A067558 -id:A067558 - cp's OEIS frontend

A276634 Sum of cubes of proper divisors of n.

0, 1, 1, 9, 1, 36, 1, 73, 28, 134, 1, 316, 1, 352, 153, 585, 1, 981, 1, 1198, 371, 1340, 1, 2556, 126, 2206, 757, 3160, 1, 4752, 1, 4681, 1359, 4922, 469, 8605, 1, 6868, 2225, 9710, 1, 12600, 1, 12052, 4257, 12176, 1, 20476, 344, 16759, 4941, 19846, 1, 26496, 1457, 25624, 6887, 24398, 1

Offset: 1

Ilya Gutkovskiy, Sep 08 2016

More generally, the Dirichlet generating function for the sum of k-th powers of proper divisors of n is zeta(s-k)*(zeta(s) - 1).

			a(10) = 1^3 + 2^3 + 5^3 = 134, because 10 has 3 proper divisors {1,2,5}.
a(11) = 1^3 = 1, because 11 has 1 proper divisor {1}.

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

Cf. A000035, A000578, A001065, A001158, A032741, A053867, A067558.

[DivisorSigma(3, n) - n^3: n in [1..70]]; // Vincenzo Librandi, Sep 09 2016

Table[DivisorSigma[3, n] - n^3, {n, 70}]

a(n) = sigma(n, 3) - n^3; \\ Michel Marcus, Sep 08 2016

a(n) = 1 if n is prime.
a(p^k) = (p^(3*k) - 1)/(p^3 - 1) for p prime.
Dirichlet g.f.: zeta(s-3)*(zeta(s) - 1).
a(n) = A001158(n) - A000578(n).
A000035(a(n)) = A053867(n).
Sum_{n=1..k} a(n) ~ k^2*(Pi^4*k^2/90 - (k + 1)^2)/4.
G.f.: -x*(1 + 4*x + x^2)/(1 - x)^4 + Sum_{k>=1} k^3*x^k/(1 - x^k). - Ilya Gutkovskiy, Mar 17 2017

A279363 Sum of 4th powers of proper divisors of n.

Original entry on oeis.org

0, 1, 1, 17, 1, 98, 1, 273, 82, 642, 1, 1650, 1, 2418, 707, 4369, 1, 7955, 1, 10898, 2483, 14658, 1, 26482, 626, 28578, 6643, 41090, 1, 62644, 1, 69905, 14723, 83538, 3027, 133923, 1, 130338, 28643, 174994, 1, 236692, 1, 249170, 57893, 279858, 1, 423794, 2402, 401267, 83603, 485810, 1, 644372, 15267, 659842, 130403, 707298, 1, 1053636

Offset: 1

Ilya Gutkovskiy, Dec 10 2016

			a(10) = 1^4 + 2^4 + 5^4 = 642, because 10 has 3 proper divisors {1,2,5}.
a(11) = 1^4 = 1, because 11 has 1 proper divisor {1}.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Proper divisors.
Index entries for sequences related to sums of divisors

Cf. A000583, A001159.

Cf. A001065, A067558, A276634, A279364.

Table[DivisorSigma[4, n] - n^4, {n, 60}]

for(n=1, 60, print1(sigma(n, 4) - n^4,", ")) \\ Indranil Ghosh, Mar 18 2017

from sympy.ntheory import divisor_sigma
print([divisor_sigma(n,4) - n**4 for n in range(1,61)]) # Indranil Ghosh, Mar 18 2017

a(n) = 1 if n is prime.
a(p^k) = (p^(4*k) - 1)/(p^4 - 1) when p is prime.
Dirichlet g.f.: zeta(s-4)*(zeta(s) - 1).
a(n) = A001159(n) - A000583(n).
G.f.: -x*(1 + 11*x + 11*x^2 + x^3)/(1 - x)^5 + Sum_{k>=1} k^4 x^k/(1 - x^k). - Ilya Gutkovskiy, Mar 18 2017
Sum_{k=1..n} a(k) ~ (Zeta(5) - 1)*n^5 / 5. - Vaclav Kotesovec, Feb 02 2019

A279847 a(n) = Sum_{k=1..n} k^2*(floor(n/k) - 1).

