A228023 - OEIS

%I #19 Aug 05 2013 18:09:37

%S 1,20,50,117,200,242,325,500,578,605,650,800,968,1025,1058,1280,1445,

%T 1476,1682,1700,2312,2340,2600,2645,3200,3362,3757,3872,4205,4232,

%U 4352,4418,4693,5618,6728,6962,7514,8228,8405,8833,9248,9425,9472,10082,10400,11045,11849,12493

%N Primitive antiharmonic numbers.

%C Antiharmonic numbers (A020487) which are not the product of an antiharmonic number and a relatively prime square > 1. Apart from the first term, a subsequence of A227771 (antiharmonic numbers that are not squares).

%C Is this sequence infinite? It seems that 4n^2 <= a(n) <= 8n^2 for n > 1, and that a(n) ~ 6n^2 as n -> infinity--see A228036 for motivation.

%C The antiharmonic mean of the divisors of a(n) is A228024(n).

%H Charles R Greathouse IV, <a href="/A228023/b228023.txt">Table of n, a(n) for n = 1..10000</a>

%e 200 = 2^3 * 5^2 is antiharmonic (since sigma_2(200)/sigma(200) = 119 is an integer) but 2^3 is not antiharmonic, so 200 is in this sequence.

%e 180 = 2^2 * 3^2 * 5 is antiharmonic but 180/3^2 = 20 is also antiharmonic, so 180 is not in the sequence.

%o (PARI) isf(f)=denominator(prod(i=1,#f~,(f[i,1]^(f[i,2]+1)+1)/(f[i,1]+1)))==1

%o nosmaller(f,startAt)=for(i=startAt,#f~,if(f[i,2]%2==0&&f[i,2],return(nosmaller(f,i+1)&&!(f[i,2]=0)&&!isf(f)&&nosmaller(f,i+1))));1

%o is(n)=my(f);isf(f=factor(n))&&nosmaller(f,1)

%Y Cf. A020487, A227771, A228024, A228036.

%K nonn

%O 1,2

%A _Charles R Greathouse IV_ and _Jonathan Sondow_, Aug 03 2013