A013954 -id:A013954 - cp's OEIS frontend

A082771 Triangular array, read by rows: t(n,k) = Sum_{d|n} d^k, 0 <= k < n.

1, 2, 3, 2, 4, 10, 3, 7, 21, 73, 2, 6, 26, 126, 626, 4, 12, 50, 252, 1394, 8052, 2, 8, 50, 344, 2402, 16808, 117650, 4, 15, 85, 585, 4369, 33825, 266305, 2113665, 3, 13, 91, 757, 6643, 59293, 532171, 4785157, 43053283, 4, 18, 130, 1134, 10642, 103158, 1015690, 10078254, 100390882, 1001953638

Offset: 1

Reinhard Zumkeller, May 21 2003

			From _R. J. Mathar_, Dec 06 2006 (Start):
The triangle may be extended to a rectangular array (A319278):
  1  1   1    1     1 1 1 1 1 1 1 ...
  2  3   5    9    17 33 65 129 257 513 1025 ...
  2  4  10   28    82 244 730 2188 6562 19684 59050 ...
  3  7  21   73   273 1057 4161 16513 65793 262657 1049601 ...
  2  6  26  126   626 3126 15626 78126 390626 1953126 9765626 ...
  4 12  50  252  1394 8052 47450 282252 1686434 10097892 60526250 ...
  2  8  50  344  2402 16808 117650 823544 5764802 40353608 282475250 ...
  4 15  85  585  4369 33825 266305 2113665 16843009 134480385 1074791425 ...
  3 13  91  757  6643 59293 532171 4785157 43053283 387440173 3486843451 ...
  4 18 130 1134 10642 103158 1015690 10078254 100390882 1001953638... (End)

T. D. Noe, Rows n=1..100, flattened
Eric Weisstein's World of Mathematics, Divisor Function.

Cf. A000005, A000203, A001157, A001158, A001159, A001160.
Cf. A013954, A013955, A013956, A013957, A013958, A013959, A013960, A013961, A013962, A013963.
Cf. A013964, A013965, A013966, A013967, A013968, A013969, A013970, A013971, A013972.

T:= (n,k)-> numtheory[sigma][k](n):
seq(seq(T(n,k), k=0..n-1), n=1..10);  # Alois P. Heinz, Oct 25 2024

T[n_, k_] := DivisorSigma[k, n];
Table[T[n, k], {n, 1, 10}, {k, 0, n-1}] // Flatten (* Jean-François Alcover, Dec 16 2021 *)

row(n) = {my(f = factor(n)); vector(n, k, sigma(f, k-1));} \\ Amiram Eldar, May 09 2025

t(n, k) = Product(((p^((e(n, p)+1)*k))-1)/(p^k-1): n=Product(p^e(n, p): p prime)), 0<=k

t(n,0) = A000005(n), t(n,n) = A023887(n).

t(n,1) = A000203(n), n>1; t(n,2) = A001157(n), n>2; t(n,3) = A001158(n), n>3.

t(n,4) = A001159(n), n>4; t(n,5) = A001160(n), n>5; t(n,6) = A013954(n), n>6.

From R. J. Mathar, Oct 29 2006: (Start)

t(2,k) = A000051(k); t(3,k) = A034472(k); t(4,k) = A001576(k);

t(5,k) = A034474(k); t(6,k) = A034488(k); t(7,k) = A034491(k);

t(8,k) = A034496(k); t(9,k) = A034513(k); t(10,k) = A034517(k);

t(11,k) = A034524(k); t(12,k) = A034660(k). (End)

Corrected by R. J. Mathar, Dec 05 2006

A245466 a(n) = sigma_1(1) + sigma_2(2) + sigma_3(3) + ... + sigma_n-1(n-1) + sigma_n(n).

Original entry on oeis.org

1, 6, 34, 307, 3433, 50883, 874427, 17717436, 405157609, 10414924259, 295726594871, 9214021138217, 312089127730471, 11424774176377721, 449318695089164129, 18896344248070459234, 846136606134407223412, 40192694877626586149007, 2018612350537940175272987

Offset: 1

Wesley Ivan Hurt, Jul 22 2014

nonn

Let sigma_k(n) represent the sum of the k-th powers of the divisors of n.
Then a(n) = Sum_{k=1..n} sigma_k(k), the partial sums of sigma_k(k) for k from 1 to n.
Partial sums of A023887.

			a(1) = 1 because sigma_1(1) = sigma(1) = 1.
a(2) = 6: sigma_1(1) + sigma_2(2) = 1 + (1^2 + 2^2) = 6.
a(3) = 34: sigma_1(1) + sigma_2(2) + sigma_3(3) = 6 + (1^3 + 3^3) = 34.
a(4) = 307: sigma_1(1) + ... + sigma_4(4) = 34 + (1^4 + 2^4 + 4^4) = 307.