Original entry on oeis.org

0, 1, 2, 7, 8, 22, 23, 44, 54, 84, 85, 151, 152, 206, 241, 326, 327, 458, 459, 605, 664, 790, 791, 1065, 1091, 1265, 1356, 1622, 1623, 2023, 2024, 2365, 2496, 2790, 2865, 3480, 3481, 3847, 4026, 4636, 4637, 5373, 5374, 6000, 6341, 6875, 6876, 7982, 8032, 8787, 9086, 9952, 9953, 11137, 11284

Offset: 1

Ilya Gutkovskiy, Dec 20 2016

Sum of all squares of proper divisors of all positive integers <= n.

Total volume of all rectangular prisms with dimensions (x, x, z) and integers x and y, such that x + y = n, 0 < x <= y, and z = floor(y/x). - Wesley Ivan Hurt, Dec 21 2020

			For n = 7 the proper divisors of the first seven positive integers are {0}, {1}, {1}, {1, 2}, {1}, {1, 2, 3}, {1} so a(7) = 0^2 + 1^2 + 1^2 + 1^2 + 2^2 + 1 ^2 + 1^2 + 2^2 + 3^2 + 1^2 = 23.

Index entries for sequences related to sums of divisors

Cf. A000290, A000330, A001157, A064602, A067558, A153485.

Table[Sum[k^2 (Floor[n/k] - 1), {k, 1, n}], {n, 55}]
Table[Sum[DivisorSigma[2, k] - k^2, {k, 1, n}], {n, 55}]

a(n) = sum(k=1, n, k^2*(floor(n/k)-1)) \\ Felix Fröhlich, Dec 20 2016

from math import isqrt
def A279847(n): return (-n*(n+1)*(2*n+1)-(s:=isqrt(n))**2*(s+1)*(2*s+1) + sum((q:=n//k)*(6*k**2+q*(2*q+3)+1) for k in range(1,s+1)))//6 # Chai Wah Wu, Oct 21 2023

G.f.: -x*(1 + x)/(1 - x)^4 + (1/(1 - x))*Sum_{k>=1} k^2*x^k/(1 - x^k).
a(n) = A064602(n) - A000330(n).
a(n) = Sum_{k=1..n} A067558(k).
a(n) = Sum_{k=1..n} (A001157(k) - A000290(k)).
a(p^k) = a(p^k-1) + (p^(2*k) - 1)/(p^2 - 1), when p is prime.
a(n) ~ ((zeta(3) - 1)/3)*n^3.
a(n) = Sum_{k=1..floor(n/2)} k^2 * floor((n-k)/k). - Wesley Ivan Hurt, Dec 21 2020

A321258 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals: A(n,k) = sigma_k(n) - n^k.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 0, 1, 1, 2, 0, 1, 1, 3, 1, 0, 1, 1, 5, 1, 3, 0, 1, 1, 9, 1, 6, 1, 0, 1, 1, 17, 1, 14, 1, 3, 0, 1, 1, 33, 1, 36, 1, 7, 2, 0, 1, 1, 65, 1, 98, 1, 21, 4, 3, 0, 1, 1, 129, 1, 276, 1, 73, 10, 8, 1, 0, 1, 1, 257, 1, 794, 1, 273, 28, 30, 1, 5

Offset: 1

Ilya Gutkovskiy, Nov 01 2018

A(n,k) is the sum of k-th powers of proper divisors of n.

			Square array begins:
  0,  0,   0,   0,   0,    0,  ...
  1,  1,   1,   1,   1,    1,  ...
  1,  1,   1,   1,   1,    1,  ...
  2,  3,   5,   9,  17,   33,  ...
  1,  1,   1,   1,   1,    1,  ...
  3,  6,  14,  36,  98,  276,  ...

