A289691 - OEIS

%I #30 Apr 15 2018 14:26:50

%S 12,30,40,45,56,63,70,105,154,165,175,176,182,195,198,208,220,231,234,

%T 260,273,275,286,297,306,325,340,351,357,374,380,385,399,418,425

%N 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.

%C From _Richard Locke Peterson_, Aug 29 2017: (Start)

%C 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.

%C 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.

%C 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)

%H Charles R Greathouse IV, <a href="/A289691/b289691.txt">Table of n, a(n) for n = 1..10000</a>

%e 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.

%t 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 *)

%Y Cf. A289484.

%K nonn

%O 1,1

%A _Richard Locke Peterson_, Jul 09 2017

%E New name from _Richard Locke Peterson_, Aug 29 2017