A020487 -id:A020487 - cp's OEIS frontend

A176797 Antiharmonic means of divisors of antiharmonic numbers.

1, 3, 7, 11, 13, 21, 21, 43, 35, 43, 61, 63, 85, 111, 77, 157, 91, 129, 119, 147, 185, 171, 273, 183, 255, 343, 231, 301, 245, 255, 333, 313, 507, 301, 455, 481, 521, 425, 471, 547, 473, 455, 813, 441, 931, 629, 559

Offset: 1

Jaroslav Krizek, Apr 26 2010

Values of sigma_2(n) / sigma(n) corresponding to terms of A020487(n), where sigma_2(n) is the sum of squares of divisors of n (A001157) and sigma(n) is the sum of the divisors of n (A000203).

			For n = 4; a(4) = 11 because A020487(4) = 16; A001157(16) = 341, A000203(n) = 31, antiharmonic means of divisors of 16 = 341 / 31 = 11.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Select[DivisorSigma[2, #]/DivisorSigma[1, #] & /@ Range[1000], IntegerQ] (* Amiram Eldar, Sep 18 2019 *)

a(n) = (A001157(A020487(n))) / (A000203(A020487(n))).

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).

A046871 Numbers k such that sigma_2(k) divides sigma_4(k).

Original entry on oeis.org

1, 4, 9, 16, 20, 25, 36, 48, 49, 64, 81, 100, 121, 144, 162, 169, 180, 196, 225, 245, 256, 289, 324, 361, 400, 432, 441, 484, 500, 529, 576, 605, 625, 648, 676, 729, 784, 841, 900, 931, 961, 980, 1024, 1089, 1156, 1200, 1225, 1280, 1296, 1369, 1444, 1521

Offset: 1

Labos Elemer

nonn

sigma_2(k) is the sum of the squares of the divisors of k (A001157).

sigma_4(k) is the sum of the 4th powers of the divisors of k (A001159).

			k = a(8) = 48 of which divisor power sums for powers 0, 1, 2, 3, 4 are 10, 124, 3410, 131068, 5732210, respectively. Here sigma_2(k) = 3410 and sigma_4(k) = 3410*1681.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Paolo P. Lava)

Cf. A001157, A001159, A020487, A020486, A046841.

[n: n in [1..1600] | IsZero(DivisorSigma(4,n) mod DivisorSigma(2,n))]; // Bruno Berselli, Apr 10 2013

Select[Range@ 1600, Divisible[DivisorSigma[4, #], DivisorSigma[2, #]] &] (* Michael De Vlieger, May 20 2017 *)

isok(n) = !(sigma(n, 4) % sigma(n, 2)); \\ Michel Marcus, May 21 2017

A144695 Numbers n such that sigma_1(n)/sigma_0(n) = c^2, c an integer.

Original entry on oeis.org

1, 7, 17, 22, 30, 31, 71, 94, 97, 115, 119, 127, 138, 154, 164, 165, 199, 210, 214, 217, 241, 260, 265, 318, 337, 343, 374, 382, 449, 497, 510, 513, 517, 527, 577, 647, 658, 668, 679, 682, 705, 745, 759, 805, 862, 881, 889, 894, 930, 966, 967, 996, 1102, 1125

Offset: 1

Ctibor O. Zizka, Sep 19 2008

A000203(n)/A000005(n) = c^2. Generalized sigma-sequences are sequences of numbers n for which sigma_r(n)/sigma_s(n) = c^t . Sigma_i(n) denotes sum of i-th powers of divisors of n; c,r,s,t positive integers. This sequence has r=1,s=0,t=2, sequence A003601 has r=1,s=0,t=1, sequence {1,21,53,85,102,110,127,217,431,....} has r=1,s=0,t=3, sequence A020487 has r=2,s=1,t=1, sequence A020486 has r=2,s=0,t=1, sequence A140480 has r=2,s=0,t=2.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Divisor function

Cf. A000005, A000203, A140480, A003601, A020487, A020486.