Eric Weisstein's World of Mathematics, Proper divisors
Index entries for sequences related to sums of divisors

Columns k=0..5 give A032741, A001065, A067558, A276634, A279363, A279364.

Cf. A109974, A285425, A286880, A321259 (diagonal).

Table[Function[k, DivisorSigma[k, n] - n^k][i - n], {i, 0, 12}, {n, 1, i}] // Flatten
Table[Function[k, SeriesCoefficient[Sum[j^k x^(2 j)/(1 - x^j), {j, 1, n}], {x, 0, n}]][i - n], {i, 0, 12}, {n, 1, i}] // Flatten

G.f. of column k: Sum_{j>=1} j^k*x^(2*j)/(1 - x^j).
Dirichlet g.f. of column k: zeta(s-k)*(zeta(s) - 1).
A(n,k) = 1 if n is prime.

A339353 G.f.: Sum_{k>=1} k^2 * x^(k*(k + 1)) / (1 - x^k).

Original entry on oeis.org

0, 1, 1, 1, 1, 5, 1, 5, 1, 5, 1, 14, 1, 5, 10, 5, 1, 14, 1, 21, 10, 5, 1, 30, 1, 5, 10, 21, 1, 39, 1, 21, 10, 5, 26, 30, 1, 5, 10, 46, 1, 50, 1, 21, 35, 5, 1, 66, 1, 30, 10, 21, 1, 50, 26, 70, 10, 5, 1, 91, 1, 5, 59, 21, 26, 50, 1, 21, 10, 79, 1, 130, 1, 5, 35, 21, 50, 50, 1, 110

Offset: 1

Ilya Gutkovskiy, Dec 01 2020

nonn

Sum of squares of divisors of n that are smaller than sqrt(n).

Index entries for sequences related to sums of divisors

Cf. A001157, A056924, A067558, A070039, A095118, A339354.

nmax = 80; CoefficientList[Series[Sum[k^2 x^(k (k + 1))/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
Table[DivisorSum[n, #^2 &, # < Sqrt[n] &], {n, 80}]

a(n) = sumdiv(n, d, if (d^2 < n, d^2)); \\ Michel Marcus, Dec 02 2020

A140362 Semiprimes pq that divide the sum of the squares of their divisors, 1+p^2+q^2+(pq)^2.

Original entry on oeis.org

10, 65, 20737

Offset: 1

Mohamed Bouhamida, Jul 22 2008, Jul 27 2008

6 is the smallest integer n which is the product of two distinct primes and which divides the sum of the cubes of the divisors of n. Are there other numbers with this property?
Using Pell equations and a Fibonacci identity, Max Alekseyev and I have shown that all terms are the product of prime Fibonacci numbers whose indices are twin primes. The first three terms are Fib(3)*Fib(5), Fib(5)*Fib(7) and Fib(11)*Fib(13). The other two known terms are Fib(431)*Fib(433) and Fib(569)*Fib(571), huge numbers that are in the b-file. The sequence probably has no additional terms. - T. D. Noe, Jul 27 2008
Let a, b, c and d be consecutive odd-indexed Fibonacci numbers. Then it can be proved that 1 + b^2 + c^2 + (bc)^2 = abcd, which shows that bc divides 1 + b^2 + c^2 + (bc)^2. Hence if b and c are prime, then bc is in this sequence. - T. D. Noe, Jul 27 2008
Empirical search suggests that A067558(a(n))/A032741(a(n)) = a(n). A032741(a(n)) = 3 for all n by definition of semiprime. A067558(a(n)) must also then be divisible by 3. a(n) can be called the n-th "perfect mean square aliquot number". - William Krier, Dec 16 2024

			10 divides (1^2 + 2^2 + 5^2) giving 3 - the number of proper divisors of semiprime 10.
65 divides (1^2 + 5^2 + 13^2) giving 3 - the number of proper divisors of semiprime 65.
20737 divides (1^2 + 89^2 + 233^2) giving 3 - the number of proper divisors of semiprime 20737.

