A005265 a(1)=3, b(n) = Product_{k=1..n} a(k), a(n+1) is the smallest prime factor of b(n)-1.

3, 2, 5, 29, 11, 7, 13, 37, 32222189, 131, 136013303998782209, 31, 197, 19, 157, 17, 8609, 1831129, 35977, 508326079288931, 487, 10253, 1390043, 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

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• nice
• hard

Comments

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

References

• R. K. Guy and R. Nowakowski, Discovering primes with Euclid, Delta (Waukesha), Vol. 5, pp. 49-63, 1975.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
• S. S. Wagstaff, Jr., Computing Euclid's primes, Bull. Institute Combin. Applications, 8 (1993), 23-32.

Links

• Sean A. Irvine, Table of n, a(n) for n = 1..61
• R. K. Guy and R. Nowakowski, Discovering primes with Euclid, Research Paper No. 260 (Nov 1974), The University of Calgary Department of Mathematics, Statistics and Computing Science.
• Des MacHale, Infinitely many proofs that there are infinitely many primes, Math. Gazette, 97 (No. 540, 2013), 495-498.
• OEIS wiki, OEIS sequences needing factors
• S. S. Wagstaff, Jr., Computing Euclid's primes, Bull. Institute Combin. Applications, 8 (1993), 23-32. (Annotated scanned copy)

Crossrefs

Cf. A000945, A000946, A005266, A084599.

Essentially the same as A084598.

Programs

Maple

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);
# Robert FERREOL, Sep 25 2019

PARI

lpf(n)=factor(n)[1,1] \\ better code exists, usually best to code in C and import
print1(A=3); for(n=2,99, a=lpf(A); print1(", "a); A*=a) \\ Charles R Greathouse IV, Apr 07 2020