Jens Kruse Andersen, Table of n, a(n) for n = 1..100

Cf. A000203, A001157-A001160, A013954-A013958, A023887.

[&+[DivisorSigma(i, i): i in [1..n]]: n in [1..20]]; // Bruno Berselli, Jul 29 2014

[n eq 1 select 1 else Self(n-1)+ DivisorSigma(n, n): n in [1..20]]; // Vincenzo Librandi, Aug 05 2015

B:= [seq(numtheory:-sigma[n](n),n=1..100)]:
seq(add(B[i],i=1..n),n=1..100); # Robert Israel, Jul 28 2014

Table[Sum[DivisorSigma[k, k], {k, n}], {n, 20}]
Accumulate[Table[DivisorSigma[n,n],{n,20}]] (* Harvey P. Dale, Apr 10 2018 *)

a(n) = sum(i=1,n,sigma(i,i))
vector(50, n, a(n)) \\ Derek Orr, Jul 27 2014

a(n) = Sum_{k=1..n} sigma_k(k).
a(1) = 1. a(n) = a(n-1) + sigma_n(n), for n > 1. - Jens Kruse Andersen, Jul 29 2014
a(n) = n + Sum_{d=2..n} (d^(d*(floor(n/d)+1))-d^d)/(d^d-1). - Chayim Lowen, Aug 04 2015

A301550 Expansion of Product_{k>=1} (1 + x^k)^(sigma_6(k)).

Original entry on oeis.org

1, 1, 65, 795, 6971, 69317, 690756, 6316950, 55729130, 484275457, 4111328940, 34029153900, 275901508917, 2197552381491, 17207716281240, 132575879110175, 1006214596929014, 7531171360277228, 55632520744009711, 405876769498808480, 2926507055330036936

Offset: 0

Vaclav Kotesovec, Mar 23 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A013954, A107742, A301544.

nmax = 30; CoefficientList[Series[Product[(1+x^k)^DivisorSigma[6, k], {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ exp(2^(5/2) * Pi * (127*Zeta(7)/15)^(1/8) * n^(7/8)/7 - Pi * (5/(127*Zeta(7)))^(1/8) * n^(1/8) / (504 * sqrt(2) * 3^(7/8))) * (127*Zeta(7)/15)^(1/16) / (2^(9/4) * n^(9/16)).

G.f.: exp(Sum_{k>=1} sigma_7(k)*x^k/(k*(1 - x^(2*k)))). - Ilya Gutkovskiy, Oct 26 2018

A013962 a(n) = sigma_14(n), the sum of the 14th powers of the divisors of n.

Original entry on oeis.org

1, 16385, 4782970, 268451841, 6103515626, 78368963450, 678223072850, 4398314962945, 22876797237931, 100006103532010, 379749833583242, 1283997101947770, 3937376385699290, 11112685048647250, 29192932133689220, 72061992352890881, 168377826559400930

Offset: 1

N. J. A. Sloane

If the canonical factorization of n into prime powers is the product of p^e(p) then sigma_k(n) = Product_p ((p^((e(p)+1)*k))-1)/(p^k-1).

Sum_{d|n} 1/d^k is equal to sigma_k(n)/n^k. So sequences A017665-A017712 also give the numerators and denominators of sigma_k(n)/n^k for k = 1..24. The power sums sigma_k(n) are in sequences A000203 (k=1), A001157-A001160 (k=2,3,4,5), A013954-A013972 for k = 6,7,...,24. - Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 05 2001

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for sequences related to sigma(n).

Cf. A000203, A001157-A001160, A013673, A013954-A013972, A017665-A017712.

