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.

A084725 Primes arising in A084724. a(n) = N-th partial product of A084724 +1.

Original entry on oeis.org

3, 13, 97, 1153, 18433, 331777, 6635521, 145981441, 3795517441, 136638627841, 5738822369281, 252508184248321, 25250818424832001, 3030098210979840001, 375732178161500160001, 60868612862163025920001
Offset: 1

Views

Author

Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), Jun 13 2003

Keywords

Crossrefs

Cf. A084722, A084723, A084724.

Extensions

More terms from David Wasserman, Jan 03 2005

A083566 a(1) = 1; for n>1, a(n) = smallest odd number > a(n-1) such that a(1)*...*a(n) + 2 is a prime.

Original entry on oeis.org

1, 3, 5, 7, 9, 15, 19, 25, 27, 53, 57, 65, 71, 87, 101, 151, 195, 247, 253, 255, 277, 289, 291, 301, 321, 355, 361, 443, 455, 461, 491, 531, 533, 541, 599, 603, 619, 635, 647, 667, 805, 817, 871, 1003, 1011, 1179, 1205, 1223, 1327, 1357, 1531, 1551, 1603, 1619
Offset: 1

Views

Author

Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), Jun 13 2003

Keywords

Links

Crossrefs

Cf. A084723, A084724, A084725, A085013.

Programs

  • Mathematica
    a[1] = 1; a[n_] := a[n] = Block[{k = a[n - 1] + 2}, While[ !PrimeQ[k*Times @@ Table[a[i], {i, 1, n - 1}] + 2], k += 2]; k]; Table[ a[n], {n, 1, 54}]
nxt[{pr_,a_}]:=Module[{k=a+2},While[!PrimeQ[pr k+2],k=k+2];{pr k,k}]; NestList[nxt,{1,1},60][[;;,2]] (* Harvey P. Dale, Jul 10 2023 *)

Extensions

Edited and extended by Robert G. Wilson v, Jun 17 2003
Edited by N. J. A. Sloane, Nov 01 2008 at the suggestion of R. J. Mathar

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

A084723 Primes arising in A083566. a(n) = n-th partial product of A083566 + 2.

Original entry on oeis.org

3, 5, 17, 107, 947, 14177, 269327, 6733127, 181794377, 9635101877, 549200806877, 35698052446877, 2534561723728127, 220506869964346877, 22271193866399034377, 3362950273826254190627, 655775303396119567171877
Offset: 1

Views

Author

Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), Jun 13 2003

Keywords

Crossrefs

Cf. A083566, A084724, A084725.

Extensions

More terms from David Wasserman, Jan 03 2005
Showing 1-4 of 4 results.

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

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