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 1-5 of 5 results.

A078708 Sum of divisors d of n such that n/d is not congruent to 0 mod 3.

Original entry on oeis.org

1, 3, 3, 7, 6, 9, 8, 15, 9, 18, 12, 21, 14, 24, 18, 31, 18, 27, 20, 42, 24, 36, 24, 45, 31, 42, 27, 56, 30, 54, 32, 63, 36, 54, 48, 63, 38, 60, 42, 90, 42, 72, 44, 84, 54, 72, 48, 93, 57, 93, 54, 98, 54, 81, 72, 120, 60, 90, 60, 126, 62, 96, 72, 127, 84, 108, 68, 126, 72, 144
Offset: 1

Views

Author

Vladeta Jovovic, Dec 18 2002

Keywords

Links

Crossrefs

Cf. A000203, A035191, A046913, A346933.
Cf. A002131 (k=2), this sequence (k=3), A285895 (k=4), A285896 (k=5).
Cf. A326399, A326400.

Programs

  • Mathematica
    f[p_, e_] := If[p == 3, 3^e, (p^(e+1)-1)/(p-1)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 30 2022 *)
  • PARI
    for(n=1,70,d=divisors(n); s=0; for(j=1,matsize(d)[2],if((n/d[j])%3>0,s=s+d[j])); print1(s,","))
  • PARI
    a(n)=sumdiv(n,d,if((n/d)%3,1,0)*d)

Formula

G.f.: Sum_{k>0} x^k*(1+x^k)^2*(1+x^(2*k))/(1-x^(3*k))^2.
a(n) = (A000203(3*n)-A000203(n))/3. - Vladeta Jovovic, Dec 22 2003
G.f.: Sum_{k>=1} k*x^k*(1 + x^k)/(1 - x^(3*k)). - Ilya Gutkovskiy, Sep 13 2019
From R. J. Mathar, May 25 2020: (Start)
a(n) = A326399(n) + A326400(n).
a(n) = A000203(n) - A000203(n/3), where A000203(.) = 0 for non-integer arguments. (End)
From Amiram Eldar, Oct 30 2022: (Start)
Multiplicative with a(3^e) = 3^e and a(p^e) = (p^(e+1)-1)/(p-1) if p != 3.
Sum_{k=1..n} a(k) ~ c * n^2, where c = 2*Pi^2/27 = 0.731081... (A346933). (End)
Dirichlet g.f.: zeta(s)*zeta(s-1)*(1-1/3^s). - Amiram Eldar, Dec 30 2022

Extensions

Extended by Klaus Brockhaus and Benoit Cloitre, Dec 20 2002

A346929 Decimal expansion of 73 * Pi^10 / 4591650240.

Original entry on oeis.org

9, 9, 9, 0, 2, 2, 5, 8, 8, 7, 7, 6, 4, 8, 6, 2, 9, 7, 9, 5, 3, 6, 4, 4, 7, 9, 5, 7, 6, 0, 3, 8, 0, 9, 7, 9, 5, 5, 9, 9, 9, 7, 5, 2, 6, 6, 9, 0, 2, 1, 5, 9, 0, 4, 2, 5, 5, 2, 4, 4, 4, 3, 4, 4, 3, 3, 8, 8, 8, 8, 0, 2, 2, 2, 6, 2, 9, 8, 2, 1, 2, 8, 4, 0, 9, 0, 6
Offset: 0

Views

Author

Sean A. Irvine, Aug 07 2021

Keywords

Examples

			0.999022588776486297953644795760380979559997526690...

References

  • L. B. W. Jolley, Summation of Series, Dover, 1961, Eq. (312).

Links

Crossrefs

Cf. A346933, A346932, A346931, A346930.

Programs

  • Mathematica
    RealDigits[73 * Pi^10 / 4591650240, 10, 120][[1]] (* Amiram Eldar, Jun 20 2023 *)

Formula

Equals 11 * 61 * 73 * Pi^10 / (2^6 * 3^15 * 5).
Equals 1 + Sum_{k>=1} ( (-1)^k/(3*k-1)^10 + (-1)^k/(3*k+1)^10 ).

A346930 Decimal expansion of 5207 * Pi^8 / 49601160.

