A001159 -id:A001159 - cp's OEIS frontend

A363606 Expansion of Sum_{k>0} x^(2*k)/(1-x^k)^6.

0, 1, 6, 22, 56, 133, 252, 484, 798, 1344, 2002, 3157, 4368, 6441, 8630, 12112, 15504, 21274, 26334, 35014, 42762, 55133, 65780, 84349, 98336, 123124, 143304, 176373, 201376, 247380, 278256, 336744, 379000, 451402, 502250, 600055, 658008, 775733, 855042

Offset: 1

Seiichi Manyama, Jun 11 2023

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A013664, A032741, A065608, A069153, A363604, A363605.

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

a[n_] := DivisorSum[n, Binomial[# + 3, 5] &]; Array[a, 40] (* Amiram Eldar, Jul 25 2023 *)

my(N=40, x='x+O('x^N)); concat(0, Vec(sum(k=1, N, x^(2*k)/(1-x^k)^6)))

a(n) = my(f = factor(n)); (sigma(f, 5) + 5*sigma(f, 4) + 5*sigma(f, 3) - 5*sigma(f, 2) - 6*sigma(f)) / 120; \\ Amiram Eldar, Dec 30 2024

G.f.: Sum_{k>0} binomial(k+3,5) * x^k/(1 - x^k).
a(n) = Sum_{d|n} binomial(d+3,5).
From Amiram Eldar, Dec 30 2024: (Start)
a(n) = (sigma_5(n) + 5*sigma_4(n) + 5*sigma_3(n) - 5*sigma_2(n) - 6*sigma_1(n)) / 120.
Dirichlet g.f.: zeta(s) * (zeta(s-5) + 5*zeta(s-4) + 5*zeta(s-3) - 5*zeta(s-2) - 6*zeta(s-1)) / 120.
Sum_{k=1..n} a(k) ~ (zeta(6)/720) * n^6. (End)

A386747 a(n) = n^2*sigma_4(n).

Original entry on oeis.org

0, 1, 68, 738, 4368, 15650, 50184, 117698, 279616, 538083, 1064200, 1771682, 3223584, 4826978, 8003464, 11549700, 17895680, 24137858, 36589644, 47046242, 68359200, 86861124, 120474376, 148036418, 206356608, 244531875, 328234504, 392263236, 514104864, 594824162

Offset: 0

Vaclav Kotesovec, Aug 01 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..9000

Cf. A000290, A001159, A386749, A386748.

[0] cat [n^2*DivisorSigma(4, n): n in [1..35]]; // Vincenzo Librandi, Aug 02 2025

Table[n^2*DivisorSigma[4, n], {n, 0, 40}]
nmax = 40; CoefficientList[Series[Sum[k^6*x^k*(1 + x^k)/(1 - x^k)^3, {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Sum_{k>=1} k^6*x^k*(1 + x^k)/(1 - x^k)^3. - Amiram Eldar, Aug 01 2025
a(n) = n^2*A001159(n).
Dirichlet g.f.: zeta(s-2)*zeta(s-6).- R. J. Mathar, Aug 03 2025

A386749 a(n) = n*sigma_4(n).

Original entry on oeis.org

0, 1, 34, 246, 1092, 3130, 8364, 16814, 34952, 59787, 106420, 161062, 268632, 371306, 571676, 769980, 1118480, 1419874, 2032758, 2476118, 3417960, 4136244, 5476108, 6436366, 8598192, 9781275, 12624404, 14528268, 18360888, 20511178, 26179320, 28629182, 35791392, 39621252

Offset: 0

Vaclav Kotesovec, Aug 01 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..9000

Cf. A064987, A328259, A281372, A282050, A386750, A282060, A386751.

Cf. A001159, A386747, A386748.

[0] cat [n*DivisorSigma(4, n): n in [1..35]]; // Vincenzo Librandi, Aug 02 2025

Table[n*DivisorSigma[4, n], {n, 0, 50}]
nmax = 50; CoefficientList[Series[x*Sum[k^5*x^(k-1)/(1 - x^k)^2, {k, 1, nmax}], {x, 0, nmax}], x]

a(n) = if (n, n*sigma(n,4), 0); \\ Michel Marcus, Aug 02 2025

G.f.: Sum_{k>=1} k^5*x^(k-1)/(1 - x^k)^2.
a(n) = n*A001159(n).
Dirichlet g.f.: zeta(s-1)*zeta(s-5). - R. J. Mathar, Aug 03 2025

A386784 a(n) = n^4*sigma_4(n).

