A020487 -id:A020487 - cp's OEIS frontend

A046870 Numbers k such that sigma_1(k) divides sigma_4(k).

1, 4, 9, 16, 20, 25, 36, 49, 50, 64, 81, 100, 117, 121, 144, 169, 180, 196, 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, 980, 1024, 1025, 1058, 1089, 1156, 1225, 1280, 1296

Offset: 1

Labos Elemer

nonn

sigma_1(n) is the sum of the divisors of n [same as sigma(n)] (A000203).
sigma_2(n) is the sum of the squares of the divisors of n (A001157).
sigma_4(n) is the sum of the 4th powers of the divisors of n (A001159).

			k = a(18) = 196 of which divisor power sums for k=0,1,2,3,4 are 9,399,51471, 8613489, 1574446419. sigma_1(k) = 399 and sigma_4(k) = 51471*30589=399*129*30589. Thus both sigma_2(k) and sigma_1(k) divide sigma_4(k).

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

Cf. A000203, A001157, A001159.

Has large overlap with A020487.

Select[Range[1300], Divisible @@ DivisorSigma[{4, 1}, #] &] (* Amiram Eldar, Jun 15 2024 *)

is(k) = {my(f = factor(k)); !(sigma(f, 4) % sigma(f)); } \\ Amiram Eldar, Jun 15 2024

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

A074632 Numbers k such that the sum of 2nd, 3rd, 4th and 5th powers of divisors of k are divisible by sum of divisors of k.

Original entry on oeis.org

1, 20, 64, 500, 729, 1024, 1280, 4096, 4352, 14580, 15625, 32000, 39168, 46656, 47360, 59049, 65536, 117649, 144640, 161024, 262144, 312500, 364500, 509184, 531441, 746496, 796797, 933120, 1000000, 1180980, 1184000, 1449216, 1771561

Offset: 1

Labos Elemer, Aug 27 2002

nonn

			For k = 20: sigma(k) = 42 ,sigma_2(k) = 546 = 13 * 42, sigma_3(k) = 9198 = 219 * 42, sigma_4(k) = 170898 = 4069 * 42, sigma_5(k) = 3304182 = 78671 * 42.

Amiram Eldar, Table of n, a(n) for n = 1..269 (terms up to 10^10)

See also A020487, A046841, A046870, A046871.

Cf. A000203, A001157, A001158, A001159, A001160.

Select[Range[2000000],And@@Divisible[DivisorSigma[Range[2,5],#], DivisorSigma[ 1,#]]&] (* Harvey P. Dale, Jan 01 2012 *)

is(k) = {my(f = factor(k), s = sigma(f)); for(k = 2, 5, if(sigma(f, k) % s, return(0))); 1; }  \\ Amiram Eldar, Jun 15 2024

A144922 Numbers k such that k*sigma_2(k)/sigma_1(k) is an integer.

Original entry on oeis.org

1, 4, 6, 9, 12, 16, 18, 20, 24, 25, 28, 36, 44, 45, 48, 49, 50, 54, 60, 64, 72, 81, 90, 92, 96, 100, 108, 112, 117, 121, 132, 140, 144, 150, 153, 162, 168, 169, 180, 192, 196, 198, 200, 204, 216, 225, 228, 234, 240, 242, 252, 256, 270, 288, 289, 294, 300

Offset: 1

Ctibor O. Zizka, Sep 25 2008

Numbers k such that k*A001157(k)/A000203(k) is an integer. This sequence is connected closely with Ore divisor numbers (A001599) and RMS numbers (A140480).

This sequence is infinite. E.g., all the numbers of the form 3*2^m, for m >= 1, are terms, since 3*2^m * sigma_2(3*2^m) / sigma_1(3*2^m) = 5 * 2^(m-1) * (2^(m+1)+1) is an integer. - Amiram Eldar, Dec 25 2024

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

Cf. A000203, A001157, A001599, A140480.

A020487 is a subsequence.

Select[Range[300],IntegerQ[(#*DivisorSigma[2,#])/DivisorSigma[1,#]]&] (* Harvey P. Dale, Oct 28 2018 *)

is(k) = my(f = factor(k)); !((k*sigma(f, 2)) % sigma(f)); \\ Amiram Eldar, Dec 25 2024

More terms from Harvey P. Dale, Oct 28 2018

A152215 Numbers k such that sigma_2(k)/(sigma_1(k)*sigma_0(k)) = c, c an integer.

