A013955 -id:A013955 - cp's OEIS frontend

A126812 Ramanujan numbers (A000594) read mod 4.

1, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0

Offset: 1

N. J. A. Sloane, Feb 25 2007

nonn

D. B. Lahiri, On Ramanujan's function tau(n) and divisor function sigma_k(n), I, Bulletin of the Calcutta Mathematical Society, Vol. 38 (1946), pp. 193-206; II, ibid., Vol. 39 (1947), pp. 33-51.

Antti Karttunen, Table of n, a(n) for n = 1..65537
H. P. F. Swinnerton-Dyer, On l-adic representations and congruences for coefficients of modular forms, pp. 1-55 of Modular Functions of One Variable III (Antwerp 1972), Lect. Notes Math., 350, 1973.

Cf. A000594, A008442, A013955, A098108, A126813, A126814.

Mod[#, 4] & /@ RamanujanTau@ Range@ 105 (* Michael De Vlieger, Nov 26 2017 *)

A126812(n) = (ramanujantau(n)%4); \\ Antti Karttunen, Nov 26 2017

a(n) == n^2 * sigma_7(n) (mod 4) (Lahiri, 1946-1947). - Amiram Eldar, Jan 04 2025

A126834 Ramanujan numbers (A000594) read mod 125.

Original entry on oeis.org

1, 101, 2, 28, 80, 77, 6, 105, 107, 80, 112, 56, 12, 106, 35, 11, 66, 57, 45, 115, 12, 62, 22, 85, 25, 87, 45, 43, 5, 35, 82, 71, 99, 41, 105, 121, 61, 45, 24, 25, 67, 87, 42, 11, 60, 97, 121, 22, 43, 25, 7, 86, 52, 45, 85, 5, 90, 5, 10, 105, 37, 32, 17, 18, 85, 124, 116, 98, 44

Offset: 1

N. J. A. Sloane, Feb 25 2007

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..10000
R. P. Bambah and S. Chowla, Congruence properties of Ramanujan’s function tau(n), Bull. Amer. Math. Soc. 53 (1947), 950-955.
H. P. F. Swinnerton-Dyer, On l-adic representations and congruences for coefficients of modular forms, pp. 1-55 of Modular Functions of One Variable III (Antwerp 1972), Lect. Notes Math., 350, 1973.

Cf. A000594, A001157, A013955.

Cf. A126832 (mod 5^1), A126833 (mod 5^2), this sequence (mod 5^3), A126835 (mod 5^4).

Mod[RamanujanTau@ #, 125] & /@ Range@ 69 (* Michael De Vlieger, Apr 26 2016 *)

a(n) = ramanujantau(n) % 125; \\ Amiram Eldar, Jan 05 2025

a(n) = (5*n^2*sigma_7(n) - 4*n*sigma_2(n)) mod 125, for n coprime to 5. - Michel Marcus, Apr 26 2016

A068024 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=7.

Original entry on oeis.org

1, 255, 3280, 43435, 97656, 998184, 960800, 6347715, 8069620, 27615060, 21435888, 184770040, 67977560, 263540112, 343123440, 866251507, 435984840, 2595218340, 943531280, 4944199260, 3308659904, 5722701624, 3559590240

Offset: 1

Vladeta Jovovic, Feb 08 2002

nonn

Cf. A067692, A068020-A068023, A068025-A068027, A000203, A001157-A001160, A013954, A013955.

CIP7 = CycleIndexPolynomial[SymmetricGroup[7], Array[x, 7]]; a[n_] := CIP7 /. x[k_] -> DivisorSigma[k, n]; Array[a, 23] (* Jean-François Alcover, Nov 04 2016 *)

1/7!*(sigma[1](n)^7 + 21*sigma[1](n)^5*sigma[2](n) + 70*sigma[1](n)^4*sigma[3](n) + 105*sigma[1](n)^3*sigma[2](n)^2 + 210*sigma[1](n)^3*sigma[4](n) + 420*sigma[1](n)^2*sigma[2](n)*sigma[3](n) + 105*sigma[1](n)*sigma[2](n)^3 + 504*sigma[1](n)^2*sigma[5](n) + 630*sigma[1](n)*sigma[2](n)*sigma[4](n) + 280*sigma[1](n)*sigma[3](n)^2 + 210*sigma[2](n)^2*sigma[3](n) + 840*sigma[1](n)*sigma[6](n) + 504*sigma[2](n)*sigma[5](n) + 420*sigma[3](n)*sigma[4](n) + 720*sigma[7](n)).

A087115 Convolution of sum of cubes of divisors with itself.

Original entry on oeis.org

0, 1, 18, 137, 650, 2350, 6860, 17609, 39870, 83976, 162382, 301070, 522886, 885284, 1424468, 2254537, 3419448, 5143987, 7448874, 10750712, 15015872, 20948610, 28373444, 38539022, 50863150, 67454492, 87209316, 113326308, 143748766, 183759900, 229271536

Offset: 1

Michael Somos, Aug 13 2003

nonn

Ramanujan's Eisenstein series: P(q) (see A006352), Q(q) (A004009), R(q) (A013973).

