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

1,1

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)

The above theorem generalizes: for any m > 1 and any k, there are finitely many k-almost-primes j (perhaps none) such that j*m is in the sequence. - Peter Munn, Feb 25 2026

Links

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Michael De Vlieger, Plot prime(i)^m | a(n) at (x,y) = (n,i), n = 1..2048, 8X vertical exaggeration, with a color function showing m = 1 in black, m = 2 in red, m = 3 in orange, ..., m = 11 in magenta.

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 *)

Crossrefs

Complement of A008578 U A392985.

Subsequences: A251728, A380438; also composites in A051037, odd composites in A080682, composites in A253784.

Sequence in context: A066071 A306307 A355170 * A248010 A067013 A141607

Adjacent sequences: A379268 A379269 A379270 * A379272 A379273 A379274

Keyword

nonn

Author

Peter Munn, Feb 18 2025

Status

approved