Original entry on oeis.org

0, 1, 272, 6642, 69888, 391250, 1806624, 5767202, 17895424, 43584723, 106420000, 214373522, 464196096, 815759282, 1568678944, 2598682500, 4581294080, 6975840962, 11855044656, 16983693362, 27343680000, 38305755684, 58309597984, 78311265122, 118861406208, 152832421875

Offset: 0

Vaclav Kotesovec, Aug 02 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..10000

Cf. A280022, A386783, A280025, A386785, A386786, A386787, A386788.

Cf. A001159, A386749, A386747, A386748.

[0] cat [n^4*DivisorSigma(4, n): n in [1..35]]; // Vincenzo Librandi, Aug 03 2025

Table[n^4*DivisorSigma[4, n], {n, 0, 40}]
nmax = 40; CoefficientList[Series[Sum[k^4*x^k*(1 + 247*x^k + 4293*x^(2*k) + 15619*x^(3*k) + 15619*x^(4*k) + 4293*x^(5*k) + 247*x^(6*k) + x^(7*k))/(1 - x^k)^9, {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Sum_{k>=1} k^4*x^k*(1 + 247*x^k + 4293*x^(2*k) + 15619*x^(3*k) + 15619*x^(4*k) + 4293*x^(5*k) + 247*x^(6*k) + x^(7*k))/(1 - x^k)^9.
a(n) = n^4*A001159(n).
Dirichlet g.f.: zeta(s-4)*zeta(s-8). - R. J. Mathar, Aug 03 2025

A046687 Numbers k such that k and sum of 4th powers of divisors of k are relatively prime.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 15, 16, 17, 18, 19, 21, 23, 25, 27, 29, 31, 32, 33, 35, 37, 39, 41, 43, 45, 47, 49, 50, 51, 53, 55, 57, 59, 61, 64, 65, 67, 69, 71, 72, 73, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 98, 99, 100, 101, 103, 105, 107, 109, 111, 113, 115, 119

Offset: 1

Labos Elemer

nonn

All even terms are either squares or doubled squares. - Ivan Neretin, Dec 30 2015

The asymptotic density of this sequence is 0 (Dressler, 1974). - Amiram Eldar, Jul 23 2020

Ivan Neretin, Table of n, a(n) for n = 1..10076
Robert E. Dressler, On a theorem of Niven, Canadian Mathematical Bulletin, Vol. 17, No. 1 (1974), pp. 109-110.

Cf. A001159, A046684, A046686.

Select[Range[120], GCD[#, DivisorSigma[4, #]] == 1 &] (* Ivan Neretin, Dec 30 2015 *)
Select[Range[150],CoprimeQ[#,DivisorSigma[4,#]]&] (* Harvey P. Dale, Jan 15 2024 *)

isok(n) = gcd(n, sigma(n, 4))  == 1; \\ Michel Marcus, Dec 20 2013

A046764 Sum of the 4th powers of the divisors of n is divisible by n.

Original entry on oeis.org

1, 34, 84, 156, 364, 492, 1092, 3444, 5617, 6396, 11234, 22468, 33628, 44772, 67404, 100884, 157276, 190978, 292084, 435708, 437164, 471828, 549687, 569772, 709937, 742612, 763912, 876252, 986076, 1099374, 1118480, 1289484, 1311492, 1419874

Offset: 1

Labos Elemer

nonn

Compare with multiply perfect numbers, A007691. Here Sum[ divisors ] is replaced by Sum[ 4th powers of divisors ].

Problem 11090 proves that this sequence is infinite. - T. D. Noe, Apr 18 2006

			n=84, Sigma[ 4,84 ] = Sum(d^4) = 53771172 = 640133*84 = 640133*n;
n=5617, Sigma[ 4,5617 ] = 995446331475844 = 5617*17722083332, a multiple of n.

