A227771 Antiharmonic numbers that are not squares.

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

Author

Jonathan Sondow, Aug 02 2013

Keywords

• nonn

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

Examples

			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.
		

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

Crossrefs

Subsequence of A020487. Cf. A000203, A001157, A176797, A227771, A227986, A228023, A228024, A228036.

Programs

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

Formula

A001157(a(n))/A000203(a(n)) = A227986(n).