A266620 a(n) = least non-divisor of n!.

2, 3, 4, 5, 7, 7, 11, 11, 11, 11, 13, 13, 17, 17, 17, 17, 19, 19, 23, 23, 23, 23, 29, 29, 29, 29, 29, 29, 31, 31, 37, 37, 37, 37, 37, 37, 41, 41, 41, 41, 43, 43, 47, 47, 47, 47, 53, 53, 53, 53, 53, 53, 59, 59, 59, 59, 59, 59, 61, 61, 67, 67, 67, 67, 67, 67, 71

Offset

1,1

Comments

It appears that a(n) = A151800(n) with the exception of n = 3. - Robert Israel, Jan 13 2016

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Formula

a(n) = min_{k >= 1} prime(k)^(1 + v(n!, prime(k))) where v(m, p) is the p-adic order of m. - Robert Israel, Jan 13 2016

a(n) = prime(pi(n) + 1) except for n = 3, in which case the least non-divisor of 3! is 4, not 5. - Alonso del Arte, Sep 23 2017

Example

For n = 4 the least non-divisor of 4! = 24 = 2^3 * 3 is 5.
For n = 5 the least non-divisor of 5! = 120 = 2^3 * 3 * 5 is 7.

Maple

N:= 100: # to get a(1)..a(N)
m:= 1 + numtheory:-pi(N):
Primes:= [seq(ithprime(i), i=1..m)]:
for i from 1 to m do pindex[Primes[i]]:= i od:
V:= Vector(m):
k:= 0:
for n from 1 to N do
  for f in ifactors(n)[2] do
    q:= pindex[f[1]];
    V[q]:= V[q] + f[2];
    k:= max(k, q);
  od:
  a[n]:= min(seq(Primes[i]^(1+V[i]), i=1..k), Primes[k+1]);
od:
seq(a[n], n=1..N); # Robert Israel, Jan 13 2016

Mathematica

Table[Complement[Range[2n], Divisors[n!]][[1]], {n, 30}] (* Alonso del Arte, Sep 23 2017 *)
Table[Block[{m = n!, k = n + 1}, While[Divisible[m, k], k++]; k], {n, 67}] (* Michael De Vlieger, Sep 23 2017 *)

Prog

(Python)
from sympy import nextprime
def A266620(n): return 4 if n == 3 else nextprime(n) # Chai Wah Wu, Feb 22 2023

Crossrefs

Cf. A007918, A007978, A066169, A115627, A151800.

Sequence in context: A348531 A237824 A227972 * A222415 A022473 A214921

Adjacent sequences: A266617 A266618 A266619 * A266621 A266622 A266623

Keyword

nonn,easy

Author

Jeffrey Shallit, Jan 01 2016

Status

approved