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.

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

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

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

Adjacent sequences: A253781 A253782 A253783 * A253785 A253786 A253787

Keyword

nonn

Author

Antti Karttunen, Jan 16 2015

Status

approved