A228023 Primitive antiharmonic numbers.

1, 20, 50, 117, 200, 242, 325, 500, 578, 605, 650, 800, 968, 1025, 1058, 1280, 1445, 1476, 1682, 1700, 2312, 2340, 2600, 2645, 3200, 3362, 3757, 3872, 4205, 4232, 4352, 4418, 4693, 5618, 6728, 6962, 7514, 8228, 8405, 8833, 9248, 9425, 9472, 10082, 10400, 11045, 11849, 12493

Offset: 1

Author

Charles R Greathouse IV and Jonathan Sondow, Aug 03 2013

Keywords

• nonn

Comments

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).

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.

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

Examples

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

Links

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

Crossrefs

Cf. A020487, A227771, A228024, A228036.

Programs

PARI

isf(f)=denominator(prod(i=1,#f~,(f[i,1]^(f[i,2]+1)+1)/(f[i,1]+1)))==1
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
is(n)=my(f);isf(f=factor(n))&&nosmaller(f,1)