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A056867
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Nilpotent numbers: n such that every group of order n is nilpotent.
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10
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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, 45, 47, 49, 51, 53, 59, 61, 64, 65, 67, 69, 71, 73, 77, 79, 81, 83, 85, 87, 89, 91, 95, 97, 99, 101, 103, 107, 109, 113, 115, 119, 121, 123, 125, 127, 128, 131, 133, 135, 137, 139
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OFFSET
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1,2
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COMMENTS
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Contains exactly the numbers n for which gcd(n,|A153038(n)|)=1 [Pazderski]. - R. J. Mathar, Apr 03 2012
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
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LINKS
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FORMULA
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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
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MATHEMATICA
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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(0)))); 1 \\ Charles R Greathouse IV, Sep 18 2012
(GAP)
IsNilpotentInt := function(n)
local f, i, j; f := PrimePowersInt(n);
for i in [1..Length(f)/2] do
for j in [1..f[2*i]] do
if Gcd(f[2*i-1]^j-1, n) > 1 then return false; fi;
od;
od;
return true;
end;
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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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