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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 1-2 of 2 results.

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

Views

Author

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

Keywords

Comments

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

Examples

			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

Links

Crossrefs

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

Programs

  • Maple
      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)];
  • Mathematica
    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 *)

Extensions

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

A073673 Rearrangement of natural numbers such that every partial product + 1 is a prime.

Original entry on oeis.org

1, 2, 3, 5, 6, 9, 4, 7, 10, 17, 12, 11, 13, 8, 19, 27, 21, 26, 20, 16, 14, 28, 22, 18, 47, 30, 31, 23, 34, 37, 41, 45, 49, 33, 36, 58, 24, 62, 39, 56, 42, 93, 54, 25, 51, 53, 15, 70, 72, 73, 46, 50, 64, 97, 55, 57, 171, 96, 79, 81, 66, 71, 132, 89, 121, 29, 61, 60, 177, 32
Offset: 1

Views

Author

Amarnath Murthy, Aug 11 2002

Keywords

Comments

From Robert G. Wilson v, Dec 24 2012: (Start)
Records: 1, 2, 3, 5, 6, 9, 10, 17, 19, 27, 28, 47, 49, 58, 62, 93, 97, 171, 177, 184, 221, 243, 470, 512, 573, 768, 856, 999, 1028, 1226, 1659, 2522, ...
Late Records: 1, 2, 3, 4, 7, 8, 14, 15, 29, 32, 35, 59, 75, ... (End)

Links

Crossrefs

Cf. A073674, A083769.

Programs

  • Mathematica
    f[s_List] := Block[{k = 1, p = Times @@ s}, While[ MemberQ[s, k] || !PrimeQ[k*p + 1], k++]; Append[s, k]]; Nest[f, {1}, 69] (* Robert G. Wilson v, Dec 24 2012 *)
  • PARI
    v=[1];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
  • Python
    from gmpy2 import is_prime
from itertools import islice
def agen(startp=1, startset=set()): # generator of terms
    aset, p, mink = startset, startp, 1
    while True:
        an = mink
        while an in aset or not is_prime(p*an + 1): an += 1
        yield an; aset.add(an); p *= an
        while mink in aset: aset.discard(mink); mink += 1
print(list(islice(agen(), 70))) # Michael S. Branicky, May 19 2023

Formula

Conjecture: n/log(n) << a(n) << n*log(n). - Thomas Ordowski, Aug 09 2017

Extensions

More terms from Sascha Kurz, Feb 01 2003
Showing 1-2 of 2 results.

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