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A055561
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Numbers n such that there are precisely 3 groups of order n.
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25
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75, 363, 609, 867, 1183, 1265, 1275, 1491, 1587, 1725, 1805, 2067, 2175, 2373, 2523, 3045, 3525, 3685, 3795, 3975, 4137, 4205, 4335, 4425, 4895, 5019, 5043, 5109, 5901, 5915, 6171, 6225, 6627, 6675, 6699, 7935, 8025, 8427, 8475, 8855, 9429, 9537, 10275
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OFFSET
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1,1
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COMMENTS
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Let gnu(n) (= A000001(n)) denote the "group number of n" defined in A000001 or in (J. H. Conway, Heiko Dietrich and E. A. O'Brien, 2008), then the sequence n -> gnu(a(n)) -> gnu(gnu(a(n))) consists of 1's. - Muniru A Asiru, Nov 19 2017
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LINKS
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Gheorghe Coserea, Table of n, a(n) for n = 1..234567, terms 1..206 from Muniru A Asiru.
H.-U. Besche, B. Eick and E. A. O'Brien, The Small Groups Library
H.-U. Besche, B. Eick and E. A. O'Brien, A Millennium Project: Constructing Small Groups, Internat. J. Algebra and Computation, 12 (2002), 623-644.
J. H. Conway, Heiko Dietrich and E. A. O'Brien, Counting groups: gnus, moas and other exotica, Math. Intell., Vol. 30, No. 2, Spring 2008.
Gordon Royle, Numbers of Small Groups
Index entries for sequences related to groups
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EXAMPLE
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For n = 75, the 3 groups of order 75 are C75, (C5 x C5) : C3, C15 x C5 and for n = 363 the 3 groups of order 363 are C363, (C11 x C11) : C3, C33 x C11 where C is the Cyclic group of the stated order. The symbols x and : mean direct and semi-direct products respectively. - Muniru A Asiru, Oct 24 2017
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PROG
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(PARI)
is(n) = {
my(p = gcd(n, eulerphi(n)), f, g);
if (isprime(p), return(n % p^2 == 0 && isprime(gcd(p+1, n))));
if (omega(p) != 2 || !issquarefree(n), return(0));
f = factor(n); g = factor(p);
1 == g[2, 1] % g[1, 1] &&
1 == sum(k=1, matsize(f)[1], f[k, 1] % g[1, 1] == 1) &&
1 == sum(k=1, matsize(f)[1], f[k, 1] % g[2, 1] == 1);
};
seq(N) = {
my(a = vector(N), k=0, n=1);
while(k < N, if(is(n), a[k++]=n); n++); a;
};
seq(43) \\ Gheorghe Coserea, Dec 12 2017
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CROSSREFS
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Cf. A000001, A003277, A054395, A054396, A054397.
Sequence in context: A158765 A226741 A223078 * A193252 A223452 A015223
Adjacent sequences: A055558 A055559 A055560 * A055562 A055563 A055564
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KEYWORD
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nonn
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AUTHOR
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Christian G. Bower, May 25 2000; Nov 12 2003; Feb 17 2006
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STATUS
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approved
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