A013957 -id:A013957 - cp's OEIS frontend

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

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

A057210 Number of fullerenes with 2n vertices (or carbon atoms), counting enantiomorphic pairs as distinct.

Original entry on oeis.org

1, 0, 1, 1, 3, 3, 10, 9, 23, 30, 66, 80, 162, 209, 374, 507, 835, 1113, 1778, 2344, 3532, 4670, 6796, 8825, 12501, 16091, 22142, 28232, 38016, 47868, 63416, 79023, 102684, 126973, 162793, 199128, 252082, 306061, 382627, 461020

Offset: 10

N. J. A. Sloane, Aug 28 2003

nonn

P. W. Fowler and D. E. Manolopoulos, An Atlas of Fullerenes, Cambridge Univ. Press, 1995, see p. 32.

Gunnar Brinkmann, Table of n, a(n) for n = 10..100 (Received Aug 18, 2006)
Philip Engel and Peter Smillie The number of non-negative curvature triangulations of S^2, arXiv:1702.02614 [math.GT], 2017.
Philip Engel, Jan Goedgebeur, and Peter Smillie, Exact enumeration of fullerenes, arXiv:2304.01655 [math.GT], 2023.

Cf. A007894, A013957.

a(n) = (809/1306069401600)*sigma_9(n) + O(n^8) where sigma_9(n) is the ninth divisor power sum, A013957. - Philip Engel, Nov 29 2017

A126833 Ramanujan numbers (A000594) read mod 25.

Original entry on oeis.org

1, 1, 2, 3, 5, 2, 6, 5, 7, 5, 12, 6, 12, 6, 10, 11, 16, 7, 20, 15, 12, 12, 22, 10, 0, 12, 20, 18, 5, 10, 7, 21, 24, 16, 5, 21, 11, 20, 24, 0, 17, 12, 17, 11, 10, 22, 21, 22, 18, 0, 7, 11, 2, 20, 10, 5, 15, 5, 10, 5, 12, 7, 17, 18, 10, 24, 16, 23, 19, 5, 22, 10, 22, 11, 0, 10, 22, 24, 5, 5

Offset: 1

N. J. A. Sloane, Feb 25 2007

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..10000
George E. Andrews and Bruce C. Berndt, Ramanujan's Unpublished Manuscript on the Partition and Tau Functions, in: Ramanujan's Lost Notebook, Part III, Springer, New York, NY, 2012.
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, A013957, A126832 (mod 5^1), this sequence (mod 5^2), A126834 (mod 5^3), A126835 (mod 5^4).

a[n_] := Mod[RamanujanTau[n], 25]; Array[a, 100] (* Amiram Eldar, Jan 04 2025 *)

a(n) = ramanujantau(n) % 25; \\ Amiram Eldar, Jan 04 2025

a(n) == n * sigma_9(n) (mod 25) (Andrews and Berndt, 2012, eq. (5.4.2), p. 98). - Amiram Eldar, Jan 04 2025

A301553 Expansion of Product_{k>=1} (1 + x^k)^(sigma_9(k)).

Original entry on oeis.org

1, 1, 513, 20197, 413669, 12445003, 372981573, 9158438541, 223776496101, 5567873958982, 132009631562091, 3018411978731059, 68171158091244082, 1512439928316217508, 32796174722883608382, 698503712498547606328, 14656105328324700415778, 302787437988353941515934

Offset: 0

Vaclav Kotesovec, Mar 23 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..995

Cf. A107742 (m=0), A192065 (m=1), A288414 (m=2), A288415 (m=3), A301548 (m=4), A301549 (m=5), A301550 (m=6), A301551 (m=7), A301552 (m=8).

Cf. A013957, A301547.

nmax = 30; CoefficientList[Series[Product[(1+x^k)^DivisorSigma[9, k], {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ exp(11 * Pi^(10/11) * (31*Zeta(11))^(1/11) * n^(10/11) / (2^(13/11) * 5^(10/11))) * (155*Zeta(11)/Pi)^(1/22) / (2^(155/264) * sqrt(11) * n^(6/11)).

G.f.: exp(Sum_{k>=1} sigma_10(k)*x^k/(k*(1 - x^(2*k)))). - Ilya Gutkovskiy, Oct 26 2018

A321813 Sum of 9th powers of odd divisors of n.

Original entry on oeis.org

1, 1, 19684, 1, 1953126, 19684, 40353608, 1, 387440173, 1953126, 2357947692, 19684, 10604499374, 40353608, 38445332184, 1, 118587876498, 387440173, 322687697780, 1953126, 794320419872, 2357947692, 1801152661464, 19684, 3814699218751

Offset: 1

N. J. A. Sloane, Nov 24 2018

Seiichi Manyama, Table of n, a(n) for n = 1..10000
J. W. L. Glaisher, On the representations of a number as the sum of two, four, six, eight, ten, and twelve squares, Quart. J. Math. 38 (1907), 1-62 (see p. 4 and p. 8).
Eric Weisstein's World of Mathematics, Odd Divisor Function.
Index entries for sequences mentioned by Glaisher.

