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 1-3 of 3 results.

A121443 Sum of divisors d of n which are odd and n/d is not divisible by 3.

Original entry on oeis.org

1, 1, 3, 1, 6, 3, 8, 1, 9, 6, 12, 3, 14, 8, 18, 1, 18, 9, 20, 6, 24, 12, 24, 3, 31, 14, 27, 8, 30, 18, 32, 1, 36, 18, 48, 9, 38, 20, 42, 6, 42, 24, 44, 12, 54, 24, 48, 3, 57, 31, 54, 14, 54, 27, 72, 8, 60, 30, 60, 18, 62, 32, 72, 1, 84, 36, 68, 18, 72, 48, 72, 9, 74, 38, 93, 20, 96, 42
Offset: 1

Views

Author

Michael Somos, Jul 30 2006, Apr 18 2007

Keywords

Comments

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

Examples

			G.f. = q + q^2 + 3*q^3 + q^4 + 6*q^5 + 3*q^6 + 8*q^7 + q^8 + 9*q^9 + 6*q^10 + ...

References

  • Nathan J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 86, Eq. (33.124).

Links

Crossrefs

Cf. A004016, A005882, A005928, A098098, A185717, A291050.

Programs

  • Magma
    A := Basis( ModularForms( Gamma0(6), 2), 80); A[2] + A[3]; /* Michael Somos, Jun 12 2014 */
  • Mathematica
    a[ n_] := If[ n < 1, 0, Sum[ d Mod[ d, 2] Boole[ Mod[ n/d, 3] > 0], {d, Divisors @n}]]; (* Michael Somos, Jun 12 2014 *)
a[ n_] := SeriesCoefficient[ q (QPochhammer[ q^3] QPochhammer[ q^6])^3 / (QPochhammer[ q] QPochhammer[ q^2]), {q, 0, n}]; (* Michael Somos, Jun 12 2014 *)
f[p_, e_] := Which[p == 2, 1, p == 3, p^e, p > 3, (p^(e + 1) - 1)/(p - 1)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 12 2020 *)
  • PARI
    {a(n) = if( n<1, 0, sumdiv(n, d, (d%2) * (n/d%3 > 0) * d))};
  • PARI
    {a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x^3 + A) * eta(x^6 + A))^3 / (eta(x + A) * eta(x^2 + A)), n))};
  • Sage
    A = ModularForms( Gamma0(6), 2, prec=80) . basis(); A[1] + A[2]; # Michael Somos, Jun 12 2014

Formula

Expansion of c(q) * c(q^2) / 9 where c(q) is a cubic AGM theta function.
Euler transform of period 6 sequence [ 1, 2, -2, 2, 1, -4, ...].
Expansion of (eta(q^3) * eta(q^6))^3 / (eta(q) * eta(q^2)) in powers of q.
Multiplicative with a(2^e) = 1, a(3^e) = 3^e, a(p^e) = (p^(e+1) - 1) / (p - 1) if p>3.
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = v^4 - u*w * (u-2*v) * (v-2*w).
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^3), A(x^6)) where f(u1, u2, u3, u6) = u1^3*u6 + 2*u2^3*u3 + 3*u2^2*u3^2 + 6*u1*u2*u3*u6 + 48*u2^2*u6^2 - 3*u1^2*u2*u6 - 3*u1*u2*u3^2 - 24*u2^2*u3*u6 - 30*u1*u2*u6^2. - Michael Somos, Apr 18 2007
G.f.: x * Product_{k>0} ((1 - x^(3*k)) * (1 - x^(6*k)))^3 / ((1 - x^k) * (1 - x^(2*k))) = Sum_{k>0} k * x^k * (1 - x^k) / (1 + x^(3*k)).
a(2*n) = a(n), a(2*n + 1) = A185717(n). a(3*n) = 3*a(n). a(6*n + 5) = 6 * A098098(n).
G.f.: Sum_{n = -inf..inf} (-1)^n*x^(3*n+1)/(1 - x^(3*n+1))^2. Cf. A124340. - Peter Bala, Jan 06 2021
Sum_{k=1..n} a(k) ~ c * n^2, where c = Pi^2/27 