A253569 Composite numbers n = p_i * p_j * p_k * ... * p_u, p_i <= p_j <= p_k <= ... <= p_u, where each successive prime factor (when sorted into a nondecreasing order) is greater than the square of the previous: (p_i)^2 < p_j, (p_j)^2 < p_k, etc.

10, 14, 22, 26, 33, 34, 38, 39, 46, 51, 57, 58, 62, 69, 74, 82, 86, 87, 93, 94, 106, 111, 118, 122, 123, 129, 134, 141, 142, 145, 146, 155, 158, 159, 166, 177, 178, 183, 185, 194, 201, 202, 205, 206, 213, 214, 215, 218, 219, 226, 235, 237, 249, 254, 262, 265, 267, 274, 278, 290

Offset: 1

Author

Antti Karttunen & Reinhard Zumkeller, Jan 09 2015

Keywords

• nonn

Comments

Numbers n = A020639(n) * A014673(n) * A054576(n), for which A020639(n)^2 < A014673(n) and either A054576(n) = 1 or A032742(n) satisfies the same condition (is the term of this sequence).

Examples

			290 = 2*5*29 is a member, because 2^2 < 5 and 5^2 < 29.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..10000

Crossrefs

Complement: A253567.

Subsequence of A002808, A005117, A088381, A251727, A245729 and A253785.

A138511 is a subsequence, from which this sequence differs for the first time at n=60, where A138511(60) = 291, while here a(60) = 290.

Cf. A000290, A001222, A020639, A032742, A014673, A054576.

Programs

Haskell

a253569 n = a253569_list !! (n-1)
a253569_list = filter f [1..] where
                    f x = (p ^ 2 < a020639 q) && (a010051' q == 1 || f q)
                          where q = div x p; p = a020639 x
-- Antti Karttunen after Reinhard Zumkeller's code for A138511, Jan 09 2015
a253569 n = a253569_list !! (n-1)
a253569_list = filter (not . f''') a002808_list where
   f''' x = p ^ 2 > a020639 q || (a010051 q == 0 && f''' q)
            where q = div x p; p = a020639 x
-- Reinhard Zumkeller, Jan 12 2015
(Scheme, with Antti Karttunen's IntSeq-library)
(define A253569 (MATCHING-POS 1 1 (lambda (n) (and (> (A001222 n) 1) (numbers-sparsely-distributed? (ifactor n))))))
(define (numbers-sparsely-distributed? lista) (cond ((null? lista) #t) ((null? (cdr lista)) #t) ((> (A000290 (car lista)) (cadr lista)) #f) (else (numbers-sparsely-distributed? (cdr lista)))))
;; Antti Karttunen, Jan 16 2015

Mathematica

cnQ[n_]:=CompositeQ[n]&&Union[Boole[#[[2]]>#[[1]]^2&/@Partition[Flatten[Table[ #[[1]], #[[2]]]&/@FactorInteger[n]],2,1]]]=={1}; Select[Range[300],cnQ] (* Harvey P. Dale, Jul 10 2023 *)