A067362 -id:A067362 - cp's OEIS frontend

A067836 Let a(1)=2, f(n)=a(1)a(2)...*a(n-1) for n>=1 and a(n)=nextprime(f(n)+1)-f(n) for n>=2, where nextprime(x) is the smallest prime > x.

2, 3, 5, 7, 13, 11, 17, 19, 23, 37, 73, 29, 31, 43, 79, 53, 83, 67, 41, 47, 179, 149, 181, 103, 71, 59, 197, 167, 109, 137, 107, 251, 101, 157, 199, 283, 211, 277, 173, 127, 269, 61, 89, 271, 151, 191, 227, 311, 409, 577, 331, 281, 313, 307, 223, 491, 439, 233, 367

Offset: 1

Frank Buss (fb(AT)frank-buss.de), Feb 09 2002

nonn

The terms are easily seen to be distinct. It is conjectured that every element is prime. Do all primes occur in the sequence?
All elements are prime and distinct through n=1000. - Robert Price, Mar 09 2013
All elements are prime and distinct through n=3724. - Dana Jacobsen, Feb 15 2015
With a(0) = 1, a(n) is the next smallest number not in the sequence such that a(n) + Product_{i=1..n-1} a(i) is prime. - Derek Orr, Jun 16 2015
A generalization of Fortunate's conjecture, cf. A005235. - M. F. Hasler, Nov 04 2024
Conjecture: there are infinitely many values of this sequence such that a(n) < n. - Davide Rotondo, Feb 28 2025

Robert Price and Dana Jacobsen, Table of n, a(n) for n = 1..3724 (first 1000 terms from Robert Price)
Carlos Rivera, Conjecture 28. Frank Buss's Conjecture, The Prime Puzzles & Problems Connection.
Carlos Rivera, Conjecture 29. The Frank Buss's B(n) function, The Prime Puzzles & Problems Connection.

Cf. A062894 has the indices of the primes in this sequence. A071290 has the sequence of f's. Also see A067362, A068192.

Cf. A005235 (Fortunate numbers).

Mathematica
```
<Jayanta Basu, Aug 10 2013 *)
```

f := 1:for n from 1 to 50 do a := nextprime(f+2)-f:f := f*a:print(a) end_for

v=[2];n=2;while(n<500,s=n+prod(i=1,#v,v[i]);if(isprime(s)&&!vecsearch(vecsort(v),n),v=concat(v,n);n=1);n++);v \\ Derek Orr, Jun 16 2015

from sympy import nextprime
def A067836_gen(): # generator of terms
    a, f = 2, 1
    yield 2
    while True:
        yield (a:=nextprime((f:=f*a)+1)-f)
A067836_list = list(islice(A067836_gen(),30)) # Chai Wah Wu, Sep 09 2023

Edited by Dean Hickerson, Mar 02 2002

Edited by Dean Hickerson and David W. Wilson, Jun 10 2002

A067363 a(n)=p-n!^3, where p is the smallest prime > n!^3+1.

Original entry on oeis.org

2, 3, 7, 5, 17, 11, 17, 23, 23, 103, 59, 17, 29, 79, 59, 23, 347, 307, 53, 227, 131, 83, 67, 223, 29, 59, 197, 83, 181, 293, 71, 71, 139, 43, 67, 103, 431, 743, 1279, 197, 419, 127, 271, 73, 229, 503, 211, 181, 1597, 151, 151, 197, 1013, 179, 587, 71, 137, 547

Offset: 1

Frank Buss (fb(AT)frank-buss.de), Jan 19 2002

nonn

The first 118 terms are primes. Are all terms prime? For n!^i, with 0

The first 2278 terms are primes. - Dana Jacobsen, May 13 2015

Dana Jacobsen, Table of n, a(n) for n = 1..2278
Cyril Banderier, Fortunate Numbers

Cf. A037153, A037153, A005235, A067362, A067364, A067365.

a[n_] := For[i=2, True, i++, If[PrimeQ[n!^3+i], Return[i]]]
spn[n_]:=Module[{c=(n!)^3},NextPrime[c+1]-c]; Array[spn,60] (* Harvey P. Dale, May 25 2023 *)

for n from 1 to 50 do f := n!^3:a := nextprime(f+2)-f:print(a) end_for

for(n=1, 100, f=n!^3; print1(nextprime(f+2)-f,", ")) \\ Dana Jacobsen, May 13 2015

use ntheory ":all"; use Math::GMP qw/:constant/; for my $n (1..500) { my $f=factorial($n)**3; say "$n ", next_prime($f+1)-$f; } # Dana Jacobsen, May 13 2015

Edited by Dean Hickerson, Mar 02 2002

A067364 a(n)=p-n!^4, where p is the smallest prime > n!^4+1.