T. D. Noe, Table of n, a(n) for n=1..5
T. Cai, D. Chen, and Y. Zhang, Perfect numbers and Fibonacci primes, arXiv:1310.0898 [math.NT], 2013-2014.
T. Cai, D. Chen, and Y. Zhang, Perfect numbers and Fibonacci primes (II), arXiv:1406.5684 [math.NT], 2014 (see case m=1 in Table 1).

Cf. A000045, A001605, A046762.

isok(n) = sigma(n, 2) - n^2 == 3*n; \\ Michel Marcus, Jun 24 2014

A279364 Sum of 5th powers of proper divisors of n.

Original entry on oeis.org

0, 1, 1, 33, 1, 276, 1, 1057, 244, 3158, 1, 9076, 1, 16840, 3369, 33825, 1, 67101, 1, 104182, 17051, 161084, 1, 290676, 3126, 371326, 59293, 555688, 1, 870552, 1, 1082401, 161295, 1419890, 19933, 2206525, 1, 2476132, 371537, 3336950, 1, 4646784, 1, 5315740, 821793, 6436376, 1, 9301876, 16808, 9868783

Offset: 1

Ilya Gutkovskiy, Dec 10 2016

			a(10) = 1^5 + 2^5 + 5^5 = 3158, because 10 has 3 proper divisors {1,2,5}.
a(11) = 1^5 = 1, because 11 has 1 proper divisor {1}.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Proper divisors.
Index entries for sequences related to sums of divisors.

Cf. A000584, A001160, A013664.

Cf. A001065, A067558, A276634, A279363.

Table[DivisorSigma[5, n] - n^5, {n, 50}]

for(n=1, 50, print1(sigma(n, 5) - n^5,", ")) \\ Indranil Ghosh, Mar 18 2017

from sympy.ntheory import divisor_sigma
print([divisor_sigma(n,5) - n**5 for n in range(1,51)]) # Indranil Ghosh, Mar 18 2017

a(n) = 1 if n is prime.
a(p^k) = (p^(5*k) - 1)/(p^5 - 1) when p is prime.
Dirichlet g.f.: zeta(s-5)*(zeta(s) - 1).
a(n) = A001160(n) - A000584(n).
G.f.: -x*(1 + 26*x + 66*x^2 + 26*x^3 + x^4)/(1 - x)^6 + Sum_{k>=1} k^5 x^k/(1 - x^k). - Ilya Gutkovskiy, Mar 18 2017
Sum_{k=1..n} a(k) ~ (zeta(6) - 1) * n^6 / 6. - Amiram Eldar, Jan 11 2025

A321259 a(n) = sigma_n(n) - n^n.

Original entry on oeis.org

0, 1, 1, 17, 1, 794, 1, 65793, 19684, 9766650, 1, 2194095090, 1, 678223089234, 30531927033, 281479271743489, 1, 150196195641350171, 1, 100000096466944316978, 558545874543637211, 81402749386839765307626, 1, 79501574308536809523296482, 298023223876953126

Offset: 1

Ilya Gutkovskiy, Nov 01 2018

nonn

a(n) is the sum of n-th powers of proper divisors of n.

Seiichi Manyama, Table of n, a(n) for n = 1..400
Eric Weisstein's World of Mathematics, Proper divisors
Index entries for sequences related to sums of divisors

Cf. A000312, A001065, A023887, A067558, A276634, A279363, A279364, A292919, A321258.

