A036013 -id:A036013 - cp's OEIS frontend

A036012 a(n) = smallest number > 1 such that a(1)a(2)...a(n) + 1 is prime.

2, 2, 3, 3, 2, 6, 3, 2, 4, 7, 7, 3, 8, 6, 2, 3, 6, 9, 6, 14, 19, 11, 4, 4, 19, 4, 13, 3, 10, 13, 15, 4, 11, 9, 2, 5, 26, 19, 52, 21, 20, 63, 4, 19, 17, 6, 29, 19, 3, 5, 51, 11, 14, 15, 7, 12, 44, 34, 7, 21, 32, 3, 22, 10, 19, 19, 7, 20, 4, 22, 4, 17, 35, 47, 40, 14, 5, 14, 36, 39, 16

Offset: 1

Russell Easterly

Except for the first term, same as A084401. - David Wasserman, Dec 22 2004

Michael S. Branicky, Table of n, a(n) for n = 1..1000 (terms 1..300 from T. D. Noe)

Equals A084716(n+1)/A084716(n).

Cf. A036013, A084716, A084717, A084718.

n := 1: while true do j := 2: while not isprime(j*n+1) do j := j+1: od: print(j): n := n*j: od:

a[1] = 2; a[n_] := a[n] = Catch[For[an = 2, True, an++, If[PrimeQ[Product[a[k], {k, 1, n - 1}]*an + 1], Throw[an]]]]; Table[a[n], {n, 1, 81}] (* Jean-François Alcover, Nov 27 2012 *)
nxt[{t_,n_}]:=Module[{k=2},While[!PrimeQ[t*k+1],k++];{t*k,k}]; NestList[ nxt,{2,2},80][[All,2]] (* Harvey P. Dale, Oct 03 2020 *)

from gmpy2 import is_prime
from itertools import count, islice
def agen(): # generator of terms
    p = 1
    while True:
        an = next(k for k in count(2) if (t:=p*k+1) == 1 or is_prime(t))
        p *= an
        yield an
print(list(islice(agen(), 81))) # Michael S. Branicky, Jan 20 2024

Conjecture: a(n) = O(n). - Thomas Ordowski, Aug 08 2017

More terms from Erich Friedman
More terms from Jud McCranie, Jan 26 2000
Description corrected by Len Smiley

A083769 a(1)=2; for n >= 2, a(n) = smallest even number such that a(1)a(2)...*a(n) + 1 is prime.

Original entry on oeis.org

2, 6, 8, 12, 16, 10, 4, 30, 26, 22, 24, 14, 50, 42, 18, 64, 46, 60, 32, 36, 20, 34, 28, 108, 48, 44, 68, 282, 90, 54, 76, 62, 180, 66, 132, 86, 74, 38, 58, 106, 120, 52, 244, 94, 100, 82, 138, 156, 98, 72, 172, 150, 248, 154, 166, 114, 162, 126, 124, 208, 222, 324, 212

Offset: 1

Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), May 06 2003

nonn

Is this a permutation of the even numbers?

For any even positive integers a_1, a_2, ..., a_n, there are infinitely many even positive integers t such that a_1 a_2 ... a_n t + 1 is prime: this follows from Dirichlet's theorem on primes in arithmetic progressions. As far as I know there is no guarantee that the sequence defined here leads to a permutation of the even numbers, i.e. there might be some even integer that never appears in the sequence. However, if the partial products a_1 ... a_n grow like 2^n n!, heuristically the probability of a_1 ... a_n t + 1 being prime is on the order of 1/log(a_1 ... a_n) ~ 1/(n log n), and since sum_n 1/(n log n) diverges we might expect that there should be infinitely many n for which some a_1 ... a_n t + 1 is prime, and thus every even integer should occur. - Robert Israel, Dec 20 2012

			2+1=3, 2*6+1=13, 2*6*8+1=97, 2*6*8*12+1=1153, etc. are primes.
After 200 terms the prime is
224198929826405912196464851358435330956778558123234657623126\
069546460095464785674042966210907411841359152393200850271694\
899718487202330385432243578646330245831108247815285116235792\
875886417750289946171599027675234787802312202111702704952223\
563058999855839876391430601719636148884060097930252529666254\
756431522481046758186320659298713737639441014068272279177710\
551232067814381240340990584869121776471244800000000000000000\
00000000000000000000000000000 (449 digits). - _Robert Israel_, Dec 21 2012

Robert Israel, Table of n, a(n) for n = 1..200

Cf. A036013, A046966, A046972, A051957, A073673, A073674, A083769, A083770, A083771, A084401, A084402, A084724, A087338.

