OFFSET
1,1
COMMENTS
If n > 1, then a(n)! is the smallest factorial such that |e - m/n| > 1/a(n)! for any integer m.
a(n)! is the second smallest factorial divisible by n.
LINKS
Mohammad K. Azarian, Euler's Number Via Difference Equations, International Journal of Contemporary Mathematical Sciences, Vol. 7, 2012, No. 22, pp. 1095 - 1102.
Jonathan Sondow, A geometric proof that e is irrational and a new measure of its irrationality, Amer. Math. Monthly 113 (2006) 637-641.
Jonathan Sondow, A geometric proof that e is irrational and a new measure of its irrationality, arXiv:0704.1282 [math.HO], 2007-2010.
FORMULA
a(n) = A002034(n) + 1.
EXAMPLE
a(6) = S(6) + 1 = 3 + 1 = 4.
MATHEMATICA
nmax = 100; Do[m=1; While[!IntegerQ[m!/n], m++]; a[n] = m+1, {n, 1, nmax}];
Array[a, nmax] (* Jean-François Alcover, Dec 04 2018 *)
PROG
(Python)
from itertools import count
from sympy import factorint, multiplicity
def A122416(n):
m = 2
for p, e in factorint(n).items():
if e<=p:
m = max(m, e*p+1)
else:
c, a = 0, max(p, (e*(p-1)//p**2)*p)
b = a-p
while b>0:
c += (b:=b//p)
for k in count(a, p):
c += multiplicity(p, k) if p>2 else (~k&k-1).bit_length()
if c+k+p > e:
m = max(m, 1+p*(max(k, e-c) if e>=p else e+1-c))
break
return m # Chai Wah Wu, Feb 26 2026
CROSSREFS
KEYWORD
nonn
AUTHOR
Jonathan Sondow, Sep 03 2006
STATUS
approved