Amiram Eldar, Table of n, a(n) for n = 1..1000 (terms 1..200 from T. D. Noe)
Florian Luca and John Ferdinands, Problem 11090: Sometimes n divides sigma_k(n), Amer. Math. Monthly 113:4 (2006), pp. 372-373.

Cf. A001159, A007691.

Do[If[Mod[DivisorSigma[4, n], n]==0, Print[n]], {n, 1, 2*10^6}]
Select[Range[1500000],Divisible[DivisorSigma[4,#],#]&] (* Harvey P. Dale, Jun 25 2014 *)

is(n)=sigma(n, 4)%n==0 \\ Charles R Greathouse IV, Feb 04 2013

Mod[ Sigma [ 4, n ], n ]=0.

More terms from Robert G. Wilson v, Jun 09 2000

A066109 Numbers k such that sigma_4(k)/sigma_2(k) is prime.

Original entry on oeis.org

4, 9, 20, 25, 169, 289, 961, 1849, 3721, 6889, 11881, 14641, 15625, 17161, 52441, 57121, 66049, 69169, 72361, 96721, 97969, 117649, 130321, 196249, 214369, 253009, 326041, 351649, 358801, 383161, 410881, 418609, 426409, 434281, 491401

Offset: 1

Labos Elemer, Dec 05 2001

nonn

Numbers k such that A001159(k)/A001157(k) is prime.
Except for the 3rd term 20, below 10000000 all the other terms are even powers of a prime. These primes are listed in A066111. It is not known whether other numbers similar to 20 exist or not.
20 is the only exception within the first 2000 terms. - Amiram Eldar, Feb 25 2025

			For k = 20: divisors(20) = {20, 10, 5, 4, 2, 1}, sigma_4 = 160000 + 10000 + 625 + 256 + 16 + 1 = 170898, sigma_2 = 400 + 100 + 25 + 16 + 4 + 1 = 546; p = 170898/546 = 73 is prime.

Amiram Eldar, Table of n, a(n) for n = 1..2000 (terms 1..250 from Harry J. Smith)

Cf. A000040, A001157, A001159, A046871, A066111, A066112.

Do[s = DivisorSigma[4, n]; z = DivisorSigma[2, n]; If[PrimeQ[s/z], Print[{n, s, z, s/z}]], {n, 1, 10000000}]
Select[Range[500000],PrimeQ[DivisorSigma[4,#]/DivisorSigma[2,#]]&] (* Harvey P. Dale, May 02 2011 *)

isok(k) = { my(f=sigma(k, 4)/sigma(k, 2)); !frac(f) && isprime(f) } \\ Harry J. Smith, Nov 16 2009

A066110 Primes of the form sigma_4(k)/sigma_2(k), arising in A066109.

Original entry on oeis.org

13, 73, 313, 601, 28393, 83233, 922561, 3416953, 13842121, 47451433, 141146281, 212601841, 234750601, 294482761, 2750006041, 3262751521, 4362404353, 4784281393, 5236041961, 9354855121, 9597826993, 13564461457, 16936647121

Offset: 1

Labos Elemer, Dec 05 2001

nonn

			For k = 20: divisors(20) = {20,10,5,4,2,1}, sigma_4(20) = 160000 + 10000 + 625 + 256 + 16 + 1 = 170898, sigma_2(20) = 400 + 100 + 25 + 16 + 4 + 1 = 546; p = 170898/546 = 73 is prime, the 2nd term.

Amiram Eldar, Table of n, a(n) for n = 1..2000 (terms 1..250 from Harry J. Smith)

Cf. A000040, A001157, A001159, A066109, A066111, A066112.

Do[s=DivisorSigma[4, n]; z=DivisorSigma[2, n]; If[PrimeQ[s/z], Print[s/z]], {n, 1, 10000000}]
Select[Table[DivisorSigma[4,n]/DivisorSigma[2,n],{n,200000}],PrimeQ] (* Harvey P. Dale, Jan 31 2022 *)

{ n=0; for (m=1, 10^9, if (frac(f=sigma(m, 4)/sigma(m, 2)), next); if (isprime(f), write("b066110.txt", n++, " ", f); if (n==250, return)) ) } \\ Harry J. Smith, Feb 01 2010

Primes of the form A001159(A066109(k))/A001157(A066109(k)).

