A037153 -id:A037153 - cp's OEIS frontend

A067362 a(n) = p - n!^2, where p is the smallest prime > n!^2+1.

2, 3, 5, 11, 7, 11, 11, 13, 23, 17, 13, 59, 23, 31, 23, 41, 59, 67, 29, 31, 103, 389, 59, 107, 47, 127, 67, 181, 101, 97, 409, 37, 61, 43, 61, 47, 263, 109, 53, 199, 167, 337, 47, 131, 127, 73, 181, 257, 191, 101, 83, 79, 181, 167, 229, 859, 421, 433, 107, 971

Offset: 1

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

nonn

The first 157 terms are primes. Are all terms prime? For n!^i, with 0

The first 200 terms are primes. - Jon Perry and Christ van Willegen, Mar 07 2003

The first 3003 terms are primes. - Dana Jacobsen, May 13 2015

Dana Jacobsen, Table of n, a(n) for n = 1..3000
Cyril Banderier, Fortunate Numbers

Cf. A037153, A037153, A005235, A067363, A067364, A067365.

a[n_] := For[i=2, True, i++, If[PrimeQ[n!^2+i], Return[i]]]
Table[p = NextPrime[(x = (n!)^2) + 1]; p - x, {n, 60}] (* Jayanta Basu, Aug 10 2013 *)

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

for(n=1,500,f=n!^2;print1(nextprime(f+2)-f, ", ")) \\  Dana Jacobsen, May 10 2015

use ntheory ":all"; use Math::GMP qw/:constant/; for my $n (1..500) { my $f=factorial($n)**2; say "$n ",next_prime($f+1)-$f; } # Dana Jacobsen, May 10 2015

Edited by Dean Hickerson, Mar 02 2002

A090786 Least nonnegative integer k such that n! + n + k + 1 is prime.

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 0, 3, 14, 7, 0, 5, 16, 53, 4, 27, 6, 13, 18, 69, 8, 9, 8, 73, 106, 15, 32, 19, 38, 193, 76, 95, 46, 3, 62, 25, 94, 273, 4, 57, 12, 19, 54, 27, 2, 193, 54, 185, 4, 33, 10, 219, 0, 17, 168, 15, 92, 49, 224, 233, 210, 707, 68, 207, 2, 127, 216, 5, 14, 61, 68, 785

Offset: 0

Frederick Magata (frederick.magata(AT)uni-muenster.de), Feb 09 2004

nonn

The (n-1) consecutive numbers n!+2, ..., n!+n (for n >= 2) are not prime. This fact implies that there are arbitrarily large gaps in the distribution of the prime numbers. However, n!+n+1 need not be a prime number. Now a(n) measures, when the next prime number after n!+n appears. Thus a(n)=0 means that n!+n+1 is prime and so on. Obviously, a(n) is parity conserving for n >= 2. I.e., if n >= 2 then 2 divides n iff 2 divides a(n).

Conjectures: By definition a(n)+n!+1 is prime, but is a(n)+n+1=A037153(n) also a prime number for all n > 2? Is the growth of b(n) := Sum_{k=0..n} a(k) quadratic, that is, is b(n)=O(n^2)?

			a(5)=1 because 5!+5+1+1=127 is prime and 126 is not.
a(7)=3 because 7!+7+3+1=5051 is prime and 5048, 5049 and 5050 are not prime.

Seiichi Manyama, Table of n, a(n) for n = 0..500

Cf. A005095, A037153, A073308.

a := proc(n) option remember;local r,m,k: r := n!+n: k := 1: m := r+1: while (not isprime(m)) do k := k+1: while (not igcd(k,n)=1) do k := k+1: od: m := r+k: od: k-1; end;
# alternatively:
a := proc(n) option remember; nextprime(n!+n)-n!-n-1; end;

lnik[n_]:=Module[{c=n!+n+1},If[PrimeQ[c],0,NextPrime[c]-c]]; Array[ lnik, 80,0] (* Harvey P. Dale, Apr 08 2019 *)

a(n) = apply(x->(nextprime(x)-x), n!+n+1); \\ Michel Marcus, Mar 21 2018

A275272 a(n) = p - n!, where p is the second smallest prime > n!.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 19, 31, 23, 19, 17, 43, 73, 41, 149, 41, 53, 61, 109, 37, 37, 71, 109, 193, 97, 173, 47, 101, 229, 163, 241, 83, 139, 103, 83, 577, 311, 47, 269, 61, 61, 107, 97, 89, 379, 149, 269, 83, 137, 167, 281, 89, 79, 443, 229, 157, 179, 563, 389

Offset: 1

Clark Kimberling, Jul 23 2016

p-n! where p = nextprime(nextprime(n!)).

