A001159 -id:A001159 - cp's OEIS frontend

A373039 a(n) = (A372966(n) - 1)/240.

0, 1, 27, 257, 1625, 6508, 24010, 65793, 177174, 391626, 893101, 1665644, 3398759, 5786411, 10531652, 16843009, 29065308, 42698935, 70764303, 100231882, 155608837, 215237342, 326294606, 426404460, 634767250, 819100920, 1162438641, 1480961067, 2084357107, 2538128133

Offset: 1

Hugo Pfoertner, May 20 2024

nonn

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

Cf. A000290, A001159, A013956.

f[p_, e_] := (p^(8*e + 4) + 1)/(p^4 + 1); a[1] = 0; a[n_] := (Times @@ f @@@ FactorInteger[n] - 1) / 240; Array[a, 30] (* Amiram Eldar, Jan 08 2025 *)

a(n) = (sigma(n^2, 8)/sigma(n^2, 4)-1)/240

From Amiram Eldar, Jan 08 2025: (Start)
Dirichlet g.f.: zeta(s) * (zeta(s-8)/zeta(s-4) - 1)/240.
Sum_{k=1..n} a(k) ~ c * n^9, where c = zeta(9)/(2160*zeta(5)) = 0.000447372... . (End)

A064377 Numbers n such that sigma_4(n) > phi(n)^5.

Original entry on oeis.org

2, 3, 4, 6, 8, 10, 12, 14, 15, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 48, 50, 54, 56, 60, 66, 70, 72, 78, 80, 84, 90, 96, 100, 102, 108, 110, 114, 120, 126, 130, 132, 138, 140, 144, 150, 156, 162, 168, 174, 180, 186, 192, 198, 204, 210, 216, 222, 228, 234, 240, 246, 252, 258, 264, 270, 276, 282, 294, 300, 306, 312, 330, 336, 342, 360, 378, 390, 396, 420, 450, 462, 480, 504, 510, 540, 546, 570, 600, 630, 660, 690, 714, 720, 750, 780, 840, 870, 930, 990, 1020, 1050, 1170, 1260, 1470, 1680, 2310

Offset: 1

Labos Elemer, Sep 27 2001

It is conjectured that there are no other solutions.

This sequence is finite, since by Grönwall's theorem sigma_4(n) <= sigma(n)^4 << (n log log n)^4 but phi(n)^5 >> (n/log log n)^5. - Charles R Greathouse IV, Nov 19 2015

Cf. A001159, A000010.

Select[Range[2400],DivisorSigma[4,#]>EulerPhi[#]^5&] (* Harvey P. Dale, Aug 20 2021 *)

is(n)=my(f=factor(n)); sigma(f, 4)>eulerphi(f)^5 \\ Charles R Greathouse IV, Nov 19 2015

Solutions to A001159(n) > phi(n)^5.

A066134 Numbers from A066112 that are neither square nor twice a square, i.e., are not in A028982 but are in A028983.

Original entry on oeis.org

48, 180, 245, 432, 500, 605, 931, 980, 1200, 1280, 1620, 1805, 2205, 2352, 2420, 3380, 3724, 3888, 3920, 4500, 4655, 5445, 5780, 5808, 6125, 6845, 7203, 7220, 7936, 8112, 8379, 8405, 8820, 9072, 9251, 9680, 10580, 10800, 11520, 11760, 12500

Offset: 1

Labos Elemer, Dec 06 2001

nonn

			180 is neither square nor twice a square, but sigma_4(180)/sigma_2(180) = 1135275414/49686 = 22849 = 73*313.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harry J. Smith)

Cf. A001157, A001159, A028982, A028983, A046871, A066109, A066110, A066111, A066112.

q[k_] := !IntegerQ[Sqrt[k]] && !IntegerQ[Sqrt[k/2]] && IntegerQ[r = DivisorSigma[4, k]/DivisorSigma[2, k]] && !PrimeQ[r]; Select[Range[12500], q] (* Amiram Eldar, Feb 23 2025 *)

{ n=0; for (m=1, 10^9, if (issquare(m) || issquare(m/2), next); if (frac(f=sigma(m, 4)/sigma(m, 2)), next); if (!isprime(f), write("b066134.txt", n++, " ", m); if (n==1000, return)) ) } \\ Harry J. Smith, Feb 02 2010

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

A077457 a(n) = sigma_4(n^4)/sigma_2(n^4).

