A256431 - OEIS

%I #45 Dec 10 2015 15:12:32

%S 1,4,18,50,294,450,578,1014,1210,2166,5202,7350,8214,10890,14450,

%T 16810,25350,28830,31974,35322,49686,54150,56454,58190,71286,77658,

%U 84966,102010,106134,111630,130050,132098,151290,159414,177870,205350,219102,223494,293046,349690,366054,402486,486798,523710,536298,613470

%N Numbers A055744(n) such that for any k < n, A055744(k) and A055744(n) do not have all their prime factors in common.

%C Subsequence of A055744.

%C These numbers can be used to find terms from A055744. For instance, 294 = 2 * 3 * 7^2, so for all a, b, c >= 0, all numbers of the form 294 * 2^a * 3^b * 7^c are in A055744.

%C No two terms in this sequence can be used to find the same term from A055744 with this method, as every term in this sequence has a different set of distinct prime factors. This is because any two numbers that do not have the same set of distinct prime factors are different, by the fundamental theorem of arithmetic.

%C Also, a complete list of terms of this sequence up to n enables the computation of all terms of A055744 up to at least n.

%C The squarefree kernel (or radical, see A007947) of the terms of this sequence is different from the radical of all lesser terms of A055744. - _Michel Marcus_, Aug 06 2015

%H David A. Corneth, <a href="/A256431/b256431.txt">Table of n, a(n) for n = 1..2553</a> (all < 10^11)

%e a(3) = 18 is in A055744 and it is the smallest number in A055744 having prime factors 2 and 3.

%e a(6) = 450 is in A055744 and it is divisible by a(3) = 18 and by a(4) = 50 but it is the smallest element in A055744 having prime factors 2, 3 and 5.

%o (PARI) isA055744(n)=my(f=factor(n)); f[,1]==factor(eulerphi(f))[,1]

%o is(n)=my(f=factor(n),r=factorback(f[,1])); for(i=1,#f~,f[i,2]--); sumdiv(f,d,isA055744(d*r))==1 \\ _Charles R Greathouse IV_, May 26 2015

%Y Cf. A007947, A055744, A256430.

%K nonn

%O 1,2

%A _David A. Corneth_, Mar 28 2015