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 1-4 of 4 results.

A084403 Primes arising in A084402. a(n) = n-th partial product of A084402 - 1.

Original entry on oeis.org

2, 5, 11, 23, 47, 191, 383, 1151, 6911, 27647, 138239, 691199, 3455999, 34559999, 138239999, 414719999, 2073599999, 16588799999, 364953599999, 4744396799999, 66421555199999, 132843110399999, 664215551999999, 3321077759999999
Offset: 1

Views

Author

Amarnath Murthy, May 31 2003

Keywords

Comments

This is a sequence similar to A090552 but here with primes: a(1) = 2; for n>=2, a(n) is the smallest prime p for which p+1 is a multiple of a(n-1) + 1. - J. M. Bergot, Oct 16 2014, revised by Wolfdieter Lang, Nov 24 2014

Examples

			3-1=2, 3*2-1=5, 3*2*2-1=11, etc. are the primes arising in A084402.

Crossrefs

Cf. A084402.

Programs

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

Extensions

More terms from David Wasserman, Dec 22 2004

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

A084718 a(n) = A084717(n+1)/A084717(n).

Original entry on oeis.org

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
Offset: 1

Views

Author

Amarnath Murthy, Jun 11 2003

Keywords

Comments

Equals A084402 without the first term. - R. J. Mathar, Sep 17 2008

Links

Crossrefs

Cf. A084402, A084716, A084717, A036012.

Extensions

Edited by Don Reble, Jun 19 2003

A084401 n-th partial product + 1 is a prime, where a(n)>1 for n>1.

Original entry on oeis.org

1, 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

Views

Author

Amarnath Murthy, May 31 2003

Keywords

Comments

Except for the first term, same as A036012. - David Wasserman, Dec 22 2004

Examples

			1+1 =2, 1*2+1=3, 1*2*2 +1=5 etc. are primes.

Crossrefs

Cf. A084402.

Programs

  • Mathematica
    a[1] = p = 1; a[n_] := a[n] = Catch[For[k = 2, True, k++, If[PrimeQ[p*k + 1], p = p*k; Throw[k]]]]; Table[a[n], {n, 1, 82}] (* Jean-François Alcover, May 14 2012 *)

Extensions

More terms from David Wasserman, Dec 22 2004
Showing 1-4 of 4 results.

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