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 A056867 Nilpotent numbers: n such that every group of order n is nilpotent. 8

%I

%S 1,2,3,4,5,7,8,9,11,13,15,16,17,19,23,25,27,29,31,32,33,35,37,41,43,

%T 45,47,49,51,53,59,61,64,65,67,69,71,73,77,79,81,83,85,87,89,91,95,97,

%U 99,101,103,107,109,113,115,119,121,123,125,127,128,131,133,135,137,139

%N Nilpotent numbers: n such that every group of order n is nilpotent.

%C Contains exactly the numbers n for which gcd(n,|A153038(n)|)=1 [Pazderski]. - _R. J. Mathar_, Apr 03 2012

%C A173557(a(n)) = 1. - _Reinhard Zumkeller_, Jun 01 2015 (should this rather be "Numbers n such that gcd(n, A173557(n)) = 1.", cf. Haskell code? _Joerg Arndt_, Nov 28 2016)

%H T. D. Noe, <a href="/A056867/b056867.txt">Table of n, a(n) for n = 1..10000</a>

%H J. Pakianathan and K. Shankar, <a href="http://www.math.ou.edu/%7Eshankar/papers/nil2.pdf">Nilpotent Numbers</a>, Amer. Math. Monthly, 107, August-September 2000, <a href="http://www.jstor.org/stable/2589118">631-634</a>.

%H G. Pazderski, <a href="http://dx.doi.org/10.1007/BF01240807">Die Ordnungen, zu denen nur Gruppen mit gegebener Eigenschaft gehören</a>, Archiv Math. 10 (1) (1959) 331.

%F n is in this sequence if p^k is not congruent to 1 mod q for any primes p and q dividing n such that p^e but not p^(e+1) divides n and k <= e. - _Charles R Greathouse IV_, Aug 27 2012

%t A153038[1] = 1; A153038[n_] := (x = 1; Do[p = f[[1]]; e = f[[2]]; x = x*Product[1 - p^s, {s, 1, e}], {f, FactorInteger[n]}]; x); A056867 = Select[Range[140], GCD[#, Abs[A153038[#]]] == 1 &] (* _Jean-François Alcover_, May 15 2012, after _R. J. Mathar_ *)

%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(0)))); 1 \\ _Charles R Greathouse IV_, Sep 18 2012

%o a056867 n = a056867_list !! (n-1)

%o a056867_list = filter (\x -> gcd x (a173557 x) == 1) [1..]

%o -- _Reinhard Zumkeller_, Jun 01 2015, Jun 28 2013

%o (GAP)

%o IsNilpotentInt := function(n)

%o local f, i, j; f := PrimePowersInt(n);

%o for i in [1..Length(f)/2] do

%o for j in [1..f[2*i]] do

%o if Gcd(f[2*i-1]^j-1, n) > 1 then return false; fi;

%o od;

%o od;

%o return true;

%o end;

%o Filtered([1..140], IsNilpotentInt); # _Gheorghe Coserea_, Dec 02 2017

%Y Cf. A003277, A051532, A056866. Complement of A056868.

%Y Cf. A027748, A124010.

%Y Cf. A173557.

%K nonn,nice,easy

%O 1,2

%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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Last modified March 25 10:23 EDT 2019. Contains 321470 sequences. (Running on oeis4.)