A289691 List of composite numbers n such that the prime decomposition of n contains an initial product of prime powers greater than a later prime, but in which no proper divisor of n is in the sequence.

12, 30, 40, 45, 56, 63, 70, 105, 154, 165, 175, 176, 182, 195, 198, 208, 220, 231, 234, 260, 273, 275, 286, 297, 306, 325, 340, 351, 357, 374, 380, 385, 399, 418, 425

Offset: 1

Author

Richard Locke Peterson, Jul 09 2017

Keywords

• nonn

Comments

From Richard Locke Peterson, Aug 29 2017: (Start)

Definition: Let the prime factorization of n be n = p1^e1*...*pj^ej*p(j+1)^e(j+1)*...*pm^em, with the primes in ascending order and the ej > 0. If in n there exists a partial product p1^e1*...*pk^ek > p(k+1) for some p(k+1) in the factorization, and also such that no proper divisor of n is also in the sequence, then n is in the sequence.

Context: This sequence is a subsequence of A289484, and might be called "A289484 primes," although not primes in the usual sense (nor do they obey a unique factorization law). Every number in A289484 is a multiple of at least one number in this sequence, and if n is in this sequence, then n and every multiple of n is in A289484 although no multiple of n(except n itself) will be in this sequence.

Properties: If n is in the sequence, then no multiple of n is in it, except n itself. No primes or prime powers, nor any composite powers, are in the sequence. A number in the sequence that is squarefree must have at least three prime divisors. Rather than being closed under multiplication, this sequence is anticlosed: No product or power of numbers in it are in the sequence. This causes it to be the minimal sequence that generates A289484. (End)

Examples

			30 = 2*3*5 is in the sequence since 2*3 > 5 and no proper divisor of 30 satisfies having an initial product greater than a later prime. Counterexamples: 42 = 2*3*7 is not in the sequence because 2*3 < 7. 60 = 2^2*3*5 is also not in the sequence, because although 2^2 > 3 (and 2^2*3 > 5, for that matter), the proper divisor 30 is already in the sequence.
		

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A289484.

Programs

Mathematica

With[{S = Select[Range[10^3], Function[f, AnyTrue[Range[Length@ f - 1], Times @@ Map[#1^#2 & @@ # &, #1] > #2[[1, 1]] & @@ TakeDrop[f, #] &]]@ FactorInteger@ # &]}, Select[#, # > 0 &] &@ Table[Function[{d, n}, First[n] Boole@ NoneTrue[d, Divisible[First@ n, #] &]] @@ TakeDrop[Take[S, n], n - 1], {n, Length@ S}]] (* Michael De Vlieger, Aug 17 2017 *)

Extensions

New name from Richard Locke Peterson, Aug 29 2017