A072861 -id:A072861 - cp's OEIS frontend

A258774 a(n) = 1 + sigma(n) + sigma(n)^2.

3, 13, 21, 57, 43, 157, 73, 241, 183, 343, 157, 813, 211, 601, 601, 993, 343, 1561, 421, 1807, 1057, 1333, 601, 3661, 993, 1807, 1641, 3193, 931, 5257, 1057, 4033, 2353, 2971, 2353, 8373, 1483, 3661, 3193, 8191, 1807, 9313, 1981, 7141, 6163, 5257, 2353

Offset: 1

Robert Price, Jun 09 2015

Robert Price, Table of n, a(n) for n = 1..10000
OEIS Wiki, Cyclotomic Polynomials at x=n, n! and sigma(n)

Cf. A000203 (sum of divisors of n).

Cf. A258775 (indices of primes in this sequence), A258776 (corresponding primes).

[1+SumOfDivisors(n)+ SumOfDivisors(n)^2: n in [1..50]]; // Vincenzo Librandi, Jun 10 2015

with(numtheory): A258774:=n->1+sigma(n)+sigma(n)^2: seq(A258774(n), n=1..100); # Wesley Ivan Hurt, Jul 09 2015

Table[1 + DivisorSigma[1, n] + DivisorSigma[1, n]^2, {n, 10000}]
Table[Cyclotomic[3, DivisorSigma[1, n]], {n, 10000}]

a(n)=my(s=sigma(n)); s^2+s+1 \\ Charles R Greathouse IV, Jun 10 2015

from sympy import divisor_sigma
def A258774(n):
    return (lambda x: x*(x+1)+1)(divisor_sigma(n)) # Chai Wah Wu, Jun 10 2015

a(n) = 1 + A000203(n) + A000203(n)^2.

a(n) = 1 + A000203(n) + A072861(n). - Omar E. Pol, Jun 19 2015

A258974 a(n) = 1 + sigma(n)^2.

Original entry on oeis.org

2, 10, 17, 50, 37, 145, 65, 226, 170, 325, 145, 785, 197, 577, 577, 962, 325, 1522, 401, 1765, 1025, 1297, 577, 3601, 962, 1765, 1601, 3137, 901, 5185, 1025, 3970, 2305, 2917, 2305, 8282, 1445, 3601, 3137, 8101, 1765, 9217, 1937, 7057, 6085, 5185, 2305

Offset: 1

Robert Price, Jun 15 2015

Robert Price, Table of n, a(n) for n = 1..10000
OEIS Wiki, Cyclotomic Polynomials at x=n, n! and sigma(n)

Cf. A000203 (sum of divisors of n).

Cf. A258976 (indices of primes in this sequence), A258977 (corresponding primes).

[(1 + DivisorSigma(1, n)^2): n in [1..50]]; // Vincenzo Librandi, Jun 16 2015

with(numtheory): A258974:=n->1+sigma(n)^2: seq(A258974(n), n=1..100); # Wesley Ivan Hurt, Jul 09 2015

Table[1 + DivisorSigma[1, n]^2, {n, 10000}]
Table[Cyclotomic[4, DivisorSigma[1, n]], {n, 10000}]

a(n)=sigma(n)^2+1 \\ Charles R Greathouse IV, Jun 18 2015

a(n) = 1 + A000203(n)^2.
a(n) = 1 + A072861(n). - Omar E. Pol, Jun 19 2015
a(n) = A002522(A000203(n)). - Michel Marcus, Jun 25 2015

A259184 a(n) = 1 - sigma(n) + sigma(n)^2.

Original entry on oeis.org

1, 7, 13, 43, 31, 133, 57, 211, 157, 307, 133, 757, 183, 553, 553, 931, 307, 1483, 381, 1723, 993, 1261, 553, 3541, 931, 1723, 1561, 3081, 871, 5113, 993, 3907, 2257, 2863, 2257, 8191, 1407, 3541, 3081, 8011, 1723, 9121, 1893, 6973, 6007, 5113, 2257, 15253

Offset: 1

Robert Price, Jun 20 2015

Robert Price, Table of n, a(n) for n = 1..10000
OEIS Wiki, Cyclotomic Polynomials at x=n, n! and sigma(n)

Cf. A000203 (sum of divisors of n).