Original entry on oeis.org

1, 4, 25, 100, 121, 256, 289, 484, 529, 841, 1156, 1600, 1681, 2116, 2209, 2809, 3025, 3364, 3481, 5041, 6400, 6724, 6889, 7225, 7921, 8836, 10201, 11236, 11449, 12100, 12769, 13225, 13924, 17161, 18225, 18496, 18769, 20164, 21025, 22201, 27556, 27889, 28900

Offset: 1

Ctibor O. Zizka, Nov 29 2008

k : A001157(k)/(A000203(k)*A000005(k)) = c, c an integer.

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

Cf. A000005, A000203, A001157, A003601, A094471, A140480, A003601, A020486, A020487

Select[Range[50000],IntegerQ[DivisorSigma[2,#]/(DivisorSigma[1,#] DivisorSigma[ 0,#])]&] (* Harvey P. Dale, Feb 12 2013 *)

isok(k) = denominator(sigma(k,2)/(sigma(k, 1)*sigma(k,0))) == 1; \\ Michel Marcus, Jul 15 2019

More terms from Harvey P. Dale, Feb 12 2013

A158276 Numbers k such that sigma_1(k) does not divide sigma_2(k).

Original entry on oeis.org

2, 3, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 17, 18, 19, 21, 22, 23, 24, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77

Offset: 1

Jaroslav Krizek, Mar 15 2009

nonn

Numbers k such that the antiharmonic mean of divisors of k is not an integer.
Antiharmonic mean of divisors of a number m = Product (p_i^e_i) is A001157(m)/A000203(m) = Product ((p_i^(e_i+1)+1)/(p_i+1)).
Numbers k such that A001157(k)/A000203(k) is not an integer.

			a(12) = 15, sigma_2(15)/sigma_1(15)=260/24 = 65/6 (not integer).

Complement of A020487.

Cf. A001157, A000203.

Select[Range[100], Mod @@ DivisorSigma[{2, 1}, #] > 0 &] (* Amiram Eldar, Mar 22 2024 *)

is(n) = {my(f = factor(n)); sigma(f, 2) % sigma(f);} \\ Amiram Eldar, Mar 22 2024

More terms from Amiram Eldar, Mar 22 2024

A280353 Numbers n such that the sum of the divisors of n divides sum of squares of divisors of n and number of divisors of n divides n.

Original entry on oeis.org

1, 9, 36, 180, 225, 441, 450, 468, 625, 1089, 1476, 1521, 1620, 1800, 2025, 2178, 2340, 2601, 3249, 3600, 4050, 4500, 4761, 5202, 5625, 6561, 7569, 8100, 8649, 8712, 9522, 10000, 12321, 13572, 15129, 15138, 16200, 16641, 19881, 20808, 22500, 23400, 25281, 26244, 28224, 28800

Offset: 1

Ilya Gutkovskiy, Jan 01 2017

Intersection of A020487 and A033950.

Numbers n such that A000203(n)|A001157(n) and A000005(n)|n.

			9 is in the sequence because 9 has 3 divisors {1,3,9}, 9/3 = 3 and (1^2 + 3^2 + 9^2)/(1 + 3 + 9) = 7 are both integer.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Index entries for sequences related to sums of divisors

Cf. A000005, A000203, A001157, A020487, A033950.

Select[Range[30000], Divisible[DivisorSigma[2, #1], DivisorSigma[1, #1]] && Divisible[#1, DivisorSigma[0, #1]] & ]
Select[Range[30000],With[{ds=DivisorSigma},Mod[ds[2,#],ds[1,#]]==Mod[#,ds[0,#]]==0&]] (* Harvey P. Dale, Nov 27 2024 *)

A301482 Composite numbers whose sum of aliquot parts divide the sum of the squares of their aliquot parts.

Original entry on oeis.org

8, 22, 27, 32, 77, 125, 128, 243, 343, 494, 512, 611, 660, 1073, 1281, 1331, 1425, 2033, 2048, 2187, 2197, 2332, 3125, 4172, 4565, 4913, 5293, 6031, 6859, 8192, 9983, 12167, 13969, 15818, 15947, 16807, 17485, 19683, 23489, 23840, 24389, 25241, 25389, 29791, 32768

Offset: 1

Paolo P. Lava, Mar 22 2018

Semiprimes in the sequence: 22, 77, 611, 1073, 2033, 5293, 6031, 9983, 13969, 15947, 23489, 25241, 40301, 49901, 50249, 51101, 56759, 65017, 71677, 85079, 97217, 98099, 99101, .... - Robert Israel, Mar 29 2018

2^k is a term for all odd k > 1. - Michael S. Branicky, Aug 22 2021

			Aliquot parts of 77 are 1, 7, 11. Then (1^2 + 7^2 + 11^2)/(1 + 7 + 11) = 171/19 = 9.

