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A323729
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Non-prime-powers k at which the variance of the first differences of the logarithms of the divisors of k, scaled by log(k), reaches a new minimum.
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0
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6, 10, 21, 28, 104, 115, 136, 329, 496, 2133, 2171, 6821, 8128, 24331, 32896, 50579, 79421, 103729, 226859, 357769, 704791, 1092521, 1224829, 2048129, 2247829, 2685341, 5177371, 6967489, 9393509, 11089121, 12648871, 13651441, 16974079, 25153171, 30663671
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OFFSET
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1,1
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COMMENTS
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For any positive integer k, each divisor d of k and its complement q=k/d have as their geometric mean the value sqrt(d*q) = sqrt(d*k/d) = sqrt(k), so log(d), log(sqrt(k)), and log(q) form an arithmetic progression; consequently, the logarithms of the divisors of k are distributed symmetrically about log(sqrt(k)) = log(k)/2.
If k is a prime power p^m (where p is prime and m >= 1), the divisors of k form the geometric progression {1, p, p^2, ..., p^m}, so the real-valued sequence consisting of their logarithms is the arithmetic progression {0, log(p), 2*log(p), ..., m*log(p)}, with constant difference log(p); if we divide that real-valued sequence by log(k), we get the rational-valued sequence {0, 1/m, 2/m, ..., 1} (an arithmetic progression with constant difference 1/m). If k is not a prime power, the differences between the logarithms of successive divisors of k will vary.
Consider the question: For which non-prime-powers k are the divisors of k distributed most evenly (on a logarithmic scale)? One way to quantify the evenness of their distribution is to scale the logarithms of the divisors by dividing them by log(k) (to fit them to the interval [0,1]) and compute the population variance of their first differences. The non-prime-powers k at which this variance reaches a new minimum are the terms of this integer sequence.
All terms in the sequence appear to be of the form p^m * q where p and q are prime and q is close to p^(m+1).
Of the first 35 terms, all but 9 are odd semiprimes of the form k = p*(p^2 - 2); cf. A240436. Such numbers have 4 divisors -- in ascending order, d_1 = 1, d_2 = p, d_3 = p^2 - 2, and d_4 = p*(p^2 - 2) -- and as k increases, the values log(d_i)/log(k) approach {0, 1/3, 2/3, 1}.
Conjectures:
1. Other than A240436(1)=4 (a prime power), A240436 is a subsequence of this sequence.
2. Only nine terms in this sequence are not semiprimes of the form p*(p^2 - 2):
10 = 2 * 5
104 = 2^3 * 13
136 = 2^3 * 17
2133 = 3^3 * 79
32896 = 2^7 * 257
and the first four perfect numbers (6, 28, 496, and 8128).
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LINKS
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EXAMPLE
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k = 115 = 5 * (5^2 - 2) = 5 * 23 has 4 divisors: 1, 5, 23, and 115. We have
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d | log(d)/log(k) | (mean = 1/3) | (diff - mean)^2
----+---------------+---------------+------------------
1 | 0.00000000000 | |
----+---------------+ 0.33919092389 | 0.000034311367177
5 | 0.33919092389 | |
----+---------------+ 0.32161815221 | 0.000137245468708
23 | 0.66080907611 | |
----+---------------+ 0.33919092389 | 0.000034311367177
115 | 1.00000000000 | |
----+---------------+---------------+------------------
sum: 0.000205868203062
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Population variance = 0.000205868203062 / 3
= 0.000068622734354
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The population variance 0.000068622734354 is smaller than that for any smaller value of k, so k=115 in the sequence.
The first several terms and their population variances are
k population variance
--- -------------------
6 0.00572866817332422
10 0.00208701124916037
21 0.00151420255078270
28 0.00025693510366524
104 0.00024474680031955
115 0.00006862273435404
136 0.00001864750547090
329 0.00001148749359549
496 0.00000258435797989
2133 0.00000130263831477
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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