A176797 -id:A176797 - cp's OEIS frontend

A176799 a(n) = possible values of A176797(m) in increasing order, where A176797(m) = antiharmonic means of divisors of antiharmonic numbers A020487.

1, 3, 7, 11, 13, 21, 35, 43, 61, 63, 77, 85, 91, 111, 119, 129, 147, 157, 171, 183, 185, 231, 245, 255, 273, 301, 313, 333, 343, 425, 441, 455, 471, 473, 481, 507, 521, 547, 559, 629, 671, 683, 741, 765, 777, 793, 813, 819, 833, 841, 845, 903, 931, 935, 1015, 1029, 1099, 1105, 1183, 1197, 1221

Offset: 1

Jaroslav Krizek, Apr 26 2010

nonn

From Robert Israel, Sep 05 2024: (Start)
According to A000203, sigma_1(m) <= (6/Pi^2)*m^(3/2) for m >= 12. Thus since sigma_2(m) > m^2, sigma_2(m)/sigma_1(m) > (Pi^2/6) * m^(1/2).
This would suggest that to find all terms <= K of this sequence we should look at sigma_2(m)/sigma_1(m) for m <= 36 * K^2/Pi^4.  But using the b-file for A004394 we may get a good upper bound for sigma_1(m)/m for m in this interval, resulting in a much smaller search region. (End)

Robert Israel, Table of n, a(n) for n = 1..1009

Cf. A000203, A004394, A020487, A176797, A327054.

# This uses the b-file for A004394
K:= 10000: # to get terms <= K
M:= 36 * K^2/Pi^4:
for i from 1 while A004394[i] < M do od:
r:= numtheory:-sigma(A004394[i])/A004394[i]:
V:= Vector(K):
for m from 1 to r*K do
  F:= numtheory:-divisors(m);
v:= add(d^2, d=F)/add(d,d=F);
if v::integer and v <= K and V[v] = 0 then V[v]:= m fi;
od:
select(v -> V[v] > 0, [$1..K]); # Robert Israel, Sep 05 2024

More terms from Robert Israel, Sep 05 2024

A176800 a(n) = all values of A176797(m) in increasing order, where A176797(m) = antiharmonic means of divisors of antiharmonic numbers A020487.

Original entry on oeis.org

1, 3, 7, 11, 13, 21, 21, 35, 43, 43, 61, 63, 77, 85, 91, 111, 119, 129, 147, 157, 171, 183, 185, 231, 245, 255, 255, 273, 301, 301

Offset: 1

Jaroslav Krizek, Apr 26 2010

nonn

A057660 a(n) = Sum_{k=1..n} n/gcd(n,k).

Original entry on oeis.org

1, 3, 7, 11, 21, 21, 43, 43, 61, 63, 111, 77, 157, 129, 147, 171, 273, 183, 343, 231, 301, 333, 507, 301, 521, 471, 547, 473, 813, 441, 931, 683, 777, 819, 903, 671, 1333, 1029, 1099, 903, 1641, 903, 1807, 1221, 1281, 1521, 2163, 1197, 2101, 1563, 1911, 1727

Offset: 1

Henry Gould, Oct 15 2000

Also sum of the orders of the elements in a cyclic group with n elements, i.e., row sums of A054531. - Avi Peretz (njk(AT)netvision.net.il), Mar 31 2001
Also inverse Moebius transform of EulerPhi(n^2), A002618.
Sequence is multiplicative with a(p^e) = (p^(2*e+1)+1)/(p+1). Example: a(10) = a(2)*a(5) = 3*21 = 63.
a(n) is the number of pairs (a, b) such that the equation ax = b is solvable in the ring (Zn, +, x). See the Mathematical Reflections link. - Michel Marcus, Jan 07 2017
From Jake Duzyk, Jun 06 2023: (Start)
These are the "contraharmonic means" of the improper divisors of square integers (inclusive of 1 and the square integer itself).
Permitting "Contraharmonic Divisor Numbers" to be defined analogously to Øystein Ore's Harmonic Divisor Numbers, the only numbers for which there exists an integer contraharmonic mean of the divisors are the square numbers, and a(n) is the n-th integer contraharmonic mean, expressible also as the sum of squares of divisors of n^2 divided by the sum of divisors of n^2. That is, a(n) = sigma_2(n^2)/sigma(n^2).
(a(n) = A001157(k)/A000203(k) where k is the n-th number such that A001157(k)/A000203(k) is an integer, i.e., k = n^2.)
This sequence is an analog of A001600 (Harmonic means of divisors of harmonic numbers) and A102187 (Arithmetic means of divisors of arithmetic numbers). (End)

