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A056868 Numbers that are not nilpotent numbers. 14
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
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
LINKS
J. Pakianathan and K. Shankar, Nilpotent Numbers, Amer. Math. Monthly, 107, August-September 2000, pp. 631-634.
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
EXAMPLE
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)
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 *)
PROG
(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]]
-- Reinhard Zumkeller, Jun 28 2013
CROSSREFS
Complement of A056867.
Subsequence of A060652; A068919 is a subsequence.
Sequence in context: A330397 A135711 A161543 * A069209 A060702 A054741
KEYWORD
nonn,nice,easy
AUTHOR
N. J. A. Sloane, Sep 02 2000
EXTENSIONS
More terms from Francisco Salinas (franciscodesalinas(AT)hotmail.com), Dec 25 2001
STATUS
approved

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Last modified February 21 15:08 EST 2024. Contains 370236 sequences. (Running on oeis4.)