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

1,2

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).

Links

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

Example

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.

Prog

(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)

Crossrefs

Cf. A020487, A227771, A228024, A228036.

Sequence in context: A182462 A231293 A227771 * A241609 A331753 A345124

Adjacent sequences: A228020 A228021 A228022 * A228024 A228025 A228026

Keyword

nonn

Author

Charles R Greathouse IV and Jonathan Sondow, Aug 03 2013

Status

approved