David M. Burton, Elementary Number Theory, Allyn and Bacon Inc., Boston MA, 1976, p. 152.
H. W. Gould and Temba Shonhiwa, Functions of GCD's and LCM's, Indian J. Math. (Allahabad), Vol. 39, No. 1 (1997), pp. 11-35.
H. W. Gould and Temba Shonhiwa, A generalization of Cesaro's function and other results, Indian J. Math. (Allahabad), Vol. 39, No. 2 (1997), pp. 183-194.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
Habib Amiri and S. M. Jafarian Amiri, Sum of element orders on finite groups of the same order, J. Algebra Appl. 10 (2011), no. 2, 187--190. MR2795731 (2012d:20050)
Habib Amiri, S. M. Jafarian Amiri, and I. M. Isaacs, Sums of element orders in finite groups Comm. Algebra 37 (2009), no. 9, 2978--2980. MR2554185 (2010i:20022)
Miriam Mahannah El-Farrah, Expectation Numbers of Cyclic Groups, MS Thesis, Western Kentucky University, August 2015.
Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 154-5.
Walther Janous, Problem 10829, Amer. Math. Monthly, 107 (2000), p. 753.
Yadollah Marefat, Ali Iranmanesh, and Abolfazl Tehranian, On the sum of element orders of finite simple groups, J. Algebra Applications, 12 (2013), #1350026.
Mathematical Reflections, Solution to Problem U338, Issue 4, 2015, p 17.
Joachim von zur Gathen, Arnold Knopfmacher, Florian Luca, Lutz G. Lucht, and Igor E. Shparlinski, Average order of cyclic groups, J. Théorie Nombres Bordeaux 16 (1) (2004) 107-123.

Cf. A000010, A000203, A000290, A001157, A018804, A050873, A051193, A054522, A057661, A061255, A065764, A078747, A174405 (partial sums), A176797, A226512.

Cf. A308471.

a057660 n = sum $ map (div n) $ a050873_row n
-- Reinhard Zumkeller, Nov 25 2013

Table[ DivisorSigma[ 2, n^2 ] / DivisorSigma[ 1, n^2 ], {n, 1, 128} ]
Table[Total[Denominator[Range[n]/n]], {n, 55}] (* Alonso del Arte, Oct 07 2011 *)
f[p_, e_] := (p^(2*e + 1) + 1)/(p + 1); a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Nov 21 2020 *)

a(n)=if(n<1,0,sumdiv(n,d,d*eulerphi(d)))

a(n)=sumdivmult(n,d, eulerphi(d)*d) \\ Charles R Greathouse IV, Sep 09 2014

from math import gcd
def A057660(n): return sum(n//gcd(n,k) for k in range(1,n+1)) # Chai Wah Wu, Aug 24 2023

from math import prod
from sympy import factorint
def A057660(n): return prod((p**((e<<1)+1)+1)//(p+1) for p,e in factorint(n).items()) # Chai Wah Wu, Aug 05 2024

a(n) = Sum_{d|n} d*A000010(d) = Sum_{d|n} d*A054522(n,d), sum of d times phi(d) for all divisors d of n, where phi is Euler's phi function.
a(n) = sigma_2(n^2)/sigma_1(n^2) = A001157(A000290(n))/A000203(A000290(n)) = A001157(A000290(n))/A065764(n). - Labos Elemer, Nov 21 2001
a(n) = Sum_{d|n} A000010(d^2). - Enrique Pérez Herrero, Jul 12 2010
a(n) <= (n-1)*n + 1, with equality if and only if n is noncomposite. - Daniel Forgues, Apr 30 2013
G.f.: Sum_{n >= 1} n*phi(n)*x^n/(1 - x^n) = x + 3*x^2 + 7*x^3 + 11*x^4 + .... Dirichlet g.f.: sum {n >= 1} a(n)/n^s = zeta(s)*zeta(s-2)/zeta(s-1) for Re s > 3.  Cf. A078747 and A176797. - Peter Bala, Dec 30 2013
a(n) = Sum_{i=1..n} numerator(n/i). - Wesley Ivan Hurt, Feb 26 2017
L.g.f.: -log(Product_{k>=1} (1 - x^k)^phi(k)) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, May 21 2018
From Richard L. Ollerton, May 10 2021: (Start)
a(n) = Sum_{k=1..n} lcm(n,k)/k.
a(n) = Sum_{k=1..n} gcd(n,k)*phi(gcd(n,k))/phi(n/gcd(n,k)). (End)
From Vaclav Kotesovec, Jun 13 2021: (Start)
Sum_{k=1..n} a(k)/k ~ 3*zeta(3)*n^2/Pi^2.
Sum_{k=1..n} k^2/a(k) ~ A345294 * n.
Sum_{k=1..n} k*A000010(k)/a(k) ~ A345295 * n. (End)
Sum_{k=1..n} a(k) ~ 2*zeta(3)*n^3/Pi^2. - Vaclav Kotesovec, Jun 10 2023

