A058000 -id:A058000 - cp's OEIS frontend

A036013 a(n) = smallest number > 1 such that a(1)a(2)...a(n) - 1 is prime (or 1).

2, 2, 2, 3, 2, 4, 2, 3, 6, 4, 5, 5, 5, 10, 4, 3, 5, 8, 22, 13, 14, 2, 5, 5, 2, 20, 9, 9, 24, 5, 26, 15, 14, 25, 25, 4, 9, 30, 9, 21, 12, 11, 10, 2, 40, 19, 8, 13, 11, 50, 3, 25, 25, 8, 5, 25, 46, 19, 47, 54, 9, 13, 14, 43, 4, 24, 28, 16, 33, 25, 152, 2, 11, 22, 6, 78, 87, 7, 10, 21

Offset: 1

Russell Easterly

Michael S. Branicky, Table of n, a(n) for n = 1..1600 (terms 1..300 from T. D. Noe)

Cf. A036012, A058000 (corresponding primes).

sng1[{t_,a_}]:=Module[{k=2},While[CompositeQ[t k-1],k++];{t*k,k}]; NestList[sng1,{2,2},80][[;;,2]] (* Harvey P. Dale, May 20 2023 *)

from sympy import isprime
from itertools import count
a,p = [2],2
for _ in range(100):
    q = next(filter(lambda x:isprime(x*p-1), count(2)))
    p = p*q
    a.append(q)
print(a) # Nicholas Stefan Georgescu, Mar 06 2023

More terms from Erich Friedman. More terms from Jud McCranie, Jan 26 2000.

A057999 a(n) is smallest prime such that a(n)-1 is a proper multiple of a(n-1)-1, with a(0) = 2.

Original entry on oeis.org

2, 3, 5, 13, 37, 73, 433, 1297, 2593, 10369, 72577, 508033, 1524097, 12192769, 73156609, 146313217, 438939649, 2633637889, 23702740993, 142216445953, 1991030243329, 37829574623233, 416125320855553, 1664501283422209, 6658005133688833, 126502097540087809, 506008390160351233

Offset: 0

Henry Bottomley, Nov 02 2000

nonn

Amiram Eldar, Table of n, a(n) for n = 0..576

Cf. A036012, A046966, A046972, A058000.

a[0] = 2; a[n_] := a[n] = Module[{k = 2*a[n - 1] - 2}, While[! PrimeQ[k + 1], k += (a[n - 1] - 1)]; k + 1]; Array[a, 25, 0] (* Amiram Eldar, Jan 19 2023 *)

a(n) = 1 + Product_{i=1..n} A036012(i) = a(n-1) * A036012(n) + 1 - A036012(n).

A346161 Prime numbers p such that the number of iterations of map A039634 required for p to reach 2 sets a new record.

Original entry on oeis.org

2, 3, 7, 23, 47, 191, 383, 1439, 2879, 11519, 23039, 261071, 1044287, 2949119, 31426559, 194224127, 1069493759, 8554807007, 31337349119, 68438456063, 136876912127, 547507648511, 8760122376191

Offset: 1

Ya-Ping Lu, Jul 08 2021

It seems that the record number of iterations for a(n) is n-1.

Alternatively, prime numbers p such that the number of odd primes encountered under iteration of A004526 sets a new record. - Martin Ehrenstein, Aug 16 2021

			Terms in this sequence are indicated in square brackets in the tree below for primes up to 97. Note that a(n) is the smallest prime of depth n-1.
                 1                 ___________[2]____________
                 |                /        /   |   \    \    \
         _______[3]__       ____ 5 _     17   19   37   67   73
        /        |   \     /     |  \     |    |
     _[7]_      13   97   11    41  43   71   79
    /  |  \      |       /  \    |
  29  31  61    53    [23]  89  83
   |                    |
  59                  [47]

Cf. A005384, A058000, A039634, A346063, A346163.

from sympy import nextprime, isprime
rec = -1; p1 = 1
while p1 < 1000000000:
    p = nextprime(p1); m = p; ct = 0
    while m > 2:
        if isprime(m): ct += 1
        m //= 2
    if ct > rec: print(p); rec = ct
    p1 = p

a(19)-a(23) from Martin Ehrenstein, Aug 22 2021

cp's OEIS Frontend

A036013 a(n) = smallest number > 1 such that a(1)a(2)...a(n) - 1 is prime (or 1).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Python

Extensions

A057999 a(n) is smallest prime such that a(n)-1 is a proper multiple of a(n-1)-1, with a(0) = 2.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A346161 Prime numbers p such that the number of iterations of map A039634 required for p to reach 2 sets a new record.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Python

Extensions