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 [This is now A350586 UNION A350322. - Charles R Greathouse IV, Jan 08 2022]
Numbers m such that A000001(m) = 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 m = 4, the 2 groups of order 4 are C4, C2 x C2; for m = 6, the 2 groups of order 6 are S3, C6; and for m = 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. Cyclic numbers A003277. Numbers m such that there are precisely k groups of order m: this sequence (k=2), A055561 (k=3), A054396 (k=4), A054397 (k=5), A135850 (k=6), A249550 (k=7), A249551 (k=8), A249552 (k=9), A249553 (k=10), A249554 (k=11), A249555 (k=12), A292896 (k=13), A294155 (k=14), A294156 (k=15), A295161 (k=16), A294949 (k=17), A298909 (k=18), A298910 (k=19), A298911 (k=20).
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, May 21 2000
EXTENSIONS
More terms from Christian G. Bower, May 25 2000
STATUS
approved