Column k=9 of A285425.
Cf. A050999, A051000, A051001, A051002, A321810 - A321816 (analog for 2nd .. 12th powers).
Cf. A321543 - A321565, A321807 - A321836 for related sequences.
Cf. A000265, A013668, A013957.

a[n_] := DivisorSum[n, #^9 &, OddQ[#] &]; Array[a, 20] (* Amiram Eldar, Dec 07 2018 *)

apply( A321813(n)=sigma(n>>valuation(n,2),9), [1..30]) \\ M. F. Hasler, Nov 26 2018

from sympy import divisor_sigma
def A321813(n): return int(divisor_sigma(n>>(~n&n-1).bit_length(),9)) # Chai Wah Wu, Jul 16 2022

a(n) = A013957(A000265(n)) = sigma_9(odd part of n); in particular, a(2^k) = 1 for all k >= 0. - M. F. Hasler, Nov 26 2018
G.f.: Sum_{k>=1} (2*k - 1)^9*x^(2*k-1)/(1 - x^(2*k-1)). - Ilya Gutkovskiy, Dec 07 2018
From Amiram Eldar, Nov 02 2022: (Start)
Multiplicative with a(2^e) = 1 and a(p^e) = (p^(9*e+9)-1)/(p^9-1) for p > 2.
Sum_{k=1..n} a(k) ~ c * n^10, where c = zeta(10)/20 = Pi^10/1871100 = 0.0500497... . (End)

A068026 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=9.

Original entry on oeis.org

1, 1023, 29524, 698027, 2441406, 36192156, 47079208, 408345795, 653757313, 2773708938, 2593742460, 26912354924, 11488207654, 51851591352, 77226922344, 222984027123, 125999618778, 848125888467, 340614792100, 1991478050562

Offset: 1

Vladeta Jovovic, Feb 08 2002

nonn

Cf. A067692, A068020-A068025, A068027, A000203, A001157-A001160, A013954-A013957.

CIP9 = CycleIndexPolynomial[SymmetricGroup[9], Array[x, 9]]; a[n_] := CIP9 /. x[k_] -> DivisorSigma[k, n]; Array[a, 20] (* Jean-François Alcover, Nov 04 2016 *)

1/9!*(sigma[1](n)^9 + 36*sigma[1](n)^7*sigma[2](n) + 168*sigma[1](n)^6*sigma[3](n) + 378*sigma[1](n)^5*sigma[2](n)^2 + 756*sigma[1](n)^5*sigma[4](n) + 2520*sigma[1](n)^4*sigma[2](n)*sigma[3](n) +
+ 1260*sigma[1](n)^3*sigma[2](n)^3 + 3024*sigma[1](n)^4*sigma[5](n) + 7560*sigma[1](n)^3*sigma[2](n)*sigma[4](n) + 3360*sigma[1](n)^3*sigma[3](n)^2 + 7560*sigma[1](n)^2*sigma[2](n)^2*sigma[3](n) +
+ 945*sigma[1](n)*sigma[2](n)^4 + 10080*sigma[1](n)^3*sigma[6](n) + 18144*sigma[1](n)^2*sigma[2](n)*sigma[5](n) + 15120*sigma[1](n)^2*sigma[3](n)*sigma[4](n) + 11340*sigma[1](n)*sigma[2](n)^2*sigma[4](n) + 10080*sigma[1](n)*sigma[2](n)*sigma[3](n)^2 + 2520*sigma[2](n)^3*sigma[3](n) + 25920*sigma[7](n)*sigma[1](n)^2 +
+ 30240*sigma[1](n)*sigma[2](n)*sigma[6](n) + 24192*sigma[1](n)*sigma[3](n)*sigma[5](n) + 11340*sigma[1](n)*sigma[4](n)^2 + 9072*sigma[2](n)^2*sigma[5](n) + 15120*sigma[2](n)*sigma[3](n)*sigma[4](n) + 2240*sigma[3](n)^3 + 25920*sigma[7](n)*sigma[2](n) + 45360*sigma[8](n)*sigma[1](n) + 20160*sigma[3](n)*sigma[6](n) + 18144*sigma[4](n)*sigma[5](n) + 40320*sigma[9](n)).

A126837 Ramanujan numbers (A000594) read mod 7^2.

