cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

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

Original entry on oeis.org

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

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Author

N. J. A. Sloane

Keywords

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

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

A084599 a(1) = 2, a(2) = 3; for n >= 2, a(n+1) is largest prime factor of (Product_{k=1..n} a(k)) - 1.

Original entry on oeis.org

2, 3, 5, 29, 79, 68729, 3739, 6221191, 157170297801581, 70724343608203457341903, 46316297682014731387158877659877, 78592684042614093322289223662773, 181891012640244955605725966274974474087, 547275580337664165337990140111772164867508038795347198579326533639132704344301831464707648235639448747816483406685904347568344407941
Offset: 1

Views

Author

Marc LeBrun, May 31 2003

Keywords

Comments

Like the Euclid-Mullin sequence A000946, but subtracting rather than adding 1 to the product.

Examples

			a(4)=29 since 2*3*5=30 and 29 is the largest prime factor of 30-1
a(5)=79 since 2*3*5*29=870 and 79 is the largest prime factor of 870-1=869=11*79.

Links

Crossrefs

Cf. A000946, A005265, A084598.
Essentially the same as A005266.

Extensions

More terms from Hugo Pfoertner, May 31 2003, using Dario Alpern's ECM.
The next term a(15) is not known. It requires the factorization of the 245-digit composite number which remains after eliminating 7 smaller factors.

A359505 a(1)=2, a(2)=3, and for n >= 3, a(n) is calculated by considering in ascending order all products P of (distinct) terms from {a(1..n-1)} until finding one where P-1 has a prime factor not in {a(1..n-1)}, in which case a(n) is the smallest such prime factor.

Original entry on oeis.org

2, 3, 5, 7, 13, 29, 17, 11, 19, 37, 41, 23, 73, 43, 47, 31, 61, 101, 109, 113, 59, 137, 71, 173, 181, 97, 193, 67, 53, 79, 157, 107, 229, 127, 257, 281, 149, 151, 103, 313, 317, 163, 167, 83, 353, 89, 373, 389, 197, 131, 199, 397, 401, 409, 139, 277, 433, 223
Offset: 1

Views

Author

Joel Brennan, Jan 24 2023

Keywords

Comments

A new prime is always found since at worst P can be the product of all primes {a(1..n-1)} and then P-1 certainly has a prime factor not among a(1..n-1).
Taking products P in ascending order generally results in smaller quantities to consider than always taking the product of all primes as done in A084598.
Conjecture: P-1 has at most one prime factor not already in the sequence, so the requirement of taking "the smallest such" is unnecessary (verified up to n=10000).

Examples

			For n=3, the primes so far are 2 and 3 but products P=2 or P=3 have P-1 = 1 or 2 which have no new prime factor. Product P = 2*3 = 6 has P-1 = 5 which is a new prime so a(3) = 5.
For n=4, the smallest product P with a new prime in P-1 is P = 3*5 = 15 for which P-1 = 14 and a(4) = 7 is its smallest new prime factor.

Links

Crossrefs

Cf. A000945, A000946, A359504
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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