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%I #68 Nov 19 2023 14:07:51
%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 A group G of order m is nilpotent iff it has a quotient group of order m/d for each divisor d of m. - _Des MacHale_ and _Bernard Schott_, Jul 15 2022
%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.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Nilpotent_group">Nilpotent group</a>.
%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 (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 Complement of A056868.
%Y Cf. A003277, A051532, A056866.
%Y Cf. A027748, A124010, 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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