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

1,1

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

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

Sequence in context: A007589 A182462 A231293 * A228023 A241609 A331753

Adjacent sequences: A227768 A227769 A227770 * A227772 A227773 A227774

Keyword

nonn

Author

Jonathan Sondow, Aug 02 2013

Status

approved