Original entry on oeis.org

2, 3, 5, 5, 7, 29, 19, 29, 181, 19, 31, 173, 79, 43, 379, 61, 101, 127, 101, 83, 37, 29, 271, 233, 109, 233, 293, 1039, 137, 241, 173, 197, 613, 1933, 277, 71, 503, 449, 1667, 53, 67, 163, 179, 211, 53, 613, 1171, 1069, 359, 199, 839, 433, 1523, 463, 677

Offset: 1

Frank Buss (fb(AT)frank-buss.de), Jan 19 2002

nonn

The first 102 terms are primes. Are all terms prime? For n!^i, with 0

The first 1865 terms are primes. - Dana Jacobsen, May 13 2015

Dana Jacobsen, Table of n, a(n) for n = 1..1865
Cyril Banderier, Fortunate Numbers

Cf. A037153, A037153, A005235, A067362, A067363, A067365.

a[n_] := For[i=2, True, i++, If[PrimeQ[n!^4+i], Return[i]]]

for n from 1 to 50 do f := n!^4:a := nextprime(f+2)-f:print(a) end_for

for(n=1, 500, f=n!^4; print1(nextprime(f+2)-f, ", ")) \\ Dana Jacobsen, May 13 2015

use ntheory ":all"; use Math::GMP qw/:constant/; for my $n (1..500) { my $f=factorial($n)**4; say "$n ", next_prime($f+1)-$f; } # Dana Jacobsen, May 13 2015

Edited by Dean Hickerson, Mar 02 2002

A067365 a(n) = p-n!^5, where p is the smallest prime > n!^5+1.

Original entry on oeis.org

2, 5, 13, 13, 7, 7, 11, 71, 23, 19, 197, 17, 101, 53, 17, 47, 73, 97, 53, 433, 251, 251, 47, 263, 281, 353, 53, 61, 179, 41, 53, 401, 449, 79, 89, 1283, 367, 2011, 139, 227, 1597, 1657, 1123, 397, 131, 727, 137, 167, 89, 379, 421, 653, 223, 373, 2221, 1447

Offset: 1

Frank Buss (fb(AT)frank-buss.de), Jan 19 2002

nonn

The first 60 terms are primes. Are all terms prime? For n!^i, with 0

The first 1592 terms are primes. - Dana Jacobsen, May 13 2015

Dana Jacobsen, Table of n, a(n) for n = 1..1592
Cyril Banderier, Fortunate Numbers

Cf. A037153, A037153, A005235, A067362, A067363, A067364.

a[n_] := For[i=2, True, i++, If[PrimeQ[n!^5+i], Return[i]]]
spf[n_]:=Module[{c=(n!)^5},NextPrime[c+1]-c]; Array[spf,60] (* Harvey P. Dale, Feb 24 2015 *)

for n from 1 to 50 do f := n!^5:a := nextprime(f+2)-f:print(a) end_for

for(n=1, 100, f=n!^5; print1(nextprime(f+2)-f, ", ")) \\ Dana Jacobsen, May 13 2015

use ntheory ":all"; use Math::GMP qw/:constant/; for my $n (1..500) { my $f=factorial($n)**5; say "$n ", next_prime($f+1)-$f; } # Dana Jacobsen, May 13 2015

Edited by Dean Hickerson, Mar 02 2002

A066889 a(n) = g(P(n)+2) - P(n), where P(n) = Product_{k=1..n} Fibonacci(k) and g(i) is the smallest prime >= i.

Original entry on oeis.org

2, 2, 3, 5, 7, 11, 17, 17, 37, 23, 47, 37, 29, 19, 47, 59, 19, 37, 71, 59, 31, 67, 239, 101, 739, 409, 43, 367, 167, 251, 73, 71, 419, 1567, 107, 83, 223, 191, 227, 449, 97, 173, 103, 523, 79, 137, 223, 1163, 661, 103, 103, 541, 227, 2383, 433, 71

Offset: 1

Frank Buss (fb(AT)frank-buss.de), Jan 22 2002

nonn

The first 169 terms are primes. Are all terms primes? See links for similar sequences.

Note that g is not the usual "nextprime" function. If the usual nextprime function is used, we get A286296.

			a(4) = 5 because Fibonacci(1)*Fibonacci(2)*Fibonacci(3)*Fibonacci(4) = 1*1*2*3 = 6, g(6+2) = 11, and 11 - 6 = 5.

Harry J. Smith, Table of n, a(n) for n = 1..169
Frank Buss, Prime Puzzles - Frank Buss's Conjecture
Frank Buss, The B(n) function

Cf. A000045, A067362, A286296 (the same except for the first two terms).

Join[{2,2},Drop[NextPrime[#+2]-#&/@FoldList[Times,Fibonacci[ Range[ 60]]],2]] (* Harvey P. Dale, May 31 2017 *)

f := 1:for n from 1 to 100 do f := f*numlib::fibonacci(n):a := nextprime(f+2)-f:print(a) end_for

{ m=1; for (n=1, 1000, m*=fibonacci(n); write("b066889.txt", n, " ", nextprime(m+2) - m) ) } \\ Harry J. Smith, Apr 05 2010

Definition and example corrected by Harvey P. Dale and N. J. A. Sloane, May 31 2017

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A067836 Let a(1)=2, f(n)=a(1)*a(2)*...*a(n-1) for n>=1 and a(n)=nextprime(f(n)+1)-f(n) for n>=2, where nextprime(x) is the smallest prime > x.

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A067363 a(n)=p-n!^3, where p is the smallest prime > n!^3+1.

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A067364 a(n)=p-n!^4, where p is the smallest prime > n!^4+1.

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A067365 a(n) = p-n!^5, where p is the smallest prime > n!^5+1.

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Mathematica

MuPAD

PARI

Perl

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A066889 a(n) = g(P(n)+2) - P(n), where P(n) = Product_{k=1..n} Fibonacci(k) and g(i) is the smallest prime >= i.

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A067836 Let a(1)=2, f(n)=a(1)a(2)...*a(n-1) for n>=1 and a(n)=nextprime(f(n)+1)-f(n) for n>=2, where nextprime(x) is the smallest prime > x.