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%I #56 Dec 20 2021 10:03:43
%S 6,10,12,14,18,20,21,22,24,26,28,30,34,36,38,39,40,42,44,46,48,50,52,
%T 54,55,56,57,58,60,62,63,66,68,70,72,74,75,76,78,80,82,84,86,88,90,92,
%U 93,94,96,98,100,102,104,105,106,108,110,111,112,114,116,117,118,120
%N Numbers that are not nilpotent numbers.
%C A number is nilpotent if every group of order n is nilpotent.
%C The sequence "Numbers of the form (k*i + 1)*k*j with i, j >= 1 and k >= 2" agrees with this for the first 146 terms but then differs. Cf. A300737. - _Gionata Neri_, Mar 11 2018
%H T. D. Noe, <a href="/A056868/b056868.txt">Table of n, a(n) for n = 1..10000</a>
%H J. Pakianathan and K. Shankar, <a href="http://www.jstor.org/stable/2589118">Nilpotent Numbers</a>, Amer. Math. Monthly, 107, August-September 2000, pp. 631-634.
%F n is in this sequence if p^k = 1 mod q for primes p and q dividing n such that p^k divides n. - _Charles R Greathouse IV_, Aug 27 2012
%e From _Bernard Schott_, Dec 19 2021: (Start)
%e There are 2 groups with order 6: C_6 that is cyclic so nilpotent, and the symmetric group S_3 that is not nilpotent, hence 6 is a term.
%e There are also 2 groups with order 10: C_10 that is cyclic so nilpotent, and the dihedral group D_10 that is not nilpotent, hence 10 is another term. (End)
%t nilpotentQ[n_] := With[{f = FactorInteger[n]}, Sum[ Boole[ Mod[p[[1]]^p[[2]], q[[1]]] == 1], {p, f}, {q, f}]] == 0; Select[ Range[120], !nilpotentQ[#]& ] (* _Jean-François Alcover_, Sep 03 2012 *)
%o (PARI) is(n)=my(f=factor(n));for(k=1,#f[,1], for(j=1,f[k,2], if(gcd(n, f[k,1]^j-1)>1, return(1)))); 0 \\ _Charles R Greathouse IV_, Sep 18 2012
%o (Haskell)
%o a056868 n = a056868_list !! (n-1)
%o a056868_list = filter (any (== 1) . pks) [1..] where
%o pks x = [p ^ k `mod` q | let fs = a027748_row x, q <- fs,
%o (p,e) <- zip fs $ a124010_row x, k <- [1..e]]
%o -- _Reinhard Zumkeller_, Jun 28 2013
%Y Complement of A056867.
%Y Subsequence of A060652; A068919 is a subsequence.
%Y Cf. A003277, A051532, A056866, A027748, A124010, A300737.
%K nonn,nice,easy
%O 1,1
%A _N. J. A. Sloane_, Sep 02 2000
%E More terms from Francisco Salinas (franciscodesalinas(AT)hotmail.com), Dec 25 2001
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