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

%I A178156 #25 Oct 17 2024 15:00:32
%S A178156 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,
%T A178156 47,53,58,59,61,62,67,71,73,74,79,82,83,86,89,94,97,101,103,106,107,
%U A178156 109,113,118,122,127,131,134,137,139,142,146,149,151,157,158,163
%N A178156 Numbers m such that (m-1)! is not divisible by m^2.
%C A178156 Union of {8, 9} and A001751.
%D A178156 G. Pólya and G. Szegő, Problems and Theorems in Analysis II (Springer 1924, reprinted 1972), Part Eight, Chap. 3, Sect. 1, Problem 133b.
%t A178156 Select[Range[200],!Divisible[(#-1)!,#^2]&] (* _Harvey P. Dale_, Mar 06 2016 *)
%o A178156 (PARI) for(m=1,3e2,if((m-1)!%m^2,print1(m", "))) \\ _Charles R Greathouse IV_, Aug 21 2011
%o A178156 (Haskell)
%o A178156 import Data.List (insert)
%o A178156 a178156 n = a178156_list !! (n-1)
%o A178156 a178156_list = insert 9 $ insert 8 a001751_list
%o A178156 -- _Reinhard Zumkeller_, Oct 14 2014
%o A178156 (Python)
%o A178156 from sympy import primepi
%o A178156 def A178156(n):
%o A178156     def bisection(f,kmin=0,kmax=1):
%o A178156         while f(kmax) > kmax: kmax <<= 1
%o A178156         while kmax-kmin > 1:
%o A178156             kmid = kmax+kmin>>1
%o A178156             if f(kmid) <= kmid:
%o A178156                 kmax = kmid
%o A178156             else:
%o A178156                 kmin = kmid
%o A178156         return kmax
%o A178156     def f(x): return int(n+x-primepi(x)-primepi(x>>1)-(x>=8)-(x>=9))
%o A178156     return bisection(f,n,n) # _Chai Wah Wu_, Oct 17 2024
%Y A178156 Cf. A046022, A000142, A174460.
%Y A178156 Cf. A001751.
%K A178156 nonn
%O A178156 1,1
%A A178156 _Reinhard Zumkeller_, Dec 17 2010
%E A178156 Entries corrected by _Charles R Greathouse IV_, Aug 21 2011