cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

A337032 a(n) = (n*sigma_9(n) - tau(n))/7 = (A282254(n) - A000594(n))/7, where tau is Ramanujan's tau, sigma_9(n) = Sum_{d divides n} d^9.

Original entry on oeis.org

0, 150, 8400, 150300, 1394400, 8656200, 40356000, 153679800, 498153600, 1431378900, 3705270000, 8863150800, 19694152800, 41402744400, 82382680800, 157380332400, 288000115200, 511088547150, 875865085200, 1465721632200, 2382961862400, 3801687211800, 5918070367200, 9075809181600
Offset: 1

Views

Author

Jianing Song, Aug 12 2020

Keywords

Comments

D. H. Lehmer shows that tau(n) == n*sigma_9(n) (mod 7), so a(n) is an integer for all n. Furthermore, if n == 3, 5, 6 (mod 7) then tau(n) == n*sigma_9(n) (mod 49). See the Wikipedia link below. It seems that the latter congruence also holds for most of the other numbers. Among the 571 numbers in [1, 1000] congruent to 0, 1, 2, 4 modulo 7, tau(n) == n*sigma_9(n) holds for 311 n's, and among the 5715 numbers in [1, 10000] congruent to 0, 1, 2, 4 modulo 7, the congruence holds for 3492 n's.
It seems that 150 divides a(n) for all n. There are no counterexamples for n <= 10000.
Number of n's in [2, N] which satisfy the higher-order congruence tau(n) == n*sigma_9(n) (mod 7^e) but not tau(n) == n*sigma_9(n) (mod 7^(e+1)):
N = 1000:
e | n == 3, 5, 6 (mod 7) | n == 0, 1, 2, 4 (mod 7) | total
---+----------------------+-------------------------+-------
1 | 0 | 260 | 260
---+----------------------+-------------------------+-------
2 | 358 | 80 | 438
---+----------------------+-------------------------+-------
3 | 45 | 195 | 240
---+----------------------+-------------------------+-------
4 | 24 | 28 | 52
---+----------------------+-------------------------+-------
5 | 2 | 5 | 7
---+----------------------+-------------------------+-------
6 | 0 | 2* | 2
* n = 686, 942.
N = 10000:
e | n == 3, 5, 6 (mod 7) | n == 0, 1, 2, 4 (mod 7) | total
---+----------------------+-------------------------+-------
1 | 0 | 2223 | 2223
---+----------------------+-------------------------+-------
2 | 3368 | 728 | 4096
---+----------------------+-------------------------+-------
3 | 466 | 2280 | 2746
---+----------------------+-------------------------+-------
4 | 397 | 384 | 781
---+----------------------+-------------------------+-------
5 | 46 | 87 | 133
---+----------------------+-------------------------+-------
6 | 6 | 12 | 18
---+----------------------+-------------------------+-------
7 | 2** | 0 | 2
** n = 5185, 9021.

Examples

			a(2) = (n*sigma_9(2) - tau(2))/7 = (2*(1^9+2^9) - (-24))/7 = 1050/7 = 150;
a(3) = (n*sigma_9(3) - tau(3))/7 = (3*(1^9+3^9) - 252)/7 = 58800/7 = 8400.

Links

Crossrefs

Cf. A000594, A282254, A027860.

Programs

  • Mathematica
    a[n_] := (n * DivisorSigma[9, n] - RamanujanTau[n]) / 7; Array[a, 24] (* Amiram Eldar, Jan 10 2025 *)
  • PARI
    a(n) = (n*sigma(n, 9) - polcoeff( x * eta(x + x * O(x^n))^24, n))/7;

A282548 Expansion of phi_{12, 1}(x) where phi_{r, s}(x) = Sum_{n, m>0} m^r * n^s * x^{m*n}.