More terms from James Sellers, Oct 16 2000

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).

A228024 Antiharmonic mean of the divisors of A228023(n) (the n-th primitive antiharmonic number).

Original entry on oeis.org

1, 13, 35, 85, 119, 185, 255, 313, 455, 481, 425, 455, 629, 841, 845, 741, 1183, 841, 1355, 1015, 1547, 1105, 1445, 2197, 1799, 2735, 3315, 2405, 3523, 2873, 2755, 3605, 4165, 4595, 4607, 5705, 5525, 5365, 7111, 7995, 5915, 7157, 6165, 8285, 5525, 9373, 10933, 11305, 7813

Offset: 1

Charles R Greathouse IV and Jonathan Sondow, Aug 03 2013

nonn

A228023 is the main entry for this sequence.

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

Cf. A228023, A227986, A176797.

a(n) = sigma_2(A228023(n))/sigma(A228023(n)).

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

Original entry on oeis.org

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)).

A327054 a(n) is the smallest number m such that the antiharmonic mean of the divisors is n, or 0 if no such m exists.

Original entry on oeis.org

1, 0, 4, 0, 0, 0, 9, 0, 0, 0, 16, 0, 20, 0, 0, 0, 0, 0, 0, 0, 25, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 50, 0, 0, 0, 0, 0, 0, 0, 49, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 81, 0, 100, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 144, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Jaroslav Krizek, Oct 06 2019

nonn

a(n) = the smallest number m such that sigma_2(n) / sigma_1(n) = A001157(m) / A000203(m) = n, or 0 if no such m exists.
Zeros occur if n is not in A176799.
See A000290, A091911 and A162538 for like sequences for geometric, arithmetic and harmonic means of the divisors.

			a(3) = 4 because 4 is the smallest number m with sigma_2(m) / sigma_1(m) = 3; sigma_2(4) / sigma_1(4) = 21 / 7 = 3.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A020487, A176797, A176799, A176800.

Cf. A000290, A004394, A091911, A162538.

A327054:=func; [A327054(n): n in[1..100]];

# This uses the b-file for A004394
# See comment at A176799
K:= 100: # to get terms <= K
M:= 36 * K^2/Pi^4:
for i from 1 while A004394[i] < M do od:
r:= numtheory:-sigma(A004394[i])/A004394[i]:
V:= Vector(K):
for m from 1 to r*K do
  F:= numtheory:-divisors(m);
  v:= add(d^2, d=F)/add(d, d=F);
  if v::integer and v <= K and V[v] = 0 then V[v]:= m fi;
od:
convert(V,list); # Robert Israel, Sep 05 2024

cp's OEIS Frontend

A176799 a(n) = possible values of A176797(m) in increasing order, where A176797(m) = antiharmonic means of divisors of antiharmonic numbers A020487.

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A176800 a(n) = all values of A176797(m) in increasing order, where A176797(m) = antiharmonic means of divisors of antiharmonic numbers A020487.

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A057660 a(n) = Sum_{k=1..n} n/gcd(n,k).

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A227771 Antiharmonic numbers that are not squares.

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A228024 Antiharmonic mean of the divisors of A228023(n) (the n-th primitive antiharmonic number).

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

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A327054 a(n) is the smallest number m such that the antiharmonic mean of the divisors is n, or 0 if no such m exists.

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