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

Author

Antti Karttunen, Jan 16 2015

Keywords

• nonn

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.

Examples

			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.
		

Links

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

Crossrefs

Complement: A253785.

Subsequences: A251726 (a(n+1) differs from A251726(n) for the first time at n=23, where a(24) = 30, while A251726(23) = 31), A251728 (semiprimes only).

Subsequence of A253567.

Cf. A000290.