A066111 Prime powers m such that sigma_4(m^2)/sigma_2(m^2) is prime.

Original entry on oeis.org

2, 3, 5, 13, 17, 31, 43, 61, 83, 109, 121, 125, 131, 229, 239, 257, 263, 269, 311, 313, 343, 361, 443, 463, 503, 571, 593, 599, 619, 641, 647, 653, 659, 701, 797, 811, 853, 953, 967, 1009, 1031, 1039, 1063, 1123, 1373, 1459, 1483, 1499, 1663, 1669, 1693

Offset: 1

Labos Elemer, Dec 06 2001

nonn

Numbers m = p^w such that A001159(m^2)/A001157(m^2) is prime, i.e., m^2 is in A066109.

Also m is the square root of a term from A066109 (omitting the term 20). Apart from 20, up to 10000000 A066109 consists of squares of prime powers.

			m=125: m^2 = 15625 = A066109(13), sigma_4(15625) = 59700165039453751, sigma_2(15625) = 254313151, sigma_4/sigma_2 = 234750601 = A066110(13) is prime. Observe also that sigma_2 is close to sigma_4/sigma_2.

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

Cf. A001157, A001159, A066109, A066110.

isok(m) = isprimepower(m) && isprime(sigma(m^2, 4)/sigma(m^2, 2)); \\ Michel Marcus, Apr 06 2020

A066284 a(n) = A066135(4*n).

Original entry on oeis.org

34, 84, 34, 84, 34, 194, 34, 84, 34, 84, 34, 228, 34, 84, 34, 84, 34, 194, 34, 84, 34, 84, 34, 228, 34, 84, 34, 84, 34, 194, 34, 84, 34, 84, 34, 386, 34, 84, 34, 84, 34, 194, 34, 84, 34, 84, 34, 228, 34, 84, 34, 84, 34, 194, 34, 84, 34, 84, 34, 228, 34, 84, 34, 84, 34, 194

Offset: 1

Labos Elemer, Dec 11 2001

nonn

a(n) <= 2p, where p = A002586(4n) is the least prime factor of (1 + 16^n). (See the Comment in A066135.) - Jonathan Sondow, Nov 23 2012

			First 3 terms correspond to entries of other sequences as follows: a(1)=A046763(2), a(2)=A055712(2), a(3)=A055716(2).
From _Michael De Vlieger_, Jul 17 2017: (Start)
First position of values, with observations pertaining to values for 1 <= n <= 3000:
    Value   Position   Observations:
    --------------------------------
       34     1        All odd.
       84     2        In A047235.
      194     6        In A017593.
      228    12
      386    36
     1282    72
     1538   144
     3084   288
   147468   576
     1956   864
  1046532  1152
    24578  2304
     3252  2880
(End)

Michael De Vlieger, Table of n, a(n) for n = 1..1000

Cf. A066135, A046761, A046762, A046763, A055709-A055717, A007691, A001159.

Table[m = 2; While[Mod[DivisorSigma[4 n, m], m] > 0, m++]; m, {n, 66}] (* Michael De Vlieger, Jul 17 2017 *)

a(n) = {n *= 4; my(m = 2); while (sigma(m, n) % m, m++); m;} \\ Michel Marcus, Oct 02 2016

a(n) = Min{x : sigma_4n(x) mod x = 0, x > 1}

cp's OEIS Frontend

A363606 Expansion of Sum_{k>0} x^(2*k)/(1-x^k)^6.

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A386747 a(n) = n^2*sigma_4(n).

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A386749 a(n) = n*sigma_4(n).

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A386784 a(n) = n^4*sigma_4(n).

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A046687 Numbers k such that k and sum of 4th powers of divisors of k are relatively prime.

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A046764 Sum of the 4th powers of the divisors of n is divisible by n.

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A066109 Numbers k such that sigma_4(k)/sigma_2(k) is prime.

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A066110 Primes of the form sigma_4(k)/sigma_2(k), arising in A066109.

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