A084599 -id:A084599 - cp's OEIS frontend

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

N. J. A. Sloane

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

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.

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)

Cf. A000945, A000946, A005266, A084599.

Essentially the same as A084598.

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

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

A005266 a(1)=3, b(n) = Product_{k=1..n} a(k), a(n+1) is the largest prime factor of (b(n)-1).

Original entry on oeis.org

3, 2, 5, 29, 79, 68729, 3739, 6221191, 157170297801581, 70724343608203457341903, 46316297682014731387158877659877, 78592684042614093322289223662773, 181891012640244955605725966274974474087, 547275580337664165337990140111772164867508038795347198579326533639132704344301831464707648235639448747816483406685904347568344407941

Offset: 1

N. J. A. Sloane

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

a(15) requires completing the factorization: 13 * 67 * 14479 * 167197 * 924769 * 2688244927 * 888838110930755119 * 14372541055015356634061816579965403 * C211 where C211=6609133306626483634448666494646737799624640616060730302142187545405582531010390290502001156883917023202671554510633460047901459959959325342475132778791495112937562941066523907603281586796876335607258627832127303. - Sean A. Irvine, Nov 10 2009

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.

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.
S. S. Wagstaff, Jr., Computing Euclid's primes, Bull. Institute Combin. Applications, 8 (1993), 23-32. (Annotated scanned copy)
Status of a(15) in factordb.com.

Cf. A000945, A000946, A005265.

Essentially the same as A084599.

with(numtheory):
a:= proc(n) option remember;
      `if`(n=1, 3, max(factorset(mul(a(i), i=1..n-1)-1)[]))
    end:
seq(a(n), n=1..10);  # Alois P. Heinz, Sep 26 2013

a[0] = 3; a[n_] := a[n] = Block[{p = Times @@ (a[#] & /@ Range[0, n - 1]) - 1}, FactorInteger[p][[-1, 1]]]; Array[a, 13] (* Robert G. Wilson v, Sep 26 2013 *)

a(14) from Joe K. Crump (joecr(AT)carolina.rr.com), Jul 26 2000

A084598 a(1) = 2, a(2) = 3; for n >= 2, a(n+1) is smallest prime factor of (Product_{k = 1..n} a(k)) - 1.

Original entry on oeis.org

2, 3, 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

Offset: 1

Marc LeBrun, May 31 2003

nonn

Like the Euclid-Mullin sequence A000945, but subtracting rather than adding 1 to the product.

The first 4 terms are identical with A084599. It starts diverging at a(5) because the factorization of 2*3*5*29 - 1 = 869 = 11*79 gives A084598(5)=11 and A084599(5)=79. - Hugo Pfoertner, Mar 31 2004

			a(4) = 29 since 2*3*5 = 30 and 29 is the smallest prime factor of 30-1.

Sean A. Irvine added terms 54 through 61, May 21 2006, giving Table of n, a(n) for n = 1..61
Dario Alpern, Factorization using the Elliptic Curve Method

Cf. A000945, A005266, A084598, A084599.

Essentially the same as A005265.

a={2,3}; q=2;
For[n=3,n<=19,n++,
    q=q*Last[a];
    AppendTo[a,Min[FactorInteger[q-1][[All,1]]]];
    ];
a (* Robert Price, Jul 17 2015 *)

More terms from Hugo Pfoertner, May 31 2003, using Dario Alpern's ECM

cp's OEIS Frontend

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

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A005266 a(1)=3, b(n) = Product_{k=1..n} a(k), a(n+1) is the largest prime factor of (b(n)-1).

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A084598 a(1) = 2, a(2) = 3; for n >= 2, a(n+1) is smallest prime factor of (Product_{k = 1..n} a(k)) - 1.

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