A000005 := proc(n) numtheory[tau](n) ; end: A000203 := proc(n) numtheory[sigma](n) ; end: isA144695 := proc(n) local s ; s := A000005(n) ; if s <> 0 then issqr(A000203(n)/s) ; else false ; fi; end: for n from 1 to 5000 do if isA144695(n) then printf("%d,",n) ; fi; od: # R. J. Mathar, Sep 20 2008

Select[Range[1125], IntegerQ @ Sqrt[DivisorSigma[1, #]/DivisorSigma[0, #]] &] (* Amiram Eldar, Nov 20 2019 *)

isok(m) = my(f=factor(m), q=sigma(f)/numdiv(f)); issquare(q) && !frac(q); \\ Michel Marcus, Mar 15 2022

More terms from R. J. Mathar, Sep 20 2008

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)

A158274 Numerators of antiharmonic means of divisors of n.

Original entry on oeis.org

1, 5, 5, 3, 13, 25, 25, 17, 7, 65, 61, 15, 85, 125, 65, 11, 145, 35, 181, 13, 125, 305, 265, 85, 21, 425, 41, 75, 421, 325, 481, 65, 305, 725, 325, 21, 685, 181, 425, 221, 841, 625, 925, 61, 91, 1325, 1105, 55, 43, 35

Offset: 1

Jaroslav Krizek, Mar 15 2009

Numbers k such that sigma_2(k)/sigma_1(k) = A001157(k)/A000203(k) are integers are in A020487.

			Antiharmonic means of divisors of n>=1: 1, 5/3, 5/2, 3, 13/2, 25/6, ...

Ivan Neretin, Table of n, a(n) for n = 1..10000

Cf. A000203, A001157, A020487, A065473, A152649, A158275 (denominators)

Table[Numerator[DivisorSigma[2, n]/DivisorSigma[1, n]], {n, 50}] (* Ivan Neretin, May 22 2015 *)

a(n) = numerator(sigma(n,2)/sigma(n)); \\ Amiram Eldar, Nov 21 2022

Antiharmonic mean of divisors of number n = Product (p_i^e_i) is sigma_2(n)/sigma_1(n) = A001157(n)/A000203(n) = Product ((p_i^(e_i+1)+1)/(p_i+1)).

Sum_{k=1..n} a(k)/A158275(k) ~ c * n^2, where c = (Pi^4/72) * Product_{p prime} (1 - (3*p-2)/(p^3)) = A152649 * A065473 = 0.387941... . - Amiram Eldar, Nov 21 2022

A158275 Denominators of antiharmonic means of divisors of n.

Original entry on oeis.org

1, 3, 2, 1, 3, 6, 4, 3, 1, 9, 6, 2, 7, 12, 6, 1, 9, 3, 10, 1, 8, 18, 12, 6, 1, 21, 2, 4, 15, 18, 16, 3, 12, 27, 12, 1, 19, 6, 14, 9, 21, 24, 22, 2, 3, 36, 24, 2, 1, 1

Offset: 1

Jaroslav Krizek, Mar 15 2009

Numbers k such that sigma_2(k)/sigma_1(k) = A001157(k)/A000203(k) are integers are in A020487.

			Antiharmonic means of divisors of n>=1: 1, 5/3, 5/2, 3, 13/2, 25/6, ...

Ivan Neretin, Table of n, a(n) for n = 1..10000

Cf. A001157, A000203, A020487, A158274 (numerators).

Table[Denominator[DivisorSigma[2, n]/DivisorSigma[1, n]], {n, 50}] (* Ivan Neretin, May 22 2015 *)

a(n) = denominator(sigma(n,2)/sigma(n)); \\ Amiram Eldar, Nov 21 2022

Antiharmonic mean of divisors of number n = Product (p_i^e_i) is sigma_2(n)/sigma_1(n) = A001157(n)/A000203(n) = Product ((p_i^(e_i+1)+1)/(p_i+1)).

a(A020487(n)) = 1. - Amiram Eldar, Nov 21 2022

A277553 Numbers k such that the sum of the divisors of k is divisible by the number of divisors of k, and the sum of the squares of the divisors of k is divisible by the sum of the divisors of k.

