A392985 Composite numbers, k, whose prime factors, viewed on a log log scale, have a large standard deviation defined with respect to bigomega(k), as specified in the comments.

14, 22, 26, 34, 38, 46, 58, 62, 69, 74, 82, 86, 87, 93, 94, 106, 111, 118, 122, 123, 129, 134, 141, 142, 146, 158, 159, 166, 177, 178, 183, 194, 201, 202, 206, 212, 213, 214, 218, 219, 226, 236, 237, 244, 249, 254, 262, 267, 268, 274, 278, 284, 291, 292, 298, 302

Offset

1,1

Comments

The complement of A379271 within the composite numbers.

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 far apart. (Note we get the same criterion irrespective of whether s is sample standard deviation or population standard deviation.)

The use of a log log scale relates to the spacing on this scale of factors of large integers. Taking the log twice of the median factors given in A281889, we get 0.094..., 0.665..., 1.803..., 2.775..., ~3.75, ... (noting that the first two values can be seen as affected by "noise" from the small terms 3 and 7).

Proper subset of A024619.

Proper subset of A080259.

The sequence is closed under application of A253550(.). So if we divide a term by its greatest prime factor and multiply by a larger prime we get another term.

Smallest powerful term is a(544653) = A001694(2684) = 1737124 = 1318^2 = 2^2 * 659^2. This term is A001597(1451) = A303606(652), a perfect power k^m with squarefree composite k and m > 1.

Smallest term also an Achilles number is A052486(811731) = 502479065288 = 2^3 * 250619^2.

The first terms divisible by p^k but not p^(k+1) for some prime p exist for all k >= 1, occur in order of k and with p = 2. For k = 1, 2, 3 these are a(1) = 14 = 2^1 * 7, a(36) = 212 = 2^2 * 53, a(22414) = 76696 = 2^3 * 9587. The first term, a(n), with k = 4, has n > 2^20.

The smallest term also in A386762 is 2^4 * prime(7505209077)^2 = 559245523686209129483536.

A term also in A383394 is: 2^6 * (2^A000043(13)-1)^4 = 2^6 * (2^521-1)^4 (659 decimal digits).

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..512, 3X horizontal and vertical exaggeration, with a color function showing m = 1 in black and m = 2 in red.

Michael De Vlieger, Plot prime(i)^m | a(n) at (x,y) = (n,i), n = 1..2048, with a color function showing m = 1 in black and m = 2 in red.

Mathematica

Select[Select[Range[360], CompositeQ], GreaterEqual @@ Map[StandardDeviation, Transpose@ MapIndexed[{Log@ Log[#1], First[#2]} &, Flatten[ConstantArray[#1, #2] & @@@ FactorInteger[#]]]] &]

Crossrefs

Cf. A001222, A002808.

See the comments for the relationships with A000043, A001597, A001694, A024619, A052486, A080259, A253550, A281889, A303606, A379271, A383394, A386762.

Sequence in context: A169804 A354811 A386317 * A092112 A306146 A318929

Adjacent sequences: A392981 A392982 A392984 * A392986 A392987 A392988

Keyword

nonn

Author

Peter Munn and Michael De Vlieger, Feb 14 2026

Status

approved