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A323729 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.

%I A323729 #7 Jan 30 2019 03:19:33
%S A323729 6,10,21,28,104,115,136,329,496,2133,2171,6821,8128,24331,32896,50579,
%T A323729 79421,103729,226859,357769,704791,1092521,1224829,2048129,2247829,
%U A323729 2685341,5177371,6967489,9393509,11089121,12648871,13651441,16974079,25153171,30663671
%N A323729 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.
%C A323729 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.
%C A323729 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.
%C A323729 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.
%C A323729 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).
%C A323729 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}.
%C A323729 Conjectures:
%C A323729 1. Other than A240436(1)=4 (a prime power), A240436 is a subsequence of this sequence.
%C A323729 2. Only nine terms in this sequence are not semiprimes of the form p*(p^2 - 2):
%C A323729      10 = 2   *   5
%C A323729     104 = 2^3 *  13
%C A323729     136 = 2^3 *  17
%C A323729    2133 = 3^3 *  79
%C A323729   32896 = 2^7 * 257
%C A323729 and the first four perfect numbers (6, 28, 496, and 8128).
%e A323729 k = 115 = 5 * (5^2 - 2) = 5 * 23 has 4 divisors: 1, 5, 23, and 115. We have
%e A323729                       |     first     |
%e A323729                       |  differences  |
%e A323729     d | log(d)/log(k) |  (mean = 1/3) |  (diff - mean)^2
%e A323729   ----+---------------+---------------+------------------
%e A323729     1 | 0.00000000000 |               |
%e A323729   ----+---------------+ 0.33919092389 | 0.000034311367177
%e A323729     5 | 0.33919092389 |               |
%e A323729   ----+---------------+ 0.32161815221 | 0.000137245468708
%e A323729    23 | 0.66080907611 |               |
%e A323729   ----+---------------+ 0.33919092389 | 0.000034311367177
%e A323729   115 | 1.00000000000 |               |
%e A323729   ----+---------------+---------------+------------------
%e A323729                                    sum: 0.000205868203062
%e A323729 .
%e A323729                   Population variance = 0.000205868203062 / 3
%e A323729                                       = 0.000068622734354
%e A323729 .
%e A323729 The population variance 0.000068622734354 is smaller than that for any smaller value of k, so k=115 in the sequence.
%e A323729 The first several terms and their population variances are
%e A323729      k  population variance
%e A323729    ---  -------------------
%e A323729      6  0.00572866817332422
%e A323729     10  0.00208701124916037
%e A323729     21  0.00151420255078270
%e A323729     28  0.00025693510366524
%e A323729    104  0.00024474680031955
%e A323729    115  0.00006862273435404
%e A323729    136  0.00001864750547090
%e A323729    329  0.00001148749359549
%e A323729    496  0.00000258435797989
%e A323729   2133  0.00000130263831477
%Y A323729 Cf. A000396, A240436.
%K A323729 nonn
%O A323729 1,1
%A A323729 _Jon E. Schoenfield_, Jan 29 2019