A046966 -id:A046966 - cp's OEIS frontend

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

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

A051957 a(n) = smallest number > a(n-1) such that a(1)a(2)...*a(n) - 1 is a prime.

Original entry on oeis.org

3, 4, 5, 6, 8, 9, 11, 14, 18, 20, 21, 27, 28, 37, 62, 77, 86, 97, 100, 107, 109, 166, 167, 181, 196, 225, 262, 266, 268, 274, 278, 290, 305, 345, 378, 385, 414, 429, 438, 442, 498, 503, 516, 531, 573, 595, 596, 611, 640, 665, 749, 794, 870

Offset: 1

Felice Russo, Dec 21 1999

Amiram Eldar, Table of n, a(n) for n = 1..1000 (terms 1..200 from T. D. Noe)

Cf. A046966, A051956.

seq={3};Do[n=Last[seq]+1;While[!PrimeQ[n Times@@seq-1],n++]; AppendTo[ seq,n];,{60}];seq (* Harvey P. Dale, Oct 21 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

A051896 a(n) = smallest palindrome > a(n-1) such that a(1)a(2)...*a(n) + 1 is prime with a(1) = 2.

Original entry on oeis.org

2, 3, 5, 6, 9, 55, 66, 77, 88, 161, 191, 313, 484, 494, 525, 747, 3223, 3993, 11711, 13431, 13731, 18881, 19691, 21012, 21112, 22422, 24242, 34443, 35353, 41114, 44244, 44844, 46664, 52225, 52925, 53935, 58385, 59895, 60806, 64146, 71917

Offset: 1

Felice Russo, Dec 21 1999

Chai Wah Wu, Table of n, a(n) for n = 1..200

Cf. A046966, A051934, A051954.

Subsequence of A002113.

nxt[{t_,a_}]:=Module[{k=a+1},While[(!PalindromeQ[k])||CompositeQ[k*t+1],k++];{t*k,k}]; NestList[nxt,{2,2},40][[All,2]] (* Harvey P. Dale, Apr 18 2022 *)

Initial conditions added to description by Chai Wah Wu, Apr 16 2021

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).

A173910 a(n) = smallest number >= a(n-1) such that a(1)a(2)...*a(n)+1 is prime; a(1)=2.

Original entry on oeis.org

2, 2, 3, 3, 3, 4, 5, 7, 10, 14, 15, 18, 30, 32, 46, 56, 58, 59, 84, 86, 99, 101, 103, 106, 122, 126, 128, 128, 136, 152, 157, 170, 190, 208, 281, 282, 284, 320, 393, 406, 459, 479, 526, 529, 530, 540, 559, 601, 639, 640, 709, 789, 828, 900, 917, 949, 1029, 1029

Offset: 1

Dmitry Kamenetsky, Mar 02 2010

nonn

Amiram Eldar, Table of n, a(n) for n = 1..700

Cf. A036012, A046966.

a[1] = 2; a[n_] := a[n] = Module[{k = a[n - 1], r = Product[a[i], {i, 1, n - 1}]}, While[! PrimeQ[k*r + 1], k++]; k]; Array[a, 60] (* Amiram Eldar, Jan 19 2023 *)

More terms from Amiram Eldar, Jan 19 2023

A274695 a(1) = 1; for n>1, a(n) = smallest number > a(n-1) such that a(1)a(2)...*a(n) + 1 is a Fibonacci number.

Original entry on oeis.org

1, 2, 6, 133, 97479304649455554938377

Offset: 1

Robert C. Lyons, Jul 04 2016

nonn

a(6) = (Fibonacci(7937)-1)/(a(2)*a(3)*a(4)*a(5)) has 1633 digits and it is thus too large to be included in Data section or in a b-file. - Giovanni Resta, Jul 05 2016

			After a(1)=1 and a(2)=2, we want m, the smallest number > 2 such that 1*2*m+1 is a Fibonacci number: this is m = 6 = a(3).

Cf. A000045, A046966.

a[1] = 1; a[n_] := a[n] = Block[{p = Times @@ Array[a, n-1], i, m}, For[i=2, ! (IntegerQ[m = (Fibonacci[i] - 1)/p] && m > a[n-1]), i++]; m]; Array[a, 6] (* Giovanni Resta, Jul 05 2016 *)

product = 1
seq = [ product ]
prev_fib_index = 0
max_n = 5
for n in range(2, max_n+1):
    fib_index = prev_fib_index + 1
    found = False
    while not found:
        fib_minus_1 = fibonacci(fib_index) - 1
        if product.divides(fib_minus_1):
            m = int( fib_minus_1 / product )
            if m > seq[-1]:
                product = product * m
                seq.append( m )
                found = True
                prev_fib_index = fib_index
                break
        fib_index += 1
print(seq)

a(5) from Giovanni Resta, Jul 05 2016

cp's OEIS Frontend

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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A051957 a(n) = smallest number > a(n-1) such that a(1)*a(2)*...*a(n) - 1 is a 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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A051896 a(n) = smallest palindrome > a(n-1) such that a(1)*a(2)*...*a(n) + 1 is prime with a(1) = 2.

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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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Formula

A173910 a(n) = smallest number >= a(n-1) such that a(1)*a(2)*...*a(n)+1 is prime; a(1)=2.

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Programs

Mathematica

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A274695 a(1) = 1; for n>1, a(n) = smallest number > a(n-1) such that a(1)*a(2)*...*a(n) + 1 is a Fibonacci number.

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Sage

Extensions

A051957 a(n) = smallest number > a(n-1) such that a(1)a(2)...*a(n) - 1 is a prime.

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

A051896 a(n) = smallest palindrome > a(n-1) such that a(1)a(2)...*a(n) + 1 is prime with a(1) = 2.

A173910 a(n) = smallest number >= a(n-1) such that a(1)a(2)...*a(n)+1 is prime; a(1)=2.

A274695 a(1) = 1; for n>1, a(n) = smallest number > a(n-1) such that a(1)a(2)...*a(n) + 1 is a Fibonacci number.