Original entry on oeis.org

0, 1, 4098, 531444, 16785412, 244140630, 2177857512, 13841287208, 68753047560, 282431130813, 1000488301740, 3138428376732, 8920506494928, 23298085122494, 56721594978384, 129747072969720, 281612482805776, 582622237229778, 1157402774071674
Offset: 0

Views

Author

Seiichi Manyama, Feb 18 2017

Keywords

Comments

Multiplicative because A013959 is. - Andrew Howroyd, Jul 25 2018

Links

Crossrefs

Cf. A064987 (phi_{2, 1}), A281372 (phi_{4, 1}), A282050 (phi_{6, 1}), A282060 (phi_{8, 1}), A282254 (phi_{10, 1}), this sequence (phi_{12, 1}).
Cf. A282549 (E_2*E_4^3), A282576 (E_2*E_6^2), A058550 (E_14).
Cf. A013670.

Programs

  • Mathematica
    Table[n * DivisorSigma[11, n], {n, 0, 18}] (* Amiram Eldar, Sep 06 2023 *)
  • PARI
    a(n) = if(n < 1, 0, n*sigma(n, 11)) \\ Andrew Howroyd, Jul 25 2018

Formula

a(n) = n*A013959(n) for n > 0.
a(n) = (441*A282549(n) + 250*A282576(n) - 691*A058550(n))/65520.
Sum_{k=1..n} a(k) ~ zeta(12) * n^13 / 13. - Amiram Eldar, Sep 06 2023
From Amiram Eldar, Oct 30 2023: (Start)
Multiplicative with a(p^e) = p^e * (p^(11*e+11)-1)/(p^11-1).
Dirichlet g.f.: zeta(s-1)*zeta(s-12). (End)

A282597 Expansion of phi_{14, 1}(x) where phi_{r, s}(x) = Sum_{n, m>0} m^r * n^s * x^{m*n}.

Original entry on oeis.org

0, 1, 16386, 4782972, 268468228, 6103515630, 78373779192, 678223072856, 4398583447560, 22876806803877, 100012207113180, 379749833583252, 1284076017413616, 3937376385699302, 11113363271818416, 29192944359852360, 72066391204823056, 168377826559400946
Offset: 0

Views

Author

Seiichi Manyama, Feb 19 2017

Keywords

Comments

Multiplicative because A013961 is. - Andrew Howroyd, Jul 25 2018

Links

Crossrefs

Cf. A064987 (phi_{2, 1}), A281372 (phi_{4, 1}), A282050 (phi_{6, 1}), A282060 (phi_{8, 1}), A282254 (phi_{10, 1}), A282548 (phi_{12, 1}), this sequence (phi_{14, 1}).
Cf. A282012 (E_4^4), A282287 (E_4*E_6^2), A282596 (E_2*E_4^2*E_6).
Cf. A013672.

Programs

  • Mathematica
    Table[n * DivisorSigma[13, n], {n, 0, 17}] (* Amiram Eldar, Sep 06 2023 *)
  • PARI
    a(n) = if(n < 1, 0, n*sigma(n, 13)) \\ Andrew Howroyd, Jul 25 2018

Formula

a(n) = n*A013961(n) for n > 0.
a(n) = (3*A282012(n) + 4*A282287(n) - 7*A282596(n))/144.
Sum_{k=1..n} a(k) ~ zeta(14) * n^15 / 15. - Amiram Eldar, Sep 06 2023
From Amiram Eldar, Oct 30 2023: (Start)
Multiplicative with a(p^e) = p^e * (p^(13*e+13)-1)/(p^13-1).
Dirichlet g.f.: zeta(s-1)*zeta(s-14). (End)

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

Views

Author

Seiichi Manyama, Jul 11 2017

Keywords

Links

Crossrefs

(-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.

Programs

Formula

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

A280021 Expansion of phi_{11, 2}(x) where phi_{r, s}(x) = Sum_{n, m>0} m^r * n^s * x^{m*n}.

