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A253784
Numbers which have no two successive prime factors (when sorted into monotonic order) where the latter prime factor would be greater than the square of the former.
6
1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 15, 16, 17, 18, 19, 21, 23, 24, 25, 27, 29, 30, 31, 32, 35, 36, 37, 41, 42, 43, 45, 47, 48, 49, 53, 54, 55, 59, 60, 61, 63, 64, 65, 67, 71, 72, 73, 75, 77, 79, 81, 83, 84, 85, 89, 90, 91, 95, 96, 97, 101, 103, 105, 107, 108, 109, 113, 115, 119, 120, 121, 125, 126, 127, 128, 131, 133, 135, 137, 139, 143, 144
OFFSET
1,2
COMMENTS
In other words, {1} together with primes and such 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 less than the square of the previous: (p_i)^2 > p_j, (p_j)^2 > p_k, etc.
Whenever gcd(a(i),a(j)) > 1, then a(i)*a(j) and lcm(a(i),a(j)) are also members of this sequence.
LINKS
EXAMPLE
1 is present as it has an empty prime factorization.
2 like all primes is present.
4 = 2*2 is present as 2^2 > 2.
9 = 3*3 is present as 3^2 > 3.
10 = 2*5 is NOT present, as 2^2 < 5.
30 = 2*3*5 is present, as 2^2 > 3 and 3^2 > 5.
PROG
(Scheme, with Antti Karttunen's IntSeq-library)
(define A253784 (MATCHING-POS 1 1 (lambda (n) (numbers-densely-distributed? (ifactor n)))))
(define (numbers-densely-distributed? lista) (cond ((null? lista) #t) ((null? (cdr lista)) #t) ((< (A000290 (car lista)) (cadr lista)) #f) (else (numbers-densely-distributed? (cdr lista)))))
CROSSREFS
Complement: A253785.
Subsequence: A251728 (semiprimes only).
Subsequence of A253567 and A251726 (a(n+1) differs from A251726(n) for the first time at n=23, where a(24) = 30, while A251726(23) = 31.)
Cf. A000290.
Sequence in context: A074779 A253567 A048197 * A342191 A251726 A362981
KEYWORD
nonn
AUTHOR
Antti Karttunen, Jan 16 2015
STATUS
approved