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URL: https://oeis.org/A227771

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A227771
Antiharmonic numbers that are not squares.
7
20, 50, 117, 180, 200, 242, 325, 450, 468, 500, 578, 605, 650, 800, 968, 980, 1025, 1058, 1280, 1300, 1445, 1476, 1620, 1682, 1700, 1800, 1872, 2178, 2312, 2340, 2420, 2450, 2600, 2645, 2925, 3200, 3362, 3380, 3757, 3872, 4050, 4100, 4205, 4232, 4352, 4418, 4500, 4693, 5200
OFFSET
1,1
COMMENTS
Given prime factorization m = product (p_i^e_i), the antiharmonic (or contraharmonic) mean of the divisors of m is sigma_2(m)/sigma_1(m) = product (p_i^(e_i+1)+1)/(p_i+1). If this is an integer, then m is called antiharmonic.
All squares are trivially antiharmonic, since (p^(2*e+1)+1)/(p+1) = p^(2*e) - p^(2*e-1) + p^(2*e-2) - ... + 1 is an integer. Sequence gives the nontrivial antiharmonic numbers.
The antiharmonic means of their divisors are A227986.
Sequence is infinite, since if n is in the sequence and gcd(n, k) = 1 then nk^2 is also in the sequence. - Charles R Greathouse IV, Aug 02 2013
Removing such terms nk^2 leaves the primitive antiharmonic numbers A228023. - Jonathan Sondow, Aug 04 2013
REFERENCES
R. Guy, Unsolved Problems in Number Theory, B2 (see harmonic number).
LINKS
Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
FORMULA
A001157(a(n))/A000203(a(n)) = A227986(n).
EXAMPLE
sigma_2(20)/sigma_1(20) = (1^2 + 2^2 + 4^2 + 5^2 + 10^2 + 20^2)/(1 + 2 + 4 + 5 + 10 + 20) = 546/42 = 13 is an integer, 20 is not a square, and no smaller number has these properties, so a(1) = 20.
PROG
(PARI) is(n)=if(issquare(n), return(0)); my(f=factor(n)); denominator(prod(i=1, #f~, (f[i, 1]^(f[i, 2]+1)+1)/(f[i, 1]+1)))==1 \\ Charles R Greathouse IV, Aug 02 2013
CROSSREFS
KEYWORD
nonn
AUTHOR
Jonathan Sondow, Aug 02 2013
STATUS
approved