A005265 - OEIS

%I M2246 #43 May 30 2026 16:39:47

%S 3,2,5,29,11,7,13,37,32222189,131,136013303998782209,31,197,19,157,17,

%T 8609,1831129,35977,508326079288931,487,10253,1390043,

%U 18122659735201507243,25319167,9512386441,85577,1031,3650460767,107,41,811,15787,89,68168743,4583,239,1283,443,902404933,64775657,2753,23,149287,149749,7895159,79,43,1409,184274081,47,569,63843643

%N a(1)=3, b(n) = Product_{k=1..n} a(k), a(n+1) is the smallest prime factor of b(n)-1.

%C Suggested by Euclid's proof that there are infinitely many primes.

%D R. K. Guy and R. Nowakowski, Discovering primes with Euclid, Delta (Waukesha), Vol. 5, pp. 49-63, 1975.

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%D S. S. Wagstaff, Jr., Computing Euclid's primes, Bull. Institute Combin. Applications, 8 (1993), 23-32.

%H Sean A. Irvine, <a href="/A005265/b005265.txt">Table of n, a(n) for n = 1..61</a>

%H R. K. Guy and R. Nowakowski, <a href="/A000945/a000945_5.pdf">Discovering primes with Euclid</a>, Research Paper No. 260 (Nov 1974), The University of Calgary Department of Mathematics, Statistics and Computing Science.

%H Des MacHale, <a href="https://doi.org/10.2307/3621650">Infinitely many proofs that there are infinitely many primes</a>, Math. Gazette, 97 (No. 540, 2013), 495-498.

%H OEIS wiki, <a href="https://oeis.org/wiki/OEIS_sequences_needing_factors">OEIS sequences needing factors</a>

%H S. S. Wagstaff, Jr., <a href="/A000945/a000945_4.pdf">Computing Euclid's primes</a>, Bull. Institute Combin. Applications, 8 (1993), 23-32. (Annotated scanned copy)

%p a :=n-> if n = 1 then 3 else numtheory:-divisors(mul(a(i),i = 1 .. n-1)-1)[2] fi:seq(a(n), n=1..15);

%p # _Robert FERREOL_, Sep 25 2019

%o (PARI) lpf(n)=factor(n)[1,1] \\ better code exists, usually best to code in C and import

%o print1(A=3); for(n=2,99, a=lpf(A); print1(", "a); A*=a) \\ _Charles R Greathouse IV_, Apr 07 2020

%Y Cf. A000945, A000946, A005266, A084599.

%Y Essentially the same as A084598.

%K nonn,nice,hard

%O 1,1

%A _N. J. A. Sloane_