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Search: id:A051532
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| A051532 |
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The Abelian orders (or Abelian numbers): n such that every group of order n is Abelian. |
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+0
13
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| 1, 2, 3, 4, 5, 7, 9, 11, 13, 15, 17, 19, 23, 25, 29, 31, 33, 35, 37, 41, 43, 45, 47, 49, 51, 53, 59, 61, 65, 67, 69, 71, 73, 77, 79, 83, 85, 87, 89, 91, 95, 97, 99, 101, 103, 107, 109, 113, 115, 119, 121, 123, 127, 131, 133, 137, 139, 141, 143, 145, 149, 151, 153, 157, 159, 161
(list; graph; listen)
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OFFSET
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1,2
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COMMENT
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Except for a(2)=2 and a(4)=4, all of the terms in the sequence are odd. This is because of the existence of a non-Abelian dihedral group of order 2n for each n>2.
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REFERENCES
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J. Pakianathan and K. Shankar, Nilpotent numbers, Amer. Math. Monthly, 107 (Aug. 2000), 631-634.
W. R. Scott, Group Theory, Dover, 1987, page 217.
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LINKS
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T. D. Noe, Table of n, a(n) for n=1..1000
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FORMULA
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n must be cube-free and its prime divisors must satisfy certain congruences.
Let the prime factorization of n be p1^e1...pr^er. Then n is in this sequence if ei<3 for all i and pi^k does not equal 1 (mod pj) for all i and j and 1 <= k <= ei. - T. D. Noe (noe(AT)sspectra.com), Mar 25 2007
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EXAMPLE
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a(4)=4 because every group of order 4 is Abelian.
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CROSSREFS
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Cf. A003277, A064899, A060652
Sequence in context: A092755 A032515 A024926 this_sequence A135785 A008732 A130520
Adjacent sequences: A051529 A051530 A051531 this_sequence A051533 A051534 A051535
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KEYWORD
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nonn,nice,easy
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AUTHOR
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Des MacHale (stmt8011(AT)bureau.ucc.ie)
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