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%I #14 Feb 03 2021 01:00:33
%S 16,27,32,48,54,64,72,80,81,96,100,108,112,125,128,135,144,147,160,
%T 162,176,189,192,200,208,216,224,240,243,250
%N Two finite groups are conformal if they have the same number of elements of each order. A natural number n is said to be a conformal order if there exist two conformal groups of order n which are not isomorphic to each other. The sequence lists the conformal orders.
%C Since a(1)= 16 and p^3 is in the sequence for any odd prime p, by taking direct products with cyclic groups we see that n belongs to the sequence if either 16 or p^3 divides n for an odd prime p. However, 72 and 147, which are not of this form, both belong to the sequence. Also, every multiple of each term in the sequence is also a term of the sequence. Conformality of groups is an equivalence relation but there seem to be virtually no known conformality invariants other than group order.
%D F. J. Budden, The Fascination of Groups, Cambridge University Press, 1969.
%H James McCarron, <a href="/A118642/b118642.txt">Table of n, a(n) for n = 1..312</a>
%e a(2)= 27 because there exist two non-isomorphic groups of order 27 each of which has one element of order one and twenty-six elements of order three.
%K hard,more,nonn
%O 1,1
%A _Des MacHale_ and _Bob Heffernan_, May 10 2006
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