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A056868
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Numbers that are not nilpotent numbers.
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14
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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
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OFFSET
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1,1
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COMMENTS
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A number is nilpotent if every group of order n is nilpotent.
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
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LINKS
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J. Pakianathan and K. Shankar, Nilpotent Numbers, Amer. Math. Monthly, 107, August-September 2000, pp. 631-634.
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FORMULA
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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
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EXAMPLE
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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)
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MATHEMATICA
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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 *)
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PROG
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(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
(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]]
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CROSSREFS
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KEYWORD
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nonn,nice,easy
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AUTHOR
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EXTENSIONS
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More terms from Francisco Salinas (franciscodesalinas(AT)hotmail.com), Dec 25 2001
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STATUS
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approved
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