A227771 -id:A227771 - cp's OEIS frontend

A227986 Antiharmonic mean of the divisors of A227771(n) (the n-th antiharmonic number that is not a square).

13, 35, 85, 91, 119, 185, 255, 245, 255, 313, 455, 481, 425, 455, 629, 559, 841, 845, 741, 765, 1183, 841, 793, 1355, 1015, 833, 935, 1295, 1547, 1105, 1443, 1505, 1445, 2197, 1785, 1799, 2735, 2041, 3315, 2405, 2135, 2523, 3523, 2873, 2755, 3605, 2191, 4165

Offset: 1

Jonathan Sondow, Aug 02 2013

nonn

For comments, references, example, and crossrefs, see A227771.

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

Cf. A227771, A176797.

a(n) = sigma_2(A227771(n))/sigma_1(A227771(n)).

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

Original entry on oeis.org

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

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

A335389 Numbers k such that k and k+1 are both antiharmonic numbers (A020487).

Original entry on oeis.org

49, 324, 1024, 1444, 1681, 2600, 9800, 265225, 332928, 379456, 421200, 1940449, 4198400, 4293184, 4739328, 8346320, 11309768, 27050400, 65918161, 203694425, 384199200, 418488849, 546717924, 2239277041, 2687489280, 4866742025, 5783450400, 6933892900, 7725003664

Offset: 1

Amiram Eldar, Jun 04 2020

nonn

Terms of this sequence k such that k and k+1 are both nonsquares (A227771) are 203694425, 4866742025, ...

Can two consecutive numbers be both primitive antiharmonic numbers (A228023)? Numbers k such that k and k+2 are both primitive antiharmonic numbers exist - the first two are 38246258 and 344321280.

			49 is a term since both 49 and 50 are antiharmonic: sigma_2(49)/sigma(49) = 43 and sigma_2(50)/sigma(50) = 35 are both integers.

Cf. A000203, A001157, A020487, A227771, A228023.

antihQ[n_] := Divisible[DivisorSigma[2, n], DivisorSigma[1, n]]; seq = {}; q1 = antihQ[1];  Do[q2 = antihQ[n]; If[q1 && q2, AppendTo[seq, n - 1]]; q1 = q2, {n, 2, 2 * 10^6}]; seq

A374170 a(n) is the least nonsquare k such that sigma_n(k) divides sigma_2n(k).

Original entry on oeis.org

20, 20, 6050, 7203

Offset: 1

Mohammed Yaseen, Jun 30 2024

a(1) = A227771(1); a(2) = A046871(5).
a(5) > 10^9 if it exists.
a(6) = 17328, a(7) = 50, a(13) = 761378.

Cf. A020487, A227771, A046871.

a(n) = my(k=1, f=factor(k)); while (issquare(k) || (sigma(f, 2*n) % sigma(f, n)), f=factor(k++)); k; \\ Michel Marcus, Jun 30 2024

cp's OEIS Frontend

A227986 Antiharmonic mean of the divisors of A227771(n) (the n-th antiharmonic number that is not a square).

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

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A228023 Primitive antiharmonic numbers.

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

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A335389 Numbers k such that k and k+1 are both antiharmonic numbers (A020487).

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Mathematica

A374170 a(n) is the least nonsquare k such that sigma_n(k) divides sigma_2n(k).

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