Original entry on oeis.org

9, 9, 6, 0, 8, 1, 1, 6, 0, 2, 1, 2, 3, 7, 1, 6, 2, 3, 0, 6, 5, 2, 3, 6, 6, 3, 2, 5, 4, 7, 3, 5, 4, 3, 2, 6, 0, 5, 6, 1, 2, 8, 7, 9, 8, 3, 7, 5, 5, 8, 4, 7, 1, 5, 7, 1, 6, 6, 7, 3, 6, 8, 2, 5, 9, 6, 4, 0, 2, 8, 7, 9, 5, 2, 6, 1, 2, 3, 5, 0, 9, 4, 7, 9, 5, 4, 9
Offset: 0

Views

Author

Sean A. Irvine, Aug 07 2021

Keywords

Examples

			0.9960811602123716230652366325473543260...

References

  • L. B. W. Jolley, Summation of Series, Dover, 1961, Eq. (312).

Links

Crossrefs

Cf. A346933, A346932, A346931, A346929.

Programs

  • Mathematica
    RealDigits[5207 * Pi^8 / 49601160, 10, 120][[1]] (* Amiram Eldar, Jun 20 2023 *)

Formula

Equals 41 * 127 * Pi^8 / (2^3 * 3^11 * 5 * 7).
Equals 1 + Sum_{k>=1} ( (-1)^k/(3*k-1)^8 + (-1)^k/(3*k+1)^8 ).

A346931 Decimal expansion of 403 * Pi^6 / 393660.

Original entry on oeis.org

9, 8, 4, 1, 9, 9, 1, 6, 9, 3, 6, 1, 4, 9, 8, 9, 7, 9, 1, 2, 7, 1, 1, 0, 7, 9, 5, 5, 8, 9, 9, 1, 3, 1, 7, 1, 8, 8, 3, 0, 6, 8, 2, 9, 1, 9, 6, 9, 3, 6, 8, 9, 6, 3, 6, 3, 4, 2, 3, 6, 9, 1, 3, 9, 7, 6, 7, 5, 6, 2, 0, 3, 0, 0, 6, 3, 1, 7, 9, 8, 0, 0, 8, 4, 8, 0, 9
Offset: 0

Views

Author

Sean A. Irvine, Aug 07 2021

Keywords

Examples

			0.9841991693614989791271107955899131718830...

References

  • L. B. W. Jolley, Summation of Series, Dover, 1961, Eq. (312).

Links

Crossrefs

Cf. A346933, A346932, A346930, A346929.

Programs

  • Mathematica
    RealDigits[403 * Pi^6 / 393660, 10, 120][[1]] (* Amiram Eldar, Jun 20 2023 *)

Formula

Equals 13 * 31 * Pi^6 / (2^2 * 3^9 * 5).
Equals 1 + Sum_{k>=1} ( (-1)^k/(3*k-1)^6 + (-1)^k/(3*k+1)^6 ).

A346932 Decimal expansion of 7 * Pi^4 / 729.

Original entry on oeis.org

9, 3, 5, 3, 4, 1, 0, 6, 6, 1, 7, 0, 1, 1, 9, 4, 2, 4, 7, 6, 6, 9, 1, 6, 7, 7, 4, 7, 8, 8, 6, 6, 3, 6, 1, 9, 6, 8, 1, 8, 8, 3, 5, 3, 8, 6, 9, 5, 1, 9, 6, 0, 9, 3, 0, 5, 9, 3, 6, 2, 1, 6, 1, 4, 9, 4, 9, 1, 9, 0, 5, 1, 0, 2, 5, 7, 7, 5, 5, 4, 6, 4, 0, 7, 2, 0, 5
Offset: 0

Views

Author

Sean A. Irvine, Aug 07 2021

Keywords

Examples

			0.9353410661701194247669167747886636196818835386951960...

References

  • L. B. W. Jolley, Summation of Series, Dover, 1961, Eq. (312).

Links

Crossrefs

Cf. A346933, A346931, A346930, A346929.

Programs

  • Mathematica
    RealDigits[7 * Pi^4 / 729, 10, 120][[1]] (* Amiram Eldar, Jun 20 2023 *)

Formula

Equals 7 * Pi^4 / 3^6.
Equals 1 + Sum_{k>=1} ( (-1)^k/(3*k-1)^4 + (-1)^k/(3*k+1)^4 ).
Showing 1-5 of 5 results.

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