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A054395
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Numbers n such that there are precisely 2 groups of order n.
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29
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4, 6, 9, 10, 14, 21, 22, 25, 26, 34, 38, 39, 45, 46, 49, 55, 57, 58, 62, 74, 82, 86, 93, 94, 99, 105, 106, 111, 118, 121, 122, 129, 134, 142, 146, 153, 155, 158, 165, 166, 169, 175, 178, 183, 194, 195, 201, 202, 203, 205, 206, 207, 214, 218, 219, 226, 231, 237
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OFFSET
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1,1
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COMMENTS
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Givens characterizes this sequence, see Theorem 5. In particular, this sequence is ({n: A215935(n) = 1} INTERSECT A005117) UNION (A060687 INTERSECT A051532). - Charles R Greathouse IV, Aug 27 2012 [This is now A350586 UNION A350322. - Charles R Greathouse IV, Jan 08 2022]
Numbers n such that A000001(n) = 2. - Muniru A Asiru, Nov 03 2017
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LINKS
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Muniru A Asiru and Gheorghe Coserea, Table of n, a(n) for n = 1..234567, terms 1..422 from Muniru A Asiru.
H. U. Besche, B. Eick and E. A. O'Brien, The Small Groups Library
Clint Givens, Orders for which there exist exactly two groups (2006)
Gordon Royle, Numbers of Small Groups
Index entries for sequences related to groups
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EXAMPLE
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For n = 4, the 2 groups of order 4 are C4, C2 x C2, for n = 6, the 2 groups of order 6 are S3, C6 and for n = 9, the 2 groups of order 9 are C9, C3 x C3 where C is the cyclic group of the stated order and S is the symmetric group of the stated degree. The symbol x means direct product. - Muniru A Asiru, Oct 24 2017
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MATHEMATICA
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Select[Range[240], FiniteGroupCount[#] == 2&]
(* or: *)
okQ[n_] := Module[{p, f}, p = GCD[n, EulerPhi[n]]; If[! PrimeQ[p], Return[False]]; If[Mod[n, p^2] == 0, Return[1 == GCD[p + 1, n]]]; f = FactorInteger[n]; 1 == Sum[Boole[Mod[f[[k, 1]], p] == 1], {k, 1, Length[f]}]];
Select[Range[240], okQ] (* Jean-François Alcover, Dec 08 2017, after Gheorghe Coserea *)
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PROG
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(GAP) A054395 := Filtered([1..2015], n -> NumberSmallGroups(n) = 2); # Muniru A Asiru, Oct 24 2017
(GAP)
IsGivensInt := function(n)
local p, f; p := GcdInt(n, Phi(n));
if not IsPrimeInt(p) then return false; fi;
if n mod p^2 = 0 then return 1 = GcdInt(p+1, n); fi;
f := PrimePowersInt(n);
return 1 = Number([1..QuoInt(Length(f), 2)], k->f[2*k-1] mod p = 1);
end;;
Filtered([1..240], IsGivensInt); # Gheorghe Coserea, Dec 04 2017
(PARI)
is(n) = {
my(p=gcd(n, eulerphi(n)), f);
if (!isprime(p), return(0));
if (n%p^2 == 0, return(1 == gcd(p+1, n)));
f = factor(n); 1 == sum(k=1, matsize(f)[1], f[k, 1]%p==1);
};
seq(N) = {
my(a = vector(N), k=0, n=1);
while(k < N, if(is(n), a[k++]=n); n++); a;
};
seq(58) \\ Gheorghe Coserea, Dec 03 2017
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CROSSREFS
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Equals A350586 UNION A350322.
Cf. A000001, A003277, A054395, A054396, A054397, A055561, A135850.
Sequence in context: A236026 A193305 A084759 * A142863 A318990 A132435
Adjacent sequences: A054392 A054393 A054394 * A054396 A054397 A054398
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KEYWORD
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nonn
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AUTHOR
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N. J. A. Sloane, May 21 2000
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EXTENSIONS
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More terms from Christian G. Bower, May 25 2000
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STATUS
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approved
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