Original entry on oeis.org

1, 20, 49, 169, 361, 500, 605, 961, 980, 1025, 1369, 1445, 1700, 1849, 2645, 3380, 3721, 4205, 4352, 4489, 4693, 5329, 6241, 7220, 8228, 8281, 8405, 9409, 9425, 10609, 11045, 11849, 11881, 12493, 12500, 14045, 14580, 14641, 15125, 16129, 17405, 17689, 18785

Offset: 1

Harvey P. Dale, Oct 19 2016

nonn

Numbers k such that A000005(k) divides A000203(k), and A000203(k) divides A001157(k).

			1369 has 3 divisors which sum to 1407; 1407 is divisible by 3; the sum of the squares of the divisors of 1369 is 1875531 which is divisible by 1407; so 1369 is a term of the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harvey P. Dale)

Cf. A000005, A000203, A001157.

Intersection of A003601 and A020487. See also A020486.

Select[Range[50000],Divisible[DivisorSigma[1,#],DivisorSigma[0,#]] && Divisible[ DivisorSigma[2,#], DivisorSigma[1,#]]&]

isok(k) = {my(f = factor(k), d = numdiv(f), s = sigma(f), s2 = sigma(f, 2)); !(s % d) && !(s2 % s);} \\ Amiram Eldar, Jan 25 2025

A210494 Biharmonic numbers: numbers m such that ( Hd(m)+Cd(m) )/2 is an integer, where Hd(m) and Cd(m) are the harmonic mean and the contraharmonic (or antiharmonic) mean of the divisors of m.

Original entry on oeis.org

1, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 35, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 119, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263

Offset: 1

Bruno Berselli, Oct 03 2013 - proposed by Umberto Cerruti (Department of Mathematics "Giuseppe Peano", University of Turin, Italy)

nonn

Equivalently, numbers m such that ( m*sigma_0(m)+sigma_2(m) ) / (2*sigma_1(m)) = (A038040(m) + A001157(m))/A074400(m) is an integer.

All odd primes belong to the sequence. In fact, if p is an odd prime, (p*sigma_0(p)+sigma_2(p))/(2*sigma_1(p)) = (p+1)/2, therefore p is a biharmonic number.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Bruno Berselli)
Marco Abrate, Stefano Barbero, Umberto Cerruti, Nadir Murru, The Biharmonic mean, arXiv:1601.03081 [math.NT], 2016, pages 6-14.
Umberto Cerruti, Numeri Armonici e Numeri Perfetti (in Italian), 2013. The sequence is on page 13.

Cf. A001599 (harmonic numbers), A020487 (antiharmonic numbers), A038040 (n*sigma_0(n)), A001157 (sigma_2(n)), A074400 (2*sigma_1(n)), A230214 (nonprime terms of A210494).

Cf. A189835.

a210494 n = a210494_list !! (n-1)
a210494_list = filter
   (\x -> (a001157 x + a038040 x) `mod` a074400 x == 0) [1..]
-- Reinhard Zumkeller, Jan 21 2014

IsInteger := func; [n: n in [1..300] | IsInteger((n*NumberOfDivisors(n)+DivisorSigma(2,n))/(2*SumOfDivisors(n)))];

Maple

with(numtheory); P:=proc(q) local a,k,n; for n from 1 to q do a:=divisors(n); if type((n*tau(n)+add(a[k]^2,k=1..nops(a)))/(2*sigma(n)),integer) then print(n); fi; od; end; P(1000); # Paolo P. Lava, Oct 11 2013

Mathematica

B[n_] := (n DivisorSigma[0, n] + DivisorSigma[2, n])/(2 DivisorSigma[1, n]); Select[Range[300], IntegerQ[B[#]] &]

PARI

isok(n) = denominator((n*sigma(n,0) + sigma(n,2))/(2*sigma(n)))==1; \\ Michel Marcus, Jan 14 2016

cp's OEIS Frontend

A176797 Antiharmonic means of divisors of antiharmonic numbers.

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A227771 Antiharmonic numbers that are not squares.

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A046871 Numbers k such that sigma_2(k) divides sigma_4(k).

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A144695 Numbers n such that sigma_1(n)/sigma_0(n) = c^2, c an integer.

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A228023 Primitive antiharmonic numbers.

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A158274 Numerators of antiharmonic means of divisors of n.

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A158275 Denominators of antiharmonic means of divisors of n.

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