Is every term a prime?

			For n = 4, we have n! = 24, so that p = 31 and a(4) = 7.

Clark Kimberling, Table of n, a(n) for n = 1..300

Cf. A037153, A005235, A275273, A275274, A000040.

Cf. A000142, A187874.

Table[NextPrime[n!, 2] - n!, {n, 1, 150}]

a(n) = nextprime(nextprime(n!+1)+1) - n!; \\ Michel Marcus, Mar 05 2022

from sympy import factorial, nextprime
def a(n): fn = factorial(n); return nextprime(nextprime(fn)) - fn
print([a(n) for n in range(1, 60)]) # Michael S. Branicky, Mar 05 2022

a(n) = A187874(n) - A000142(n). - Michel Marcus, Mar 05 2022

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

A037152 Smallest prime > n!+1.

Original entry on oeis.org

3, 5, 11, 29, 127, 727, 5051, 40343, 362897, 3628811, 39916817, 479001629, 6227020867, 87178291219, 1307674368043, 20922789888023, 355687428096031, 6402373705728037, 121645100408832089, 2432902008176640029

Offset: 1

Jud McCranie

nonn

Main entry for this sequence is A037153.

Vincenzo Librandi, Table of n, a(n) for n = 1..100

Cf. A000142, A037153, A038507, A151800.

NextPrime[Range[20]!+1] (* Harvey P. Dale, Apr 08 2012 *)

makelist(next_prime(n!+1), n,  1, 20); /* Bruno Berselli, May 20 2011 */

for(n=1,100,print1(nextprime(n!+2),", ")); /* Joerg Arndt, May 21 2011 */

a(n) = A151800(A000142(n)+1) = A000142(n) + A037153(n).

a(n) = A151800(A038507(n)). - Michel Marcus, Feb 18 2024

Extended by Ray Chandler, Mar 07 2010

A069941 Number of primes p such that n! <= p <= n! + n^2.

Original entry on oeis.org

1, 3, 3, 3, 4, 5, 5, 4, 7, 7, 9, 6, 5, 8, 4, 9, 10, 14, 8, 16, 14, 14, 7, 6, 16, 12, 12, 15, 13, 12, 9, 12, 12, 17, 13, 6, 12, 18, 15, 13, 15, 17, 15, 23, 19, 12, 13, 19, 18, 22, 20, 19, 16, 17, 19, 19, 23, 20, 18, 19, 23, 24, 19, 15, 19, 20, 26, 18, 24, 22, 24, 25, 24, 16, 23

Offset: 1

Benoit Cloitre, May 04 2002

Conjecture: if n>=2 there are at least 3 primes p such that n!<=p<=n!+n^2 (or stronger: for n>1, a(n) > log(n)). This is stronger than the conjecture described in A037151(n). Because if n!+k is prime, k composite, k=A*B, where A and B must have, each one, at least one prime factor>n (if not: A=q*A' q<=n then n!+k is divisible by q), hence k>n^2. Also stronger (but more restrictive) than the Schinzel conjecture: "for m large enough there's at least one prime p such that m <= p <= m + log(m)^2" since n^2 < log(n!)^2 for n>5.

For the n-th term we have a(n) = pi(n!+n^2) - pi(n!), where pi(x) is the prime counting function. However, pi(n!) is difficult to compute for n>25. The Prime Number Theorem states that pi(x) and Li(x), the logarithmic integral, are asymptotically equal. Hence we can approximate a(n) by Li(n!+n^2) - Li(n!). These approximate values of a(n) are plotted as the red curve in the "Theoretical versus Actual" plot. By the way, using x/log(x) as approximation for Li(x) would change the curve by at most 1 unit. - T. D. Noe, Mar 06 2010

T. D. Noe, Table of n, a(n) for n=1..300
T. D. Noe, Theoretical versus Actual

Cf. A037151, A037153, A014085.

Table[Length[Select[Range[n!,n!+n^2], PrimeQ]], {n,100}] (* T. D. Noe, Mar 06 2010 *)

for(n=1,75,print1(sum(k=n!,n!+n^2,isprime(k)),","))

Showing 1-10 of 19 results. Next

cp's OEIS Frontend

A087202 a(n) is the smallest m such that m > A037153(n) and n!+ m is prime.

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A033932 Least positive m such that n! + m is prime.

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A067362 a(n) = p - n!^2, where p is the smallest prime > n!^2+1.

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A090786 Least nonnegative integer k such that n! + n + k + 1 is prime.

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A275272 a(n) = p - n!, where p is the second smallest prime > n!.

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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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