[DivisorSigma(n, n) - n^n: n in [1..30]]; // Vincenzo Librandi, Nov 02 2018

Table[DivisorSigma[n, n] - n^n, {n, 25}]
nmax = 25; Rest[CoefficientList[Series[Sum[(k x)^(2 k)/(1 - (k x)^k), {k, 1, nmax}], {x, 0, nmax}], x]]

a(n) = sigma(n, n) - n^n; \\ Michel Marcus, Nov 02 2018

G.f.: Sum_{k>=1} (k*x)^(2*k)/(1 - (k*x)^k).
a(n) = A023887(n) - A000312(n).
a(n) = A321258(n,n).
a(n) = 1 if n is prime.

A177404 Numbers n such that n^2 minus (sum of the squares of the proper divisors of n) is a prime number.

Original entry on oeis.org

2, 4, 8, 9, 18, 25, 32, 49, 64, 72, 81, 100, 121, 162, 242, 256, 289, 361, 392, 400, 484, 512, 529, 576, 648, 729, 784, 1058, 1296, 1352, 1444, 1681, 1849, 1922, 2048, 2116, 2500, 3136, 3872, 4418, 4608, 4624, 5184, 5776, 5832, 6889, 7225, 7396, 7744

Offset: 1

Claudio Meller, Dec 10 2010

nonn

Sum of squares of proper divisors of n is given by A067558.

			9 has proper divisors 3 and 1; it is in the list because 9^2-(3^2+1^2) = 71 is a prime number.

Harvey P. Dale, Table of n, a(n) for n = 1..200

Cf. A067558.

a067558:=func< n | n le 1 select 0 else &+[ D[k]^2: k in [1..#D-1] ] where D is Divisors(n) >; [ n: n in [1..10000] | IsPrime(n^2-a067558(n)) ];

Select[Range[8000],PrimeQ[#^2-Total[Most[Divisors[#]]^2]]&] (* Harvey P. Dale, Dec 14 2011 *)

A201879 Numbers n such that sigma_2(n) - n^2 is a square.

Original entry on oeis.org

1, 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 30, 31, 37, 41, 43, 47, 53, 59, 61, 67, 70, 71, 73, 79, 83, 89, 97, 101, 102, 103, 107, 109, 113, 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

Offset: 1

Michel Lagneau, Dec 06 2011

nonn

Numbers n such that sum of the square of proper (or aliquot) divisors of n is a square.

All primes are in this sequence. Nonprimes in the sequence are 1, 30, 70, 102, 282, 286, 646, 730, 920, 1242, ... - Charles R Greathouse IV, Dec 06 2011

			a(12)=30 because the aliquot divisors of 30 are  1, 2, 3, 5, 6, 10, 15, the sum of whose squares is 1^2 + 2^2 + 3^2 + 5^2 + 6^2 + 10^2 + 15^2 = 400 = 20^2.

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

Cf. A000290, A001065, A001157, A073040.

A067558 := proc(n)
    numtheory[sigma][2](n)-n^2 ;
end proc:
isA201879 := proc(n)
    issqr(A067558(n)) ;
end proc:
for n from 1 to 300 do
    if isA201879(n) then
        printf("%d,",n);
    end if;
end do: # R. J. Mathar, Dec 07 2011

Select[Range[400], IntegerQ[Sqrt[DivisorSigma[2, #]-#^2]]&]

is(n)=issquare(sigma(n,2)-n^2) \\ Charles R Greathouse IV, Dec 06 2011

{n: A067558(n) in A000290}. - R. J. Mathar, Dec 07 2011

cp's OEIS Frontend

A276634 Sum of cubes of proper divisors of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A279363 Sum of 4th powers of proper divisors of n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula

A279847 a(n) = Sum_{k=1..n} k^2*(floor(n/k) - 1).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula

A321258 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals: A(n,k) = sigma_k(n) - n^k.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A339353 G.f.: Sum_{k>=1} k^2 * x^(k*(k + 1)) / (1 - x^k).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

A140362 Semiprimes pq that divide the sum of the squares of their divisors, 1+p^2+q^2+(pq)^2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A279364 Sum of 5th powers of proper divisors of n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI