A253784 Numbers which have no two successive prime factors (when sorted into nondecreasing order) where the latter prime factor is 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 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, so has no two prime factors to be the "two successive prime factors" in the definition.
2 (like all primes) is present for the same general reason.
10 = 2*5 is NOT present, as 2^2 < 5.
15 = 3*5 is present, as 3^2 > 5.
25 = 5*5 (like all squares of primes) is present, as 5^2 > 5.
30 = 2*3*5 is present, as 2^2 > 3 and 3^2 > 5.
66 = 2*3*11 is NOT present, as 3^2 < 11.

Mathematica

Select[Range[144], If[PrimeNu[#] < 2, True, AllTrue[Partition[FactorInteger[#][[;; , 1]], 2, 1], #1^2 > #2 & @@ # &]] &] (* Michael De Vlieger, Feb 15 2026 *)

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.

Subsequences: A000961, 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), A380995.

Subsequence of A253567, A008578 U A379271.

Sequence in context: A074779 A253567 A048197 * A342191 A251726 A362981

Adjacent sequences: A253781 A253782 A253783 * A253785 A253786 A253787

Keyword

nonn,easy

Author

Antti Karttunen, Jan 16 2015

Status

approved