A056868 Numbers that are not nilpotent numbers.
6, 10, 12, 14, 18, 20, 21, 22, 24, 26, 28, 30, 34, 36, 38, 39, 40, 42, 44, 46, 48, 50, 52, 54, 55, 56, 57, 58, 60, 62, 63, 66, 68, 70, 72, 74, 75, 76, 78, 80, 82, 84, 86, 88, 90, 92, 93, 94, 96, 98, 100, 102, 104, 105, 106, 108, 110, 111, 112, 114, 116, 117, 118, 120
Offset: 1
Examples
From _Bernard Schott_, Dec 19 2021: (Start) 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. 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)
Links
- T. D. Noe, Table of n, a(n) for n = 1..10000
- J. Pakianathan and K. Shankar, Nilpotent Numbers, Amer. Math. Monthly, 107, August-September 2000, pp. 631-634.
Crossrefs
Programs
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Haskell
a056868 n = a056868_list !! (n-1) a056868_list = filter (any (== 1) . pks) [1..] where pks x = [p ^ k `mod` q | let fs = a027748_row x, q <- fs, (p,e) <- zip fs $ a124010_row x, k <- [1..e]] -- Reinhard Zumkeller, Jun 28 2013 -
Mathematica
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 *) -
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
Formula
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
Extensions
More terms from Francisco Salinas (franciscodesalinas(AT)hotmail.com), Dec 25 2001
Comments