Original entry on oeis.org

1, 205, 5905, 52429, 375601, 1210525, 5649505, 13421773, 38742049, 76998205, 212601841, 309593245, 810932305, 1158148525, 2217923905, 3435973837, 6951703105, 7942120045, 16936647121, 19692384829, 33360327025, 43583377405, 78163228705, 79255569565, 146719125601

Offset: 1

Benoit Cloitre, Nov 30 2002

sigma_y(n^x) divides sigma_x(n^x) for all n if y divides x.

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

Cf. A001157, A001159, A013667, A057660, A077454, A077455, A077456.

f[p_, e_] := (p^(8*e+2) + 1)/(p^2 + 1); a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 25] (* Amiram Eldar, Sep 09 2020 *)

a(n)=sumdiv(n^4,d,d^4)/sumdiv(n^4,d,d^2)

a(n) = my(f=factor(n^4)); sigma(f, 4)/sigma(f, 2); \\ Michel Marcus, Sep 09 2020

a(n) = A001159(n^4)/A001157(n^4).
Multiplicative with a(p^e) = (p^(8*e+2) + 1)/(p^2 + 1). - Amiram Eldar, Sep 09 2020
Sum_{k=1..n} a(k) ~ c * n^9, where c = (zeta(9)/9) * Product_{p prime} (1 - 1/p^3 + 1/p^5 - 1/p^7) = 0.09549806119... . - Amiram Eldar, Oct 28 2022

A078553 Largest prime dividing sigma(4,n).

Original entry on oeis.org

17, 41, 13, 313, 41, 1201, 257, 73, 313, 7321, 41, 14281, 1201, 313, 41, 41761, 73, 3833, 313, 1201, 7321, 139921, 257, 601, 14281, 193, 1201, 353641, 313, 1129, 241, 7321, 41761, 1201, 73, 10529, 3833, 14281, 313, 10313, 1201, 521, 7321, 313

Offset: 2

Labos Elemer, Dec 05 2002

nonn

			Observe nontrivial frequent occurrence of several primes like 73,313,14281, etc.

Amiram Eldar, Table of n, a(n) for n = 2..10000

Cf. A006530, A001157-A001160, A071190.

gpf[n_] := FactorInteger[n][[-1, 1]]; a[n_] := gpf[DivisorSigma[4, n]]; Array[a, 50, 2] (* Amiram Eldar, Aug 01 2019 *)

a(n) = A006530(A001159(n)).

A319278 Square array sigma_k(n) read down antidiagonals: sum of the k-th powers of the divisors of n.

Original entry on oeis.org

1, 1, 3, 1, 5, 4, 1, 9, 10, 7, 1, 17, 28, 21, 6, 1, 33, 82, 73, 26, 12, 1, 65, 244, 273, 126, 50, 8, 1, 129, 730, 1057, 626, 252, 50, 15, 1, 257, 2188, 4161, 3126, 1394, 344, 85, 13, 1, 513, 6562, 16513, 15626, 8052, 2402, 585, 91, 18, 1, 1025, 19684, 65793, 78126, 47450, 16808, 4369, 757, 130, 12

Offset: 1

R. J. Mathar, Sep 16 2018

Equals the square array A082771 without its first column.

			The array starts in row n=1 with columns k>=1 as:
     1      1      1      1      1      1       1        1
     3      5      9     17     33     65     129      257
     4     10     28     82    244    730    2188     6562
     7     21     73    273   1057   4161   16513    65793
     6     26    126    626   3126  15626   78126   390626
    12     50    252   1394   8052  47450  282252  1686434
     8     50    344   2402  16808 117650  823544  5764802
    15     85    585   4369  33825 266305 2113665 16843009

Columns: A000203, A001157, A001158, A001159.
Rows: A000012, A000051, A034472, A001576, A034474, A034488.
Cf. A082771, A023887 (diagonal), A109974, A319194 (partial column sums).

T[n_, k_] := DivisorSigma[k, n];
Table[T[n-k+1, k], {n, 1, 11}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Dec 16 2021 *)

sigma_k(n) = sum_{d|n} d^k.

A321598 a(n) = Sum_{d|n} d*binomial(d+2,3).

Original entry on oeis.org

1, 9, 31, 89, 176, 375, 589, 1049, 1516, 2384, 3147, 4823, 5916, 8437, 10406, 14105, 16474, 22380, 25271, 33264, 37810, 47683, 52901, 68183, 73301, 91100, 100174, 122197, 130356, 161750, 169137, 205593, 219162, 259242, 272714, 330524, 338144, 400719, 421686, 493424

Offset: 1

Ilya Gutkovskiy, Nov 14 2018

Inverse Möbius transform of A002417.

Amiram Eldar, Table of n, a(n) for n = 1..10000
N. J. A. Sloane, Transforms.

Cf. A000292, A000335, A001157, A001158, A001159, A002417, A013663, A059358, A278403.