[DivisorSigma(14, n): n in [1..20]]; // Vincenzo Librandi, Sep 10 2016

DivisorSigma[14,Range[20]] (* Harvey P. Dale, Mar 10 2013 *)

my(N=99, q='q+O('q^N)); Vec(sum(n=1, N, n^14*q^n/(1-q^n))) \\ Altug Alkan, Sep 10 2016

a(n) = sigma(n, 14); \\ Amiram Eldar, Oct 29 2023

[sigma(n,14) for n in range(1,16)] # Zerinvary Lajos, Jun 04 2009

G.f.: Sum_{k>=1} k^14*x^k/(1-x^k). - Benoit Cloitre, Apr 21 2003
Dirichlet g.f.: zeta(s-14)*zeta(s). - Ilya Gutkovskiy, Sep 10 2016
From Amiram Eldar, Oct 29 2023: (Start)
Multiplicative with a(p^e) = (p^(14*e+14)-1)/(p^14-1).
Sum_{k=1..n} a(k) = zeta(15) * n^15 / 15 + O(n^16). (End)

A013966 a(n) = sigma_18(n), the sum of the 18th powers of the divisors of n.

Original entry on oeis.org

1, 262145, 387420490, 68719738881, 3814697265626, 101560344351050, 1628413597910450, 18014467229220865, 150094635684419611, 1000003814697527770, 5559917313492231482, 26623434909949071690, 112455406951957393130, 426880482624234915250, 1477891883850485076740

Offset: 1

N. J. A. Sloane

If the canonical factorization of n into prime powers is the product of p^e(p) then sigma_k(n) = Product_p ((p^((e(p)+1)*k))-1)/(p^k-1).

Seiichi Manyama, Table of n, a(n) for n = 1..10000
Index entries for sequences related to sigma(n).

Cf. A000203, A001157-A001160, A013677, A013954-A013972, A017665-A017712.

[DivisorSigma(18,n): n in [1..50]]; // G. C. Greubel, Nov 03 2018

Table[DivisorSigma[18,n],{n,50}] (* Vladimir Joseph Stephan Orlovsky, Mar 11 2009 *)

a(n)=sigma(n,18) \\ Charles R Greathouse IV, Apr 28 2011

[sigma(n,18)for n in range(1,13)] # Zerinvary Lajos, Jun 04 2009

G.f.: Sum_{k>=1} k^18*x^k/(1-x^k). - Benoit Cloitre, Apr 21 2003
From Amiram Eldar, Oct 29 2023: (Start)
Multiplicative with a(p^e) = (p^(18*e+18)-1)/(p^18-1).
Dirichlet g.f.: zeta(s)*zeta(s-18).
Sum_{k=1..n} a(k) = zeta(19) * n^19 / 19. + O(n^20). (End)

A013971 a(n) = sigma_23(n), the sum of the 23rd powers of the divisors of n.

Original entry on oeis.org

1, 8388609, 94143178828, 70368752566273, 11920928955078126, 789730317205170252, 27368747340080916344, 590295880727458217985, 8862938119746644274757, 100000011920928963466734, 895430243255237372246532, 6624738056749922960468044, 41753905413413116367045798

Offset: 1

N. J. A. Sloane

If the canonical factorization of n into prime powers is the product of p^e(p) then sigma_k(n) = Product_p ((p^((e(p)+1)*k))-1)/(p^k-1).

Seiichi Manyama, Table of n, a(n) for n = 1..10000 (first 1000 terms from Harvey P. Dale)
Index entries for sequences related to sigma(n).

Cf. A000203, A001157-A001160, A013954-A013972, A017665-A017712.

[DivisorSigma(23,n): n in [1..30]]; // G. C. Greubel, Nov 03 2018

DivisorSigma[23,Range[15]] (* Harvey P. Dale, May 02 2016 *)

vector(30, n, sigma(n,23)) \\ G. C. Greubel, Nov 03 2018

[sigma(n,23)for n in range(1,12)] # Zerinvary Lajos, Jun 04 2009

G.f.: Sum_{k>=1} k^23*x^k/(1-x^k). - Benoit Cloitre, Apr 21 2003
From Amiram Eldar, Oct 29 2023: (Start)
Multiplicative with a(p^e) = (p^(23*e+23)-1)/(p^23-1).
Dirichlet g.f.: zeta(s)*zeta(s-23).
Sum_{k=1..n} a(k) = zeta(24) * n^24 / 24 + O(n^25). (End)

A055710 Numbers k such that k | sigma_6(k).

