

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, AugustSeptember 2000, pp. 631634.


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

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[#]& ] (* JeanFrançois 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]^j1)>1, return(1)))); 0 \\ Charles R Greathouse IV, Sep 18 2012
(Haskell)
a056868 n = a056868_list !! (n1)
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]]


CROSSREFS



KEYWORD

nonn,nice,easy


AUTHOR



EXTENSIONS

More terms from Francisco Salinas (franciscodesalinas(AT)hotmail.com), Dec 25 2001


STATUS

approved



