A379271 Composite numbers, k, whose prime factors, viewed on a log log scale, have a small standard deviation defined with respect to bigomega(k), as specified in the comments.

4, 6, 8, 9, 10, 12, 15, 16, 18, 20, 21, 24, 25, 27, 28, 30, 32, 33, 35, 36, 39, 40, 42, 44, 45, 48, 49, 50, 51, 52, 54, 55, 56, 57, 60, 63, 64, 65, 66, 68, 70, 72, 75, 76, 77, 78, 80, 81, 84, 85, 88, 90, 91, 92, 95, 96, 98, 99, 100, 102, 104, 105, 108, 110, 112, 114, 115, 116, 117, 119, 120, 121, 124, 125, 126, 128

Offset: 1

Author

Peter Munn, Feb 18 2025

Keywords

• nonn

Comments

Composite numbers k (written as a product of primes p_1 * p_2 * ... * p_m) such that s( {log(log(p_i)) : 1 <= i <= m} ) < s( {i : 1 <= i <= m} ), where s is standard deviation and m = bigomega(k).

Loosely described, these are numbers whose prime factors, including repetitions, are relatively close together. (Note we get the same criterion irrespective of whether s is sample standard deviation or population standard deviation.)

The author's intent is to divide the set of composite numbers into 2 parts whose asymptotic densities differ at most by a small factor. So his choice of criterion was guided by particular information relating to the statistics of prime factors of large numbers.

From Charles R Greathouse IV, May 19 2025: (Start)

For example, semiprimes p*q with p <= q are in this sequence if (and only if) q < p^e where e = 2.71... is the base of the natural logarithm.

For any m, there are finitely many primes p (perhaps none) such that p*m is in the sequence. (End)

Crossrefs

Subsequences: A251728, the composites in A253784, A380438.

Programs

Mathematica

Select[Select[Range[128], CompositeQ], Less @@ Map[StandardDeviation, Transpose@ MapIndexed[{Log@ Log[#1], First[#2]} &, Flatten[ConstantArray[#1, #2] & @@@ FactorInteger[#] ] ] ] &] (* Michael De Vlieger, May 04 2025 *)