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Search: id:A118642
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| A118642 |
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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. |
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1
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| 16, 27, 32, 48, 54, 64, 72, 80, 81, 96, 100, 108, 112, 125, 128, 135, 144, 147, 160, 162, 176, 189, 192, 200, 208, 216, 224, 240, 243, 250
(list; graph; listen)
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OFFSET
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1,1
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COMMENT
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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.
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REFERENCES
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F. J. Budden, The Fascination of Groups, Cambridge University Press, 1969.
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EXAMPLE
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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.
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CROSSREFS
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Adjacent sequences: A118639 A118640 A118641 this_sequence A118643 A118644 A118645
Sequence in context: A038353 A043126 A043906 this_sequence A088247 A032610 A067650
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KEYWORD
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hard,nonn
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AUTHOR
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Des MacHale and Robert Heffernan (d.machale(AT)ucc.ie), May 10 2006
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