Cf. A259185 (indices of primes in this sequence), A259186 (corresponding primes).

with(numtheory): A259184:=n->1-sigma(n)+sigma(n)^2: seq(A259184(n), n=1..100); # Wesley Ivan Hurt, Jul 09 2015

Table[1 - DivisorSigma[1, n] + DivisorSigma[1, n]^2, {n, 10000}]
Table[Cyclotomic[6, DivisorSigma[1, n]], {n, 10000}]

a(n) = polcyclo(6, sigma(n)); \\ Michel Marcus, Jun 25 2015

a(n) = 1 - A000203(n) + A000203(n)^2.
a(n) = 1 - A000203(n) + A072861(n). - Omar E. Pol, Jun 20 2015
a(n) = A002061(A000203(n)). - Michel Marcus, Jun 25 2015

A272024 Number of partitions of the sum of the divisors of n.

Original entry on oeis.org

1, 3, 5, 15, 11, 77, 22, 176, 101, 385, 77, 3718, 135, 1575, 1575, 6842, 385, 31185, 627, 53174, 8349, 17977, 1575, 966467, 6842, 53174, 37338, 526823, 5604, 5392783, 8349, 1505499, 147273, 386155, 147273, 64112359, 26015, 966467, 526823, 56634173, 53174, 118114304, 75175, 26543660, 12132164, 5392783

Offset: 1

Omar E. Pol, Apr 19 2016

nonn

Also number of partitions of the total number of parts in the partitions of n into equal parts.

Note that one of the partitions of the sum of the divisors of n is also the list of divisors of n in decreasing order, see example.

			For n = 9 the sum of the divisors of 9 is 1 + 3 + 9 = 13 and the number of partitions of 13 is A000041(13) = 101, so a(9) = 101.
Note that one of the 101 partitions of 13 is [9, 3, 1] and it is also the list of divisors of 9 in decreasing order.

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

Cf. A000041, A000203, A056538, A058699, A072861, A139041, A272209.

Table[PartitionsP@ DivisorSigma[1, n], {n, 46}] (* Michael De Vlieger, Apr 19 2016 *)

a(n) = numbpart(sigma(n)); \\ Michel Marcus, Apr 19 2016

a(n) = p(sigma(n)) = A000041(A000203(n)).

A356535 a(n) = Sum_{k=1..n} sigma_2(k)^2.

Original entry on oeis.org

1, 26, 126, 567, 1243, 3743, 6243, 13468, 21749, 38649, 53533, 97633, 126533, 189033, 256633, 372914, 457014, 664039, 795083, 1093199, 1343199, 1715299, 1996199, 2718699, 3142500, 3865000, 4537400, 5639900, 6348864, 8038864, 8964308, 10827533, 12315933, 14418433

Offset: 1

Vaclav Kotesovec, Aug 11 2022

nonn

Partial sums of A356533.

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

Cf. A001157, A057434, A061502, A072379, A356536.

Cf. A127473, A035116, A072861, A356533, A356534.

Table[Sum[DivisorSigma[2, k]^2, {k, 1, n}], {n, 1, 40}]

a(n) = sum(k=1, n, sigma(k, 2)^2); \\ Michel Marcus, Aug 11 2022

a(n) ~ 189 * zeta(3)^2 * zeta(5) * n^5 / Pi^6.

A356536 a(n) = Sum_{k=1..n} sigma_3(k)^2.

Original entry on oeis.org

1, 82, 866, 6195, 22071, 85575, 203911, 546136, 1119185, 2405141, 4179365, 8357301, 13188505, 22773721, 35220505, 57132266, 81279662, 127696631, 174756231, 259359435, 352134859, 495847003, 643907227, 912211627, 1160305628, 1551633152, 1969426752, 2600039296

Offset: 1

Vaclav Kotesovec, Aug 11 2022

nonn

Partial sums of A356534.

In general, for m>0, Sum_{k=1..n} sigma_m(k)^2 ~ zeta(2*m+1) * zeta(m+1)^2 * n^(2*m+1) / ((2*m+1) * zeta(2*m+2)).

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

Cf. A001158, A057434, A061502, A072379, A356535.

Cf. A127473, A035116, A072861, A356533, A356534.

Table[Sum[DivisorSigma[3, k]^2, {k, 1, n}], {n, 1, 40}]
Accumulate[DivisorSigma[3,Range[40]]^2] (* This program is much more efficient than the first program above. *) (* Harvey P. Dale, Feb 27 2023 *)

a(n) = sum(k=1, n, sigma(k, 3)^2); \\ Michel Marcus, Aug 11 2022

a(n) ~ zeta(7) * n^7 / 6.

A379812 a(n) = sigma_1(n) * sigma_2(n).

