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 A054395 Numbers n such that there are precisely 2 groups of order n. 24
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS 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 Numbers n such that A000001(n) = 2. - Muniru A Asiru, Nov 03 2017 LINKS 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 EXAMPLE 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 MATHEMATICA 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 *) PROG (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 CROSSREFS 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 KEYWORD nonn AUTHOR N. J. A. Sloane, May 21 2000 EXTENSIONS More terms from Christian G. Bower, May 25 2000 STATUS approved

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Last modified September 29 10:54 EDT 2020. Contains 337428 sequences. (Running on oeis4.)