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A282572
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Integers that are a product of Mersenne numbers A000225, (i.e., product of numbers of the form 2^n - 1).
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8
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1, 3, 7, 9, 15, 21, 27, 31, 45, 49, 63, 81, 93, 105, 127, 135, 147, 189, 217, 225, 243, 255, 279, 315, 343, 381, 405, 441, 465, 511, 567, 651, 675, 729, 735, 765, 837, 889, 945, 961, 1023, 1029, 1143, 1215, 1323, 1395, 1519, 1533, 1575, 1701, 1785, 1905, 1953, 2025, 2047, 2187, 2205, 2295, 2401
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OFFSET
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1,2
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COMMENTS
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Odd orders of finite abelian groups that appear as the group of units in a commutative ring (Chebolu and Lockridge, see A296241). - Jonathan Sondow, Dec 15 2017
Actually, the Chebolu and Lockridge paper states that this sequence gives all odd numbers that are possible numbers of units in a (commutative or non-commutative) ring (Ditor's theorem). Concretely, if k = (2^(e_1)-1)*(2^(e_2)-1)*...(2^(e_r)-1) is a term, let R = (F_2)^s X F_(2^(e_1)) X F_(2^(e_2)) X ... X F_(2^(e_r)) for s >= 0, then the number of units in R is k. - Jianing Song, Dec 23 2021
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LINKS
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EXAMPLE
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63 = 1*3^3*7, 81 = 1*3^4, 93 = 1*3*31, 105 = 1*7*15, 41013 = 1*3^3*7^2*31.
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MAPLE
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d:= 15: # for terms < 2^d
N:= 2^d:
S:= {1}:
for m from 2 to d do
r:= 2^m-1;
k:= ilog[r](N);
V:= S;
for i from 1 to k do
V:= select(`<`, map(`*`, V, r), N);
S:= S union V
od;
od:
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MATHEMATICA
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lmt = 2500; a = b = Array[2^# - 1 &, Floor@ Log2@ lmt]; k = 2; While[k < Length@ a, e = 1; While[e < Floor@ Log[ a[[k]], lmt], b = Union@ Join[b, Select[ a[[k]]^e*b, # < 1 + lmt &]]; e++]; k++]; b (* Robert G. Wilson v, Feb 23 2017 *)
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PROG
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(PARI) forstep(x=1, 1000000, 2, t=x; forstep(n=20, 2, -1, m=2^n-1; while(t%m==0, t=t\m)); if(t==1, print1(x, ", "))) \\ Dmitry Petukhov, Feb 23 2017
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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