Original entry on oeis.org

1, 15, 40, 147, 156, 600, 400, 1275, 1183, 2340, 1464, 5880, 2380, 6000, 6240, 10571, 5220, 17745, 7240, 22932, 16000, 21960, 12720, 51000, 20181, 35700, 32800, 58800, 25260, 93600, 30784, 85995, 58560, 78300, 62400, 173901, 52060, 108600, 95200, 198900, 70644

Offset: 1

Amiram Eldar, Jan 03 2025

Srinivasa Ramanujan, Collected papers, edited by G. H. Hardy et al., Chelsea, 1962, chapter 17, pp. 133-135.

Amiram Eldar, Table of n, a(n) for n = 1..10000
R. Balasubramanian and K. Ramachandra, The place of an identity of Ramanujan in prime number theory, Proceedings of the Indian Academy of Sciences, Section A, Vol. 83. No. 4 (1976), pp. 156-165.
Yoichi Motohashi, On an identity of Ramanujan, Hardy-Ramanujan Journal, Vol. 41 (2019), pp. 48-49.
Srinivasa Ramanujan, Some formulae in the analytic theory of numbers, Messenger of Mathematics, Vol. 45 (1916), pp. 81-84.
Bertram Martin Wilson, Proofs of some formulae enunciated by Ramanujan, Proceedings of the London Mathematical Society, Vol. s2-21, No. 1 (1923), pp. 235-255.

Cf. A000203, A001157, A072861, A086666, A092346, A158274, A158275, A356533, A356534, A379813, A379814.

a[n_] := Times @@ DivisorSigma[{1, 2}, n]; Array[a, 50]

a(n) = {my(f = factor(n)); sigma(f) * sigma(f, 2);}

a(n) = A000203(n) * A001157(n).
Multiplicative with a(p^e) = (p^(e+1)-1) * (p^(2*e+2)-1) / ((p-1) * (p^2-1)).
Dirichlet g.f.: zeta(s) * zeta(s-1) * zeta(s-2) * zeta(s-3) / zeta(2*s-3).
In general, Dirichlet g.f. of  sigma_i(n) * sigma_j(n): zeta(s) * zeta(s-i) * zeta(s-j) * zeta(s-i-j) / zeta(2*s-i-j) (Ramanujan, 1916).
Sum_{k=1..n}  a(k) ~ c * n^4 / 4, where c = zeta(2) * zeta(3) * zeta(4) / zeta(5) = zeta(3) * Pi^6 / (540*zeta(5)) = 2.06386841111121962734... .
In general, Sum_{k=1..n}  sigma_i(k) * sigma_j(k) ~ c(i,j) * n^(i+j+1) / (i+j+1), for i, j >= 1, where c(i,j) = zeta(i+1) * zeta(j+1) * zeta(i+j+1) / zeta(i+j+2).
G.f.: Sum_{k>=1} Sum_{l>=1} k*l^2*x^lcm(k, l)/(1 - x^lcm(k, l)). - Miles Wilson, Jul 10 2025

A379813 a(n) = sigma_1(n) * sigma_3(n).

Original entry on oeis.org

1, 27, 112, 511, 756, 3024, 2752, 8775, 9841, 20412, 15984, 57232, 30772, 74304, 84672, 145111, 88452, 265707, 137200, 386316, 308224, 431568, 292032, 982800, 488281, 830844, 817600, 1406272, 731700, 2286144, 953344, 2359287, 1790208, 2388204, 2080512, 5028751

Offset: 1

Amiram Eldar, Jan 03 2025

See A379812 for more links and Ramanujan's general formula.

Srinivasa Ramanujan, Collected papers, edited by G. H. Hardy et al., Chelsea, 1962, chapter 17, pp. 133-135.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Srinivasa Ramanujan, Some formulae in the analytic theory of numbers, Messenger of Mathematics, Vol. 45 (1916), pp. 81-84.

Cf. A000203, A001158, A013663, A072861, A091258, A091259, A092348, A092345, A356533, A356534, A379812, A379814.

a[n_] := Times @@ DivisorSigma[{1, 3}, n]; Array[a, 50]

a(n) = {my(f = factor(n)); sigma(f) * sigma(f, 3);}

a(n) = A000203(n) * A001158(n).
Multiplicative with a(p^e) = (p^(e+1)-1) * (p^(3*e+3)-1) / ((p-1) * (p^3-1)).
Dirichlet g.f.: zeta(s) * zeta(s-1) * zeta(s-3) * zeta(s-4) / zeta(2*s-4).
Sum_{k=1..n}  a(k) ~ c * n^5 / 5, where c = 7 * zeta(5) / 4 = 1.81462357150089737107... .