			G.f. = x^2 + 18*x^3 + 137*x^4 + 650*x^5 + 2350*x^6 + 6860*x^7 + 17609*x^8 + ...

Jean-Pierre Serre, A Course in Arithmetic, Springer-Verlag, 1973, Chapter VII, Section 4., p. 93.

Seiichi Manyama, Table of n, a(n) for n = 1..1000

Cf. A004009.

Cf. A001158 (sigma_3), A013955 (sigma_7). [Ridouane Oudra, Apr 22 2020]

with(numtheory); f:=n->add( sigma[3](k)*sigma[3](n-k),k=1..n-1);

a[ n_] := If[ n < 1, 0, (DivisorSigma[ 7, n] - DivisorSigma[ 3, n]) / 120]; (* Michael Somos, Oct 08 2017 *)

{a(n) = if( n<1, 0, (sigma(n, 7) - sigma(n, 3)) / 120)};

{a(n) = if( n<1, 0, sum(m=1, n-1, sigma(m, 3) * sigma(n-m, 3)))};

G.f.: (Sum_{k>0} k^3 * x^k / (1 - x^k))^2.
a(n) = (sigma_7(n) - sigma_3(n)) / 120.
G.f.: ((Q(x) - 1) / 240)^2 where Q() is a Ramanujan Eisenstein series.
Dirichlet g.f.: zeta(s) * (zeta(s-7) - zeta(s-3)) / 120. - Amiram Eldar, Jan 11 2025
Sum_{k=1..n} a(k) ~ Pi^8 * n^8 / 9072000. - Vaclav Kotesovec, Aug 20 2025

A279917 a(n) = Sum_{k=1..n-1} sigma_5(k)*sigma_7(n-k).

Original entry on oeis.org

0, 1, 162, 6689, 121250, 1296406, 9613604, 54550049, 252178758, 992204376, 3424910566, 10615778966, 30047257318, 78751366604, 193075366412, 446840757793, 982597838280, 2066025009763, 4171924730922, 8127978871064, 15324653827568, 28059983210370

Offset: 1

Seiichi Manyama, Dec 23 2016

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..1000

Cf. A001160, A013955, A013961, A279888.

a[n_] := (DivisorSigma[13, n] + 20 * DivisorSigma[7, n] - 21 * DivisorSigma[5, n]) / 10080; Array[a, 25] (* Amiram Eldar, Jan 07 2025 *)

a(n) = {my(f = factor(n)); (sigma(f, 13) + 20 * sigma(f, 7) - 21 * sigma(f, 5)) / 10080;} \\ Amiram Eldar, Jan 07 2025

a(n) = (sigma_13(n)+20*sigma_7(n)-21*sigma_5(n))/10080.

A279964 a(n) = Sum_{k=1..n-1} sigma_3(k)*sigma_7(n-k).

Original entry on oeis.org

0, 1, 138, 3377, 39890, 297550, 1623980, 7065329, 25808790, 82305816, 234756742, 611706830, 1474831246, 3334313204, 7118797268, 14485772017, 28206850488, 52921773667, 95877425634, 168644231672, 288301373792, 481166453010, 784226941604, 1253068878542, 1962356045590

Offset: 1

Seiichi Manyama, Dec 23 2016

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..1000

Cf. A000594, A001158, A001160, A013955, A279889.

Table[Sum[DivisorSigma[3, k] * DivisorSigma[7, n - k], {k, 1, n - 1}], {n, 1, 25}] (* Indranil Ghosh, Mar 12 2017 *)
a[n_] := (1800 * RamanujanTau[n] + 273 * DivisorSigma[11, n] - 1382 * DivisorSigma[7, n] - 691 * DivisorSigma[3, n]) / 331680; Array[a, 25] (* Amiram Eldar, Jan 07 2025 *)

a(n) = sum(k=1, n-1, sigma(k, 3)*sigma(n-k,7)); \\ Michel Marcus, Dec 24 2016

a(n) = {my(f = factor(n)); (1800 * ramanujantau(n) + 273 * sigma(f, 11) - 1382 * sigma(f, 7) - 691 * sigma(f, 3)) / 331680;} \\ Amiram Eldar, Jan 07 2025

36*tau(n) = 5*sigma_3(n) + 10*sigma_7(n) + 21*sigma_5(n) + 2400*a(n) - 5292*A279889(n).

A289744 Coefficients in expansion of q*E'_8 where E_8 is the Eisenstein Series (A008410).

Original entry on oeis.org

480, 123840, 3150720, 31704960, 187502400, 812885760, 2767107840, 8116473600, 20671878240, 48375619200, 102892268160, 208111357440, 391550752320, 713913822720, 1230765753600, 2077817249280, 3348363579840, 5333344585920, 8152110268800, 12384908524800

Offset: 1

Seiichi Manyama, Jul 11 2017

nonn

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

(-1)^(k/2)*q*E'_{k}: A076835 (k=2), A145094 (k=4), A145095 (k=6), this sequence (k=8), A289745 (k=10), A289746 (k=14).

Cf. A008410 (E_8), A013955, A282060, A289638.

