A046972 -id:A046972 - cp's OEIS frontend

A144728 a(n) is the smallest positive integer such that b * (Product_{k=1..n} a(k)) + 1 is prime, with a(n) > a(n-1) for n >= 2, and b = 6.

1, 2, 3, 5, 9, 12, 16, 22, 25, 29, 31, 35, 47, 57, 61, 66, 79, 81, 108, 114, 148, 163, 172, 185, 198, 203, 205, 236, 265, 275, 282, 294, 312, 344, 359, 377, 397, 398, 411, 427, 431, 493, 512, 589, 647, 648, 660, 708, 719, 765, 887, 911, 916, 935, 1062, 1093, 1102

Offset: 1

Artur Jasinski, Sep 19 2008

nonn

Cf. A046966, A046972, A144717, A144718, A144722, A144723, A144724, A144725, A144726, A144727, A144729, A144730, A144731.

k = 6; a = {}; Do[If[PrimeQ[k n + 1], k = k n; AppendTo[a, n]], {n, 1, 3000}]; a

Definition corrected by Georg Fischer, Jun 18 2021

A144730 a(n) is the smallest positive integer m such that b * (Product_{k=1..n} a(k)) + 1 is prime, with b = 7.

Original entry on oeis.org

4, 7, 13, 19, 33, 35, 36, 43, 48, 55, 59, 62, 87, 129, 149, 153, 159, 190, 228, 231, 245, 265, 266, 269, 284, 300, 329, 331, 340, 347, 372, 432, 449, 450, 461, 485, 496, 500, 514, 544, 560, 565, 594, 598, 605, 614, 639, 677, 684, 734, 736, 794, 804, 813, 882

Offset: 1

Artur Jasinski, Sep 19 2008

nonn

A046966, A046972, A144717, A144718, A144722, A144723, A144724, A144725, A144726, A144727, A144728, A144729, A144731

k = 7; a = {}; Do[If[PrimeQ[k n + 1], k = k n; AppendTo[a, n]], {n, 1, 3000}]; a (*Artur Jasinski*)

Definition corrected by Georg Fischer, Jun 18 2021

A144731 Primes arising in A144730.

Original entry on oeis.org

29, 197, 2549, 48413, 1597597, 55915861, 2012970961, 86557751281, 4154772061441, 228512463379201, 13482235339372801, 835898591041113601, 72723177420576883201, 9381289887254417932801

Offset: 1

Artur Jasinski, Sep 19 2008; corrected Sep 19 2008

nonn

Cf. A046966, A046972, A144717, A144718, A144722, A144723, A144724, A144725, A144726, A144727, A144728, A144729, A144730.

Typo in definition corrected by Arkadiusz Wesolowski, Aug 22 2011

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

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).

A081947 Smallest prime which is one more than the product of n distinct numbers.

Original entry on oeis.org

2, 3, 7, 31, 181, 1009, 6481, 45361, 453601, 3991681, 39916801, 566092801, 9500198401, 114960384001, 1976041267201, 24845812992001, 502146957312001, 8109673360588801, 147254595231744001, 2688996956405760001

Offset: 1

Amarnath Murthy, Apr 02 2003

nonn

			a(5) = 181 = 1*2*3*5*6 + 1.

Cf. A046972.

More terms from Sam Alexander, Oct 20 2003

Corrected and extended by David Wasserman, Jul 20 2004

cp's OEIS Frontend

A144728 a(n) is the smallest positive integer such that b * (Product_{k=1..n} a(k)) + 1 is prime, with a(n) > a(n-1) for n >= 2, and b = 6.

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A144730 a(n) is the smallest positive integer m such that b * (Product_{k=1..n} a(k)) + 1 is prime, with b = 7.

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A144731 Primes arising in A144730.

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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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Maple

Mathematica

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.

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Mathematica

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A081947 Smallest prime which is one more than the product of n distinct numbers.

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Author

Keywords

Examples

Crossrefs

Extensions

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