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A226476
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Numbers n with the property that, if tau(n) = k = number of divisors of n, and the d(i) are the divisors [arranged in increasing order], then the sum 1/d(k) + 1/d(k-1) + 1/d(k-2) + ... + 1/d(q) is an integer for some q.
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1
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1, 6, 24, 28, 120, 496, 672, 2016, 4320, 4680, 8128, 8190, 26208, 30240, 32760, 42336, 45864, 392448, 523776, 714240, 1571328, 2178540, 8910720, 17428320, 20427264, 23569920, 29795040, 33550336, 34369920, 45532800, 61900800
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OFFSET
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1,2
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COMMENTS
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By convention, for n = 1, a(1) = 1 with q = 1.
The corresponding pairs (tau(n), q) are (1, 1), (4, 2), (8, 3), (6, 2), (16, 2), (10, 2), (24, 2), (36, 6), (48, 3), (48, 3), (14, 2), (48, 6), (72, 3), (96, 2), (96, 2), (72, 7), (72, 7), (72, 5), (80, 2), (120, 8), (120, 6), (216, 2), (384, 3), (432, 3), (240, 3), (320, 2), (360, 5), (26, 2), (384, 5), (384, 2), (288, 9).
Properties of this sequence:
q = 2 if n = 1, 6, 28, 120, 496, 672, 8128, ... is a multiply-perfect number (see A007691 where it is conjectured that this sequence is infinite), which would imply that this sequence is also infinite because A007691 is a subsequence.
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LINKS
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EXAMPLE
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24 is in the sequence because the divisors of 24 are 1, 2, 3, 4, 6, 8, 12, 24, and the sum 1/24 + 1/12 + 1/8 + 1/6 + 1/4 + 1/3 = 1.
28 is in the sequence because 28 is a multiply-perfect number: the divisors are 1, 2, 4, 7, 14, 28, and the sum of the reciprocals of all the divisors is 1/28 + 1/14 + 1/7 + 1/4 + 1/2 + 1 = 2.
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MAPLE
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with(numtheory): for n from 1 to 10000000 do:x:=divisors(n):n1:=nops(x):s:=0:ii:=0:for q from n1 by -1 to 1 while(ii=0) do:s:=s+1/x[q]:if s=floor(s) then ii:=1: printf(`%d, `, n):else fi:od:od:
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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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