Table[Sum[d Binomial[d + 2, 3], {d, Divisors[n]}], {n, 40}]
nmax = 40; Rest[CoefficientList[Series[Sum[x^k (1 + 3 x^k)/(1 - x^k)^5, {k, 1, nmax}], {x, 0, nmax}], x]]
Table[(2 DivisorSigma[2, n] + 3 DivisorSigma[3, n] + DivisorSigma[4, n])/6, {n, 40}]

a(n) = my(f = factor(n)); (sigma(f, 4) + 3*sigma(f, 3) + 2*sigma(f, 2)) / 6; \\ Amiram Eldar, Jan 02 2025

G.f.: Sum_{k>=1} x^k*(1 + 3*x^k)/(1 - x^k)^5.
G.f.: Sum_{k>=1} k*A000292(k)*x^k/(1 - x^k).
L.g.f.: -log(Product_{k>=1} (1 - x^k)^A000292(k)) = Sum_{n>=1} a(n)*x^n/n.
Dirichlet g.f.: (zeta(s-4) + 3*zeta(s-3) + 2*zeta(s-2))*zeta(s)/6.
a(n) = (2*sigma_2(n) + 3*sigma_3(n) + sigma_4(n))/6.
a(n) = Sum_{d|n} A002417(d).
Sum_{k=1..n} a(k) ~ zeta(5) * n^5 / 30. - Vaclav Kotesovec, Feb 02 2019

A347160 Sum of 4th powers of distinct prime divisors of n that are <= sqrt(n).

Original entry on oeis.org

0, 0, 0, 16, 0, 16, 0, 16, 81, 16, 0, 97, 0, 16, 81, 16, 0, 97, 0, 16, 81, 16, 0, 97, 625, 16, 81, 16, 0, 722, 0, 16, 81, 16, 625, 97, 0, 16, 81, 641, 0, 97, 0, 16, 706, 16, 0, 97, 2401, 641, 81, 16, 0, 97, 625, 2417, 81, 16, 0, 722, 0, 16, 2482, 16, 625, 97, 0, 16, 81, 3042

Offset: 1

Ilya Gutkovskiy, Aug 20 2021

nonn

Cf. A001159, A005065, A005068, A063962, A097974, A098002, A347143, A347158, A347159.

Table[DivisorSum[n, #^4 &, # <= Sqrt[n] && PrimeQ[#] &], {n, 1, 70}]
nmax = 70; CoefficientList[Series[Sum[Prime[k]^4 x^(Prime[k]^2)/(1 - x^Prime[k]), {k, 1, nmax}], {x, 0, nmax}], x] // Rest

G.f.: Sum_{k>=1} prime(k)^4 * x^(prime(k)^2) / (1 - x^prime(k)).

A347172 Sum of 4th powers of odd divisors of n that are < sqrt(n).

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 82, 1, 1, 82, 1, 1, 82, 1, 1, 82, 1, 1, 82, 1, 1, 82, 1, 1, 707, 1, 1, 82, 1, 626, 82, 1, 1, 82, 626, 1, 82, 1, 1, 707, 1, 1, 82, 1, 626, 82, 1, 1, 82, 626, 2402, 82, 1, 1, 707, 1, 1, 2483, 1, 626, 82, 1, 1, 82, 3027, 1, 82, 1, 1, 707

Offset: 1

Ilya Gutkovskiy, Aug 21 2021

nonn

Cf. A001159, A051001, A333805, A333807, A347142, A347161, A347162.

Table[DivisorSum[n, #^4 &, # < Sqrt[n] && OddQ[#] &], {n, 1, 75}]
nmax = 75; CoefficientList[Series[Sum[(2 k - 1)^4 x^(2 k (2 k - 1))/(1 - x^(2 k - 1)), {k, 1, nmax}], {x, 0, nmax}], x] // Rest

G.f.: Sum_{k>=1} (2*k - 1)^4 * x^(2*k*(2*k - 1)) / (1 - x^(2*k - 1)).

cp's OEIS Frontend

A373039 a(n) = (A372966(n) - 1)/240.

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A064377 Numbers n such that sigma_4(n) > phi(n)^5.

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A066134 Numbers from A066112 that are neither square nor twice a square, i.e., are not in A028982 but are in A028983.

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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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A077457 a(n) = sigma_4(n^4)/sigma_2(n^4).

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A078553 Largest prime dividing sigma(4,n).

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A319278 Square array sigma_k(n) read down antidiagonals: sum of the k-th powers of the divisors of n.

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A321598 a(n) = Sum_{d|n} d*binomial(d+2,3).

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