A143237 Triangle read by rows, T(n, k) = A000203(n)*A000203(k), for n >= 1, 1 <= k <= n.

Original entry on oeis.org

1, 3, 9, 4, 12, 16, 7, 21, 28, 49, 6, 18, 24, 42, 36, 12, 36, 48, 84, 72, 144, 8, 24, 32, 56, 48, 96, 64, 15, 45, 60, 105, 90, 180, 120, 225, 13, 39, 52, 91, 78, 156, 104, 195, 169, 18, 54, 72, 126, 108, 216, 144, 270, 234, 324, 12, 36, 48, 84, 72, 144, 96, 180, 156, 216, 144

Offset: 1

Gary W. Adamson, Aug 01 2008

			First few rows of the triangle =
   1;
   3,  9;
   4, 12, 16;
   7, 21, 28,  49;
   6, 18, 24,  42, 36;
  12, 36, 48,  84, 72, 144;
   8, 24, 32,  56, 48,  96,  64;
  15, 45, 60, 105, 90, 180, 120, 225;
  13, 39, 52,  91, 78, 156, 104, 195, 169;
  ...
T(6,3) = 48 = sigma(6)*sigma(3) = 12*4

G. C. Greubel, Rows n = 1..50 of the triangle, flattened

Cf. A000203, A024916, A072861 (right diagonal), A130208, A143238 (row sums).

A143237:= func< n,k | DivisorSigma(1,n)*DivisorSigma(1,k) >;
[A143237(n,k): k in [1..n], n in [1..15]]; // G. C. Greubel, Sep 12 2024

A143237[n_, k_]:= DivisorSigma[1,n]*DivisorSigma[1,k];
Table[A143237[n,k], {n,15}, {k,n}]//Flatten (* G. C. Greubel, Sep 12 2024 *)

def A143237(n,k): return sigma(n,1)*sigma(k,1)
flatten([[A143237(n,k) for k in range(1,n+1)] for n in range(1,16)]) # G. C. Greubel, Sep 12 2024

Triangle read by rows, A130208 * A000012 * A130208, for 1 <= k <= n, n >= 1.
T(n, k) = sigma(n)*sigma(k), where sigma(n) = A000203(n).
Sum_{k=1..n} T(n, k) = A143238(n) (row sums).

New title by G. C. Greubel, Sep 12 2024

A232355 Numbers k such that sigma(triangular(k)) = sigma(k)^2.

Original entry on oeis.org

1, 11, 695, 991, 2839, 3707, 9347, 10703, 12847, 27089, 42251, 56419, 74671, 115289, 168739, 191051, 219295, 233729, 280111, 300731, 326899, 353651, 430859, 611799, 642991, 661715, 1035827, 1116607, 1181579, 1234519, 1365491, 1485035, 1777099, 1854671, 1905875

Offset: 1

Alex Ratushnyak, Nov 22 2013

nonn

Subsequence of A116990. - Michel Marcus, Jun 13 2015

			11 is in the sequence because sigma(11*12/2) = sigma(66) = 144 = 12^2 = sigma(11)^2.

Amiram Eldar, Table of n, a(n) for n = 1..100

Cf. A000203 (sigma(n): sum of divisors of n), A000217 (triangular(n): = n*(n+1)/2).
Cf. A074285 (sigma(triangular(n))), A072861 (sigma(n)^2).
Cf. A116990 (indices of triangular numbers whose sum of divisors is square).

[n: n in [1..7*10^5] | SumOfDivisors(n*(n+1) div 2) eq SumOfDivisors(n)^2]; // Vincenzo Librandi, Jun 13 2015

Select[Range@1000000, DivisorSigma[1, #]^2==DivisorSigma[1, (# (# + 1)/2)] &] (* Vincenzo Librandi, Jun 13 2015 *)

isok(n) = sigma(n)^2 == sigma(n*(n+1)/2); \\ Michel Marcus, Nov 23 2013

More terms from Michel Marcus, Nov 23 2013

cp's OEIS Frontend

A258774 a(n) = 1 + sigma(n) + sigma(n)^2.

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A258974 a(n) = 1 + sigma(n)^2.

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A259184 a(n) = 1 - sigma(n) + sigma(n)^2.

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A272024 Number of partitions of the sum of the divisors of n.

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A356535 a(n) = Sum_{k=1..n} sigma_2(k)^2.

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A356536 a(n) = Sum_{k=1..n} sigma_3(k)^2.

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A379812 a(n) = sigma_1(n) * sigma_2(n).

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