(p) = p, p prime;
%C A230504 a(2*n-1) = A060264(n-1).
%H A230504 Reinhard Zumkeller, <a href="/A230504/b230504.txt">Table of n, a(n) for n = 1..10000</a>
%H A230504 Eric S. Rowland, <a href="https://arxiv.org/abs/0710.3217">A simple prime-generating recurrence</a>, arXiv:0710.3217 [math.NT], 2007-2008.
%e A230504 n = 1 -> 1 + GCD(1,2) = 1+1 = 2 = prime(1) = a(1);
%e A230504 n = 2 = prime(1) = a(2);
%e A230504 n = 3 = prime(2) = a(3);
%e A230504 n = 4 -> 4+GCD(4,2) = 4+2 = 6 -> 6+GCD(6,3) = 6+3 = 9 -> 9+GCD(9,4) = 9+1 = 10 -> 10+GCD(10,5) = 10+5 = 15 -> 15+GCD(15,6) = 15+3 = 18 -> 18+GCD(18,7) = 18+1 = 19 = prime(8) = a(4) = A084662(7);
%e A230504 n = 5 = prime(3) = a(5) = A134736(1);
%e A230504 n = 6 -> 6+GCD(6,2) = 6+2 = 8 -> 8+GCD(8,3) = 8+1 = 9 -> 9+GCD(9,4) = 9+1 = 10 -> 10+GCD(10,5) = 10+5 = 15 -> 15+GCD(15,6) = 15+3 = 18 -> 18+GCD(18,7) = 18+1 = 19 = prime(8) = a(6);
%e A230504 n = 7 = prime(4) = a(7) = A106108(1);
%e A230504 n = 8 -> 8+GCD(8,2) = 8+2 = 10 -> 10+GCD(10,3) = 10+1 = 11 = prime(5) = a(8) = A084663(3);
%e A230504 n = 9 -> 9+GCD(9,2) = 9+2 = 11 = prime(5) = a(9);
%e A230504 n = 10 -> 10+GCD(10,2) = 10+2 = 12 -> 12+GCD(12,3) = 12+3 = 15 -> 15+GCD(15,4) = 15+1 = 16 -> 16+GCD(16,5) = 16+1 = 17 = prime(7) = a(10);
%e A230504 n = 11 = prime(5) = a(11);
%e A230504 n = 12 -> 12+GCD(12,2) = 12+2 = 14 -> 14+GCD(14,3) = 14+1 = 15 -> 15+GCD(15,4) = 15+1 = 16 -> 16+GCD(16,5) = 16+1 = 17 = prime(7) = a(10).
%t A230504 a[n_] := Module[{r}, If[PrimeQ[n], n, r[1]=n; r[k_] := r[k] = r[k-1] + GCD[k, r[k-1]]; For[k=1, True, k++, If[PrimeQ[r[k]], Return[r[k]]]]]];
%t A230504 Array[a, 66] (* _Jean-François Alcover_, Dec 03 2018 *)
%o A230504 (Haskell)
%o A230504 a230504 n = head $ filter ((== 1) . a010051') rs where
%o A230504                    rs = n : zipWith (+) rs (zipWith gcd rs [2..])
%o A230504 (Python)
%o A230504 from math import gcd
%o A230504 from itertools import count, accumulate
%o A230504 from sympy import isprime
%o A230504 def A230504(n): return next(filter(isprime,accumulate(count(2),lambda x,y:x+gcd(x,y),initial=n))) # _Chai Wah Wu_, Mar 15 2023
%Y A230504 Cf. A084662, A134736, A106108, A084663.
%K A230504 nonn
%O A230504 1,1
%A A230504 _Reinhard Zumkeller_, Nov 15 2013