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
Examples
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.
Links
- Robert Israel, Table of n, a(n) for n = 1..10000
Programs
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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 *) -
Python
from sympy import nextprime def A266620(n): return 4 if n == 3 else nextprime(n) # Chai Wah Wu, Feb 22 2023
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
Comments