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A253569
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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.
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5
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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
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OFFSET
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1,1
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COMMENTS
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LINKS
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EXAMPLE
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290 = 2*5*29 is a member, because 2^2 < 5 and 5^2 < 29.
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MATHEMATICA
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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 *)
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PROG
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(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
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
(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)))))
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CROSSREFS
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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.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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