Original entry on oeis.org

1, 25, 7, 47, 28, 28, 14, 4, 37, 14, 22, 35, 21, 7, 0, 31, 28, 43, 0, 42, 0, 11, 46, 28, 32, 35, 28, 21, 23, 0, 0, 31, 7, 14, 0, 24, 46, 0, 0, 14, 14, 0, 37, 5, 7, 23, 42, 21, 0, 16, 0, 7, 36, 14, 28, 7, 0, 36, 42, 0, 14, 0, 28, 7, 0, 28, 15, 42, 28, 0, 2, 1, 14, 23, 28, 0, 14, 0, 39, 35, 46

Offset: 1

N. J. A. Sloane, Feb 25 2007

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..10000
Oddmund Kolberg, Note on Ramanujan's Function tau(n), Mathematica Scandinavica, Vol. 10 (1962), pp. 171-172; alternative link.
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, A013957, A126836 (mod 7^1), this sequence (mod 7^2), A126838 (mod 7^3).

a[n_] := Mod[RamanujanTau[n], 49]; Array[a, 100] (* Amiram Eldar, Jan 05 2025 *)

a(n) = ramanujantau(n) % 49; \\ Amiram Eldar, Jan 05 2025

a(n) == n * sigma_9(n) (mod 7^2) if Legendre symbol (n,7) = A175629(n) = -1 (Kolberg, 1962). - Amiram Eldar, Jan 05 2025

A279888 a(n) = Sum_{k=1..n-1} sigma_3(k)*sigma_5(n-k).

Original entry on oeis.org

0, 1, 42, 569, 4250, 22006, 88004, 293369, 845358, 2186376, 5145646, 11282966, 23143198, 45179324, 83905292, 150271993, 258816840, 433786483, 704268402, 1119633944, 1733618768, 2640037170, 3931060364, 5777392406, 8325691750, 11873200964, 16643954724, 23133008124

Offset: 1

Seiichi Manyama, Dec 22 2016

nonn

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

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

Cf. A001158, A001160, A013957, A279917.

Table[Sum[If[k == 0, 0, DivisorSigma[3, k]] DivisorSigma[5, n - k], {k, 0, n - 1}], {n, 28}] (* Michael De Vlieger, Dec 22 2016 *)
a[n_] := (11 * DivisorSigma[9, n] - 21 * DivisorSigma[5, n] + 10 * DivisorSigma[3, n]) / 5040; Array[a, 30] (* Amiram Eldar, Jan 07 2025 *)

a(n) = {my(f = factor(n)); (11 * sigma(f, 9) - 21 * sigma(f, 5) + 10 * sigma(f, 3)) / 5040;} \\ Amiram Eldar, Jan 07 2025

a(n) = (11*sigma_9(n)-21*sigma_5(n)+10*sigma_3(n))/5040.

A289745 Coefficients in expansion of -q*E'_10 where E_10 is the Eisenstein Series (A013974).

Original entry on oeis.org

264, 270864, 15589728, 277365792, 2578126320, 15995060928, 74573467584, 284022573120, 920557851048, 2645157604320, 6847480097568, 16379004749184, 36394641851568, 76512377741184, 152243515448640, 290839114879104, 532222389723024, 944492355175248

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), A289744 (k=8), this sequence (k=10), A289746 (k=14).

Cf. A013957, A013974 (E_10), A282254, A289639.

Table[264*n*DivisorSigma[9, n], {n, 1, 18}] (* Jean-François Alcover, Feb 27 2018 *)

a(n) = 264*n*sigma(n, 9); \\ Michel Marcus, Feb 27 2018

a(n) = 264*A282254(n) = 264*n*A013957(n).

A094468 Numbers k such that sum of 9th powers of divisors of k is divisible by the square of Euler-phi of k.

Original entry on oeis.org

1, 2, 3, 6, 14, 42, 3810, 26670, 34162, 41256, 48546, 87096, 102486, 131934, 210482, 288792, 315723, 318990, 430122, 529848, 609672, 631446, 979830, 1023366, 1203960, 1473374, 1683126, 1920699, 2210061, 2241934, 2506086, 2549610

Offset: 1

Labos Elemer, May 19 2004

nonn

Amiram Eldar, Table of n, a(n) for n = 1..420

Cf. A015763, A000010, A091285, A013957.

Select[Range[2600000],Divisible[Total[Divisors[#]^9],(EulerPhi[#])^2]&]  (* Harvey P. Dale, Mar 04 2011 *)

for(n=1,10000000,if(Mod(sigma(n,9),eulerphi(n)^2)==0,print1(n,","))) \\ C. Ronaldo

A013957(k)/A000010(k)^2 is an integer.

More terms from C. Ronaldo (aga_new_ac(AT)hotmail.com), Jan 18 2005

cp's OEIS Frontend

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

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A057210 Number of fullerenes with 2n vertices (or carbon atoms), counting enantiomorphic pairs as distinct.

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A126833 Ramanujan numbers (A000594) read mod 25.

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A301553 Expansion of Product_{k>=1} (1 + x^k)^(sigma_9(k)).

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A321813 Sum of 9th powers of odd divisors of n.

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A068026 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=9.

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A126837 Ramanujan numbers (A000594) read mod 7^2.

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A279888 a(n) = Sum_{k=1..n-1} sigma_3(k)*sigma_5(n-k).

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A289745 Coefficients in expansion of -q*E'_10 where E_10 is the Eisenstein Series (A013974).