A020487 Antiharmonic numbers: numbers k such that sigma_1(k) divides sigma_2(k).

1, 4, 9, 16, 20, 25, 36, 49, 50, 64, 81, 100, 117, 121, 144, 169, 180, 196, 200, 225, 242, 256, 289, 324, 325, 361, 400, 441, 450, 468, 484, 500, 529, 576, 578, 605, 625, 650, 676, 729, 784, 800, 841, 900, 961, 968, 980, 1024, 1025, 1058, 1089, 1156, 1225, 1280, 1296

Offset: 1

Author

David W. Wilson

Keywords

• nonn

Comments

Numbers k such that antiharmonic mean of divisors of k is an integer. Antiharmonic mean of divisors of number m = Product (p_i^e_i) is A001157(m)/A000203(m) = Product ((p_i^(e_i+1)+1)/(p_i+1)). So a(n) = k, for some n, if A001157(k)/A000203(k) is an integer. - Jaroslav Krizek, Mar 09 2009

Squares are antiharmonic, since (p^(2*e+1)+1)/(p+1) = p^(2*e) - p^(2*e-1) + p^(2*e-2) - ... + 1 is an integer. The nonsquare antiharmonic numbers are A227771. They include the primitive antiharmonic numbers A228023, except for its first term. - Jonathan Sondow, Aug 02 2013

Sequence is infinite, see A227771. - Charles R Greathouse IV, Sep 02 2013

The term "antiharmonic" is also known as "contraharmonic". - Pahikkala Jussi, Dec 11 2013

Examples

			a(3) = 9 = 3^2; antiharmonic mean of divisors of 9 is (3^(2+1) + 1)/(3 + 1) = 7; 7 is an integer. - _Jaroslav Krizek_, Mar 09 2009
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000 (term 1..1000 from Paolo P. Lava)
• Marco Abrate, Stefano Barbero, Umberto Cerruti, and Nadir Murru, The Biharmonic mean, Mathematical Reports, Vol. 18(68), No. 4 (2016), pp. 483-495, preprint, arXiv:1601.03081 [math.NT], 2016.

Crossrefs

Cf. A001157, A000203, A227771, A228023, A228024, A228036.

Programs

Haskell

a020487 n = a020487_list !! (n-1)
a020487_list = filter (\x -> a001157 x `mod` a000203 x == 0) [1..]
-- Reinhard Zumkeller, Jan 21 2014

Magma

[n: n in [1..1300] | IsZero(DivisorSigma(2,n) mod DivisorSigma(1,n))]; // Bruno Berselli, Apr 10 2013

Mathematica

Select[Range[2000], Divisible[DivisorSigma[2, #], DivisorSigma[1, #]]&] (* Jean-François Alcover, Nov 14 2017 *)

PARI

is(n)=sigma(n,2)%sigma(n)==0 \\ Charles R Greathouse IV, Jul 02 2013

Python

from sympy import divisor_sigma
def ok(n): return divisor_sigma(n, 2)%divisor_sigma(n, 1) == 0
print([k for k in range(1, 1300) if ok(k)]) # Michael S. Branicky, Feb 25 2024

Python

# faster for producing initial segment of sequence
from math import prod
from sympy import factorint
def ok(n):
    f = factorint(n)
    sigma1 = prod((p**(  e+1)-1)//(p-1)    for p, e in f.items())
    sigma2 = prod((p**(2*e+2)-1)//(p**2-1) for p, e in f.items())
    return sigma2%sigma1 ==  0
print([k for k in range(1, 1300) if ok(k)]) # Michael S. Branicky, Feb 25 2024