  N := 200: # number of terms desired
P := 2:
a[1] := 2:
C := {seq(2*j, j = 2 .. 10)}:
Cmax := 20:
for n from 2 to N do
   for t in C do
      if isprime(t*P+1) then
        a[n]:= t;
        P:= t*P;
        C:= C minus {t};
        break;
      end if;
   end do;
   while not assigned(a[n]) do
     t0:= Cmax+2;
     Cmax:= 2*Cmax;
     C:= C union {seq(j, j=t0 .. Cmax, 2)};
     for t from t0 to Cmax by 2 do
       if isprime(t*P+1) then
         a[n]:= t;
         P:= t*P;
         C:= C minus {t};
         break;
       end if
     end do;
   end do;
end do;
[seq(a[n],n=1..N)];

f[s_List] := Block[{k = 2, p = Times @@ s}, While[ MemberQ[s, k] || !PrimeQ[k*p + 1], k += 2]; Append[s, k]]; Nest[f, {2}, 62] (* Robert G. Wilson v, Dec 24 2012 *)

More terms from David Wasserman, Nov 23 2004
Edited by N. J. A. Sloane, Dec 20 2012
Comment edited, Maple code and additional terms by Robert Israel, Dec 20 2012

A084402 n-th partial product - 1 is a prime, where a(n) > 1.

Original entry on oeis.org

3, 2, 2, 2, 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, 77, 23

Offset: 1

Amarnath Murthy, May 31 2003

nonn

a(n) exists for each n by Dirichlet's theorem. - Charles R Greathouse IV, Oct 31 2014

Same as A036013 for n > 4. - Georg Fischer, Oct 19 2018

			3-1 = 2, 3*2-1 = 5, 3*2*2 -1 = 11, etc. are primes.

Amiram Eldar, Table of n, a(n) for n = 1..586 (terms 1..300 from Georg Fischer)

Cf. A036013, A084401.

lista(nn) = {v = vector(nn); for (n=1, nn, v[n] = 2; while (! isprime(prod(i=1, n, v[i])-1), v[n]++); print1(v[n], ", "););} \\ Michel Marcus, Oct 31 2014

first(n) = my(res=vector(n), pp=1, t); for(i = 1, n, t = 2; while(!isprime(pp * t - 1), t++); pp *= t; res[i]=t); res \\ David A. Corneth, Oct 19 2018

More terms from David Wasserman, Dec 22 2004

A058000 a(n) is smallest prime such that a(n)+1 is a proper multiple of a(n-1)+1 [with a(1)=1].

Original entry on oeis.org

1, 3, 7, 23, 47, 191, 383, 1151, 6911, 27647, 138239, 691199, 3455999, 34559999, 138239999, 414719999, 2073599999, 16588799999, 364953599999, 4744396799999, 66421555199999, 132843110399999, 664215551999999, 3321077759999999

Offset: 1

Henry Bottomley, Nov 02 2000

nonn

Cf. A036013, A057999.

a(n)= Product{i=1_n}[A036013(i)]-1 = a(n-1)*A036013(n)-1+A036013(n).

A083275 a(n) = smallest number not occurring earlier such that a(1)a(2)...*a(n) - 1 is prime.

Original entry on oeis.org

3, 1, 2, 4, 7, 5, 6, 11, 12, 10, 13, 17, 14, 16, 15, 8, 18, 23, 21, 26, 22, 27, 41, 30, 20, 57, 32, 29, 24, 65, 42, 38, 28, 63, 35, 19, 58, 31, 36, 61, 45, 37, 33, 69, 53, 67, 127, 40, 95, 25, 86, 48, 39, 72, 70, 79, 54, 74, 91, 125, 85, 94, 46, 9, 80, 60, 119, 167, 139, 90, 49

Offset: 1

Vladeta Jovovic, Jun 01 2003

nonn

Alois P. Heinz and Giovanni Resta, Table of n, a(n) for n = 1..1200 (first 300 terms from Alois P. Heinz)

Cf. A051957, A036013, A073673, A259262.

b:= proc() false end:
m:= proc(n) option remember; a(n)*m(n-1) end: m(0):=1:
a:= proc(n) option remember; local k; for k while b(k)
      or not isprime(k*m(n-1)-1) do od; b(k):=true; k
    end:
seq(a(n), n=1..80); # Alois P. Heinz, Jun 17 2015

p=1; L={}; Do[k=1; While[ MemberQ[L, k] || !PrimeQ[p*k - 1], k++]; p *= k; AppendTo[L, k], {30}]; L (* Giovanni Resta, Jun 23 2015 *)

v=[3];n=1;while(n<100,s=-1+n*prod(i=1,#v,v[i]);if(isprime(s)&&!vecsearch(vecsort(v),n),v=concat(v,n);n=0);n++);v \\ Derek Orr, Jun 16 2015

Name corrected by Derek Orr, Jun 16 2015

cp's OEIS Frontend

A036012 a(n) = smallest number > 1 such that a(1)a(2)...a(n) + 1 is prime.

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A083769 a(1)=2; for n >= 2, a(n) = smallest even number such that a(1)*a(2)*...*a(n) + 1 is prime.

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A084402 n-th partial product - 1 is a prime, where a(n) > 1.

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Author

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Comments

Examples

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Programs

PARI

PARI

Extensions

A058000 a(n) is smallest prime such that a(n)+1 is a proper multiple of a(n-1)+1 [with a(1)=1].

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Author

Keywords

Crossrefs

Formula

A083275 a(n) = smallest number not occurring earlier such that a(1)*a(2)*...*a(n) - 1 is prime.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Extensions

A083769 a(1)=2; for n >= 2, a(n) = smallest even number such that a(1)a(2)...*a(n) + 1 is prime.

A083275 a(n) = smallest number not occurring earlier such that a(1)a(2)...*a(n) - 1 is prime.