A228024 -id:A228024 - cp's OEIS frontend

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

David W. Wilson

nonn

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

			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

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.

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

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

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

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

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

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

# 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

A227771 Antiharmonic numbers that are not squares.

Original entry on oeis.org

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

Jonathan Sondow, Aug 02 2013

nonn

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

			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.

R. Guy, Unsolved Problems in Number Theory, B2 (see harmonic number).

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

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

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

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

A228023 Primitive antiharmonic numbers.

Original entry on oeis.org

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

Charles R Greathouse IV and Jonathan Sondow, Aug 03 2013

nonn

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).

			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.

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

Cf. A020487, A227771, A228024, A228036.

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)

A228036 (10^n)-th primitive antiharmonic number.

Original entry on oeis.org

1, 605, 51005, 5837732, 599407380, 60462121402

Offset: 0

Jonathan Sondow and Charles R Greathouse IV, Aug 04 2013

nonn

We conjecture that lim_{n->oo} a(n)/100^n = lim_{n->oo} A228023(n)/n^2 = 6. This is supported by the values a(n)/(10^n)^2 = 6.05, 5.10, 5.84, 5.99, 6.05 for n = 1..5, as well as by the values of A228023(n)/n^2.

Cf. A020487, A227771, A228023, A228024.

a(n) = A228023(10^n).

a(5) from Charles R Greathouse IV, Sep 03 2013

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

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A227771 Antiharmonic numbers that are not squares.

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A228023 Primitive antiharmonic numbers.

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A228036 (10^n)-th primitive antiharmonic number.

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