Amiram Eldar, Table of n, a(n) for n = 1..1009 (terms below 10^9, terms 1..100 from Paolo P. Lava)

Cf. A001065, A001157, A020487, A067558.

Contains A056824.

with(numtheory): P:=proc(n)
if not isprime(n) and frac((add(p^2,p=divisors(n))-n^2)/(sigma(n)-n))=0
then n; fi; end: seq(P(i),i=2..35*10^3);

aQ[n_] := CompositeQ[n] && Divisible[DivisorSigma[2, n] - n^2, DivisorSigma[1, n] - n]; Select[Range[33000], aQ] (* Amiram Eldar, Aug 17 2019 *)

isok(n) = (n!=1) && !isprime(n) && (((sigma(n,2) - n^2) % (sigma(n) - n)) == 0); \\ Michel Marcus, Mar 23 2018

from sympy import divisors
def ok(n):
    divs = divisors(n)[:-1]
    return len(divs) > 1 and sum(d**2 for d in divs)%sum(divs) == 0
print(list(filter(ok, range(4, 32769)))) # Michael S. Branicky, Aug 22 2021

A176803 a(n) = the smallest natural numbers m such that product of antiharmonic mean of the divisors of n and antiharmonic mean of the divisors of m are integers, a(n) = 0 if no such number exists.

Original entry on oeis.org

1, 4, 0, 1, 4, 0, 0, 4, 1, 100, 0, 0, 9, 0, 0, 1, 100, 4, 0, 1, 0, 0, 0, 0, 1, 25, 0, 0, 325, 0

Offset: 1

Jaroslav Krizek, Apr 26 2010

nonn

Antiharmonic mean of the divisors of number n is rational number b(n) = A001157(n) / A000203(n) = A158274(n) / A158275(n). a(n) = 1 for infinitely many n. a(n) = 1 for numbers from A020487: a(A020487(n)) = 1. a(n) = 1 iff A158275(n) = 1. a(n) = 0 for infinitely many n. a(n) = 0 for even A158275(n).

			For n = 10; b(10) = 65/9, a(n) = 100 because b(100) = 63; 65/9 * 63 = 455 (integer).

A192286 Antiharmonic numbers using anti-divisors: numbers n such that sigma(n) divides sigma_2(n), where sigma(n) is the sum of anti-divisors of n and sigma_2(n) the sum of squares of anti-divisors of n.

Original entry on oeis.org

3, 4, 6, 9, 36, 54, 96, 216, 576, 1212, 1296, 1582, 2171, 3129, 3599, 26847, 45914, 69984, 76393, 91013, 137173, 176678, 182559, 183087, 236196, 393216, 497664, 3823898, 28697814, 31850496, 46572031, 47992961, 83951616, 84934656, 95969521, 126310141, 472250381

Offset: 1

Paolo P. Lava, Jul 28 2011

nonn

			Anti-divisors of 1212 are 5, 8, 24, 25, 97, 485, 808 and their sum is 1452. The sum of the squares of anti-divisors is 898788 and 898788/1452=619.

Cf. A020487, A066272.

with(numtheory);
P:=proc(n)
local a,b,i,k;
for i from 3 to n do
  a:=0; b:=0;
  for k from 2 to i-1 do
    if abs((i mod k)- k/2) < 1 then a:=a+k; b:=b+k^2; fi;
  od;
  if trunc(b/a)=b/a then print(i); fi;
od;
end:
P(200000);

Like A020487 but using anti-divisors.

4, 9, 36, 576, 1296, etc. are antiharmonic both with divisors and anti-divisors.

a(22)-a(37) from Donovan Johnson, Sep 22 2011

cp's OEIS Frontend

A046870 Numbers k such that sigma_1(k) divides sigma_4(k).

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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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A074632 Numbers k such that the sum of 2nd, 3rd, 4th and 5th powers of divisors of k are divisible by sum of divisors of k.

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A144922 Numbers k such that k*sigma_2(k)/sigma_1(k) is an integer.

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A152215 Numbers k such that sigma_2(k)/(sigma_1(k)*sigma_0(k)) = c, c an integer.

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A158276 Numbers k such that sigma_1(k) does not divide sigma_2(k).

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A280353 Numbers n such that the sum of the divisors of n divides sum of squares of divisors of n and number of divisors of n divides n.

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A301482 Composite numbers whose sum of aliquot parts divide the sum of the squares of their aliquot parts.

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A192286 Antiharmonic numbers using anti-divisors: numbers n such that sigma(n) divides sigma_2(n), where sigma(n) is the sum of anti-divisors of n and sigma_2(n) the sum of squares of anti-divisors of n.