A240436 -id:A240436 - cp's OEIS frontend

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.

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

Offset: 1

Jon E. Schoenfield, Jan 29 2019

nonn

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).

			k = 115 = 5 * (5^2 - 2) = 5 * 23 has 4 divisors: 1, 5, 23, and 115. We have
                      |     first     |
                      |  differences  |
    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
.
                  Population variance = 0.000205868203062 / 3
                                      = 0.000068622734354
.
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

Cf. A000396, A240436.

cp's OEIS Frontend

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.

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