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A178156
Numbers m such that (m-1)! is not divisible by m^2.
4
2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 17, 19, 22, 23, 26, 29, 31, 34, 37, 38, 41, 43, 46, 47, 53, 58, 59, 61, 62, 67, 71, 73, 74, 79, 82, 83, 86, 89, 94, 97, 101, 103, 106, 107, 109, 113, 118, 122, 127, 131, 134, 137, 139, 142, 146, 149, 151, 157, 158, 163
OFFSET
1,1
COMMENTS
Union of {8, 9} and A001751.
REFERENCES
G. Pólya and G. Szegő, Problems and Theorems in Analysis II (Springer 1924, reprinted 1972), Part Eight, Chap. 3, Sect. 1, Problem 133b.
MATHEMATICA
Select[Range[200], !Divisible[(#-1)!, #^2]&] (* Harvey P. Dale, Mar 06 2016 *)
PROG
(PARI) for(m=1, 3e2, if((m-1)!%m^2, print1(m", "))) \\ Charles R Greathouse IV, Aug 21 2011
(Haskell)
import Data.List (insert)
a178156 n = a178156_list !! (n-1)
a178156_list = insert 9 $ insert 8 a001751_list
-- Reinhard Zumkeller, Oct 14 2014
(Python)
from sympy import primepi
def A178156(n):
def bisection(f, kmin=0, kmax=1):
while f(kmax) > kmax: kmax <<= 1
while kmax-kmin > 1:
kmid = kmax+kmin>>1
if f(kmid) <= kmid:
kmax = kmid
else:
kmin = kmid
return kmax
def f(x): return int(n+x-primepi(x)-primepi(x>>1)-(x>=8)-(x>=9))
return bisection(f, n, n) # Chai Wah Wu, Oct 17 2024
CROSSREFS
KEYWORD
nonn
AUTHOR
Reinhard Zumkeller, Dec 17 2010
EXTENSIONS
Entries corrected by Charles R Greathouse IV, Aug 21 2011
STATUS
approved