Original entry on oeis.org

1, 10, 26, 60, 65, 130, 150, 228, 260, 442, 650, 780, 876, 988, 1105, 1140, 1460, 1690, 1950, 2210, 2850, 2964, 3211, 3796, 4380, 4420, 4940, 5070, 5475, 5548, 6010, 6422, 8840, 9633, 10950, 11050, 11388, 11972, 12350, 12818, 13260, 13756, 14820, 16644

Offset: 1

Robert G. Wilson v, Jun 09 2000

nonn

sigma_6(k) is the sum of the 6th powers of the divisors of k (A013954).

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

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..450 from Harvey P. Dale)
Florian Luca and John Ferdinands, Problem 11090: Sometimes n divides sigma_k(n), Amer. Math. Monthly 113:4 (2006), pp. 372-373.

Cf. A013954.

Cf. A055709, A055711, A055712, A055713.

Do[If[Mod[DivisorSigma[6, n], n]==0, Print[n]], {n, 1, 25000}]
Select[Range[20000],Divisible[DivisorSigma[6,#],#]&] (* Harvey P. Dale, Jun 04 2015 *)

is(n)=sigma(n,6)%n==0 \\ Charles R Greathouse IV, Feb 04 2013

A211347 Numbers n such that n = sigma_k(m) for some k >= 1.

Original entry on oeis.org

1, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 17, 18, 20, 21, 24, 26, 28, 30, 31, 32, 33, 36, 38, 39, 40, 42, 44, 48, 50, 54, 56, 57, 60, 62, 63, 65, 68, 72, 73, 74, 78, 80, 82, 84, 85, 90, 91, 93, 96, 98, 102, 104, 108, 110, 112, 114, 120, 121, 122

Offset: 1

Jon Perry, Feb 05 2013

nonn

Sigma_k(n) = Sum[d|n, d^k].
Sigma_0(n) can be any positive integer and so is ignored in this sequence.
The asymptotic density of this sequence is 0 (Niven, 1951, Rao and Murty, 1979). - Amiram Eldar, Jul 23 2020

			Sigma_2(4) = 1 + 4 + 16 = 21 so 21 is in the sequence.

Giovanni Resta, Table of n, a(n) for n = 1..10000
Ivan Niven, The asymptotic density of sequences, Bull. Amer. Math. Soc., Vol. 57 (1951), pp. 420-434.
R. Sita Rama Chandra Rao and G. Sri Rama Chandra Murty, On a theorem of Niven, Canadian Mathematical Bulletin, Vol 22, No. 1 (1979), pp. 113-115.
Eric W. Weisstein, MathWorld: Divisor function

Cf. A000203, A001157, A001158, A001159, A001160.
Cf. A013954, A013955, A013956, A013957, A013958, A013959, A013960, A013961, A013962, A013963, A013964, A013965, A013966, A013967, A013968, A013969, A013970, A013971, A013972.
Cf. A000005.

upto[n_] := Select[Union@Flatten[{1, DivisorSigma[Range@Max[1,Floor@Log[#,n]], #] & /@ Range[2,n]}], # <= n &]; upto[122] (* Giovanni Resta, Feb 05 2013 *)

list(lim)=if(lim<3, return(if(lim<1,[],[1]))); my(v=List([1])); for(k=1,logint((lim\=1)-1,2), forfactored(m=2,sqrtnint(lim-1,k), my(t=sigma(m,k)); if(t<=lim, listput(v,t)))); Set(v) \\ Charles R Greathouse IV, Apr 09 2022

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cp's OEIS Frontend

A017667 Numerator of sum of -2nd powers of divisors of n.

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A068027 Z(S_m; sigma[1](n), sigma[2](n),..., sigma[m](n)) where Z(S_m; x_1,x_2,...,x_m) is the cycle index of the symmetric group S_m and sigma[k](n) is the sum of k-th powers of divisors of n; m=10.

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A082771 Triangular array, read by rows: t(n,k) = Sum_{d|n} d^k, 0 <= k < n.

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A245466 a(n) = sigma_1(1) + sigma_2(2) + sigma_3(3) + ... + sigma_n-1(n-1) + sigma_n(n).

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A301550 Expansion of Product_{k>=1} (1 + x^k)^(sigma_6(k)).

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A013962 a(n) = sigma_14(n), the sum of the 14th powers of the divisors of n.

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A013966 a(n) = sigma_18(n), the sum of the 18th powers of the divisors of n.

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