Original entry on oeis.org

0, 1, 2052, 177156, 4202512, 48828150, 363524112, 1977326792, 8606744640, 31382654013, 100195363800, 285311670732, 744500215872, 1792160394206, 4057474577184, 8650199741400, 17626613022976, 34271896307922, 64397206034676, 116490258898580, 205200886312800
Offset: 0

Views

Author

Seiichi Manyama, Feb 22 2017

Keywords

Comments

Multiplicative because A013957 is. - Andrew Howroyd, Jul 23 2018

Links

Crossrefs

Cf. A282097 (phi_{3, 2}), A282099 (phi_{5, 2}), A282751 (phi_{7, 2}), A282753 (phi_{9, 2}), this sequence (phi_{11, 2}).
Cf. A282549 (E_2*E_4^3), A282792 (E_2^2*E_4*E_6), A282576 (E_2*E_6^2), A058550 (E_4^2*E_6 = E_14).
Cf. A013957 (sigma_9(n)), A282254 (n*sigma_9(n)), this sequence (n^2*sigma_9(n)).
Cf. A013668 (zeta(10)).

Programs

  • Mathematica
    Table[If[n>0, n^2 * DivisorSigma[9, n], 0], {n, 0, 20}] (* Indranil Ghosh, Mar 12 2017 *)
  • PARI
    for(n=0, 20, print1(if(n==0, 0, n^2 * sigma(n, 9)),", ")) \\ Indranil Ghosh, Mar 12 2017

Formula

a(n) = n^2*A013957(n) for n > 0.
a(n) = (6*A282549(n) - 5*A282792(n) + 4*A282576(n) - 5*A058550(n))/1728.
Sum_{k=1..n} a(k) ~ zeta(10) * n^12 / 12. - Amiram Eldar, Sep 06 2023
From Amiram Eldar, Oct 30 2023: (Start)
Multiplicative with a(p^e) = p^(2*e) * (p^(9*e+9)-1)/(p^9-1).
Dirichlet g.f.: zeta(s-2)*zeta(s-11). (End)

A282777 Expansion of phi_{16, 1}(x) where phi_{r, s}(x) = Sum_{n, m>0} m^r * n^s * x^{m*n}.

Original entry on oeis.org

0, 1, 65538, 43046724, 4295098372, 152587890630, 2821196197512, 33232930569608, 281483566907400, 1853020317992013, 10000305176108940, 45949729863572172, 184889914172333328, 665416609183179854, 2178019803670969104, 6568408813691796120
Offset: 0

Views

Author

Seiichi Manyama, Feb 21 2017

Keywords

Comments

Multiplicative because A013963 is. - Andrew Howroyd, Jul 25 2018

References

  • George E. Andrews and Bruce C. Berndt, Ramanujan's lost notebook, Part III, Springer, New York, 2012. See p. 212.

Links

Crossrefs

Cf. A064987 (phi_{2, 1}), A281372 (phi_{4, 1}), A282050 (phi_{6, 1}), A282060 (phi_{8, 1}), A282254 (phi_{10, 1}), A282548 (phi_{12, 1}), A282597 (phi_{14, 1}), this sequence (phi_{16, 1}).
Cf. A282546 (E_2*E_4^4), A282000 (E_4^3*E_6), A282547 (E_2*E_4*E_6^2), A282253 (E_6^3).
Cf. A013674.

Programs

  • Mathematica
    Table[If[n==0, 0, n * DivisorSigma[15, n]], {n, 0, 15}] (* Indranil Ghosh, Mar 11 2017 *)
  • PARI
    for(n=0, 15, print1(if(n==0, 0, n * sigma(n, 15)), ", ")) \\ Indranil Ghosh, Mar 11 2017

Formula

a(n) = n*A013963(n) for n > 0.
a(n) = (2156*A282546(n) - 4156*A282000(n) + 8000*A282547(n)/3 - 2000*A282253(n)/3)/16320.
Sum_{k=1..n} a(k) ~ zeta(16) * n^17 / 17. - Amiram Eldar, Sep 06 2023
From Amiram Eldar, Oct 30 2023: (Start)
Multiplicative with a(p^e) = p^e * (p^(15*e+15)-1)/(p^15-1).
Dirichlet g.f.: zeta(s-1)*zeta(s-16). (End)
Showing 1-6 of 6 results.

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