Table[480*n*DivisorSigma[7, n], {n, 1, 20}] (* Jean-François Alcover, Feb 27 2018 *)

a(n) = 480 * n * sigma(n, 7); \\ Amiram Eldar, Jan 07 2025

a(n) = 480*A282060(n) = 480*n*A013955(n).

A259673 a(n) = sigma_(prime(n))(n).

Original entry on oeis.org

1, 9, 244, 16513, 48828126, 13062296532, 232630513987208, 144115462954287105, 8862938119746644274757, 100000000186264514923632574038, 191943424957750480504146841291812, 8505622499882988712256991112913772434548, 4695452425098908797088971409337422035076128814

Offset: 1

Ilya Gutkovskiy, Jul 03 2015

Anders Hellström, Table of n, a(n) for n = 1..50
Index entries for sequences related to sums of divisors

Cf. A000203 (sigma(n)), A000040 (prime(n)), A023887 (sigma_n(n)).

Cf. A001157 (sigma_2), A001158 (sigma_3), A001160 (sigma_5), A013955 (sigma_7).

[DivisorSigma(NthPrime(n),n):n in [1..15]]; // Vincenzo Librandi, Jul 15 2015

a:= n-> numtheory[sigma][ithprime(n)](n):
seq(a(n), n=1..15);  # Alois P. Heinz, Feb 10 2020

a[n_] := DivisorSigma[Prime[n], n]; Array[a, 13]
(* Second program: *)
a[n_] := SeriesCoefficient[Sum[k^Prime[n]*x^k/(1-x^k), {k, 1, n}], {x, 0, n}]; Array[a, 13] (* Jean-François Alcover, Sep 29 2017, from 2nd formula *)

a(n) = sigma(n, prime(n)); \\ Michel Marcus, Jul 03 2015

from sympy import divisor_sigma, prime
def A259673(n):
    return divisor_sigma(n,prime(n)) # Chai Wah Wu, Jul 20 2015

a(n) = sigma_(A000040(n))(n).

a(n) = [x^n] Sum_{k>=1} k^prime(n)*x^k/(1 - x^k). - Ilya Gutkovskiy, Sep 26 2017

a(11) and a(12) from Anders Hellström, Jul 14 2015

A055701 Numbers k such that k | sigma_7(k) - phi(k)^7.

Original entry on oeis.org

1, 2, 10, 12, 26, 42, 172, 456, 588, 1326, 3315, 3635, 6468, 6720, 12102, 12922, 15288, 17836, 18810, 21756, 32984, 36108, 36603, 40572, 41748, 72905, 78120, 137004, 195216, 291060, 295176, 923212, 978014, 989604, 1009800, 1015768, 1069712, 1602692, 1711024

Offset: 1

Robert G. Wilson v, Jun 09 2000

nonn

sigma_7(k) is the sum of the 7th powers of the divisors of k (A013955).

Amiram Eldar, Table of n, a(n) for n = 1..120

Cf. A000010, A013955.

Do[If[Mod[DivisorSigma[7, n]-EulerPhi[n]^7, n]==0, Print[n]], {n, 1, 10^5}]
Select[Range[2*10^6],Divisible[DivisorSigma[7,#]-EulerPhi[#]^7,#]&] (* Harvey P. Dale, Dec 02 2016 *)

isok(n) = !((sigma(n, 7) - eulerphi(n)^7) % n); \\ Michel Marcus, Mar 02 2014

Definition corrected and more terms from Michel Marcus, Mar 02 2014

A081865 a(n) = sigma_7(2n-1).

Original entry on oeis.org

1, 2188, 78126, 823544, 4785157, 19487172, 62748518, 170939688, 410338674, 893871740, 1801914272, 3404825448, 6103593751, 10465138360, 17249876310, 27512614112, 42637932336, 64340198544, 94931877134, 137293757384, 194754273882, 271818611108, 373845175782

Offset: 1

Benoit Cloitre, Apr 12 2003

nonn

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

Cf. A013955.

DivisorSigma[7, Range[1, 45, 2]] (* Amiram Eldar, Aug 17 2019 *)

a(n) = sigma(2*n-1, 7); \\ Michel Marcus, Dec 04 2013

Sum_{k=1..n} a(k) ~ c * n^8, where c = 17 * Pi^8 / 10080 = 16.00248... . - Amiram Eldar, Jan 07 2025

Three more terms from Michel Marcus, Dec 04 2013

cp's OEIS Frontend

A126812 Ramanujan numbers (A000594) read mod 4.

Views

Author

Keywords

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A126834 Ramanujan numbers (A000594) read mod 125.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A068024 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=7.

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

A087115 Convolution of sum of cubes of divisors with itself.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Formula

A279917 a(n) = Sum_{k=1..n-1} sigma_5(k)*sigma_7(n-k).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A279964 a(n) = Sum_{k=1..n-1} sigma_3(k)*sigma_7(n-k).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A289744 Coefficients in expansion of q*E'_8 where E_8 is the Eisenstein Series (A008410).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A259673 a(n) = sigma_(prime(n))(n).

Views

Author

Keywords