A178156 Numbers m such that (m-1)! is not divisible by m^2.

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

Author

Reinhard Zumkeller, Dec 17 2010

Keywords

• nonn

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.

Crossrefs

Cf. A046022, A000142, A174460.

Cf. A001751.

Programs

Haskell

import Data.List (insert)
a178156 n = a178156_list !! (n-1)
a178156_list = insert 9 $ insert 8 a001751_list
-- Reinhard Zumkeller, Oct 14 2014

Mathematica

Select[Range[200],!Divisible[(#-1)!,#^2]&] (* Harvey P. Dale, Mar 06 2016 *)

PARI

for(m=1,3e2,if((m-1)!%m^2,print1(m", "))) \\ Charles R Greathouse IV, Aug 21 2011

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

Extensions

Entries corrected by Charles R Greathouse IV, Aug 21 2011