A253569 - OEIS

%I #25 Jul 10 2023 10:43:33

%S 10,14,22,26,33,34,38,39,46,51,57,58,62,69,74,82,86,87,93,94,106,111,

%T 118,122,123,129,134,141,142,145,146,155,158,159,166,177,178,183,185,

%U 194,201,202,205,206,213,214,215,218,219,226,235,237,249,254,262,265,267,274,278,290

%N 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.

%C 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).

%H Antti Karttunen, <a href="/A253569/b253569.txt">Table of n, a(n) for n = 1..10000</a>

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

%t 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 *)

%o (Haskell)

%o a253569 n = a253569_list !! (n-1)

%o a253569_list = filter f [1..] where

%o                     f x = (p ^ 2 < a020639 q) && (a010051' q == 1 || f q)

%o                           where q = div x p; p = a020639 x

%o -- _Antti Karttunen_ after _Reinhard Zumkeller_'s code for A138511, Jan 09 2015

%o a253569 n = a253569_list !! (n-1)

%o a253569_list = filter (not . f''') a002808_list where

%o    f''' x = p ^ 2 > a020639 q || (a010051 q == 0 && f''' q)

%o             where q = div x p; p = a020639 x

%o -- _Reinhard Zumkeller_, Jan 12 2015

%o (Scheme, with Antti Karttunen's IntSeq-library)

%o (define A253569 (MATCHING-POS 1 1 (lambda (n) (and (> (A001222 n) 1) (numbers-sparsely-distributed? (ifactor n))))))

%o (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)))))

%o ;; _Antti Karttunen_, Jan 16 2015

%Y Complement: A253567.

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

%Y 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.

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

%K nonn

%O 1,1

%A _Antti Karttunen_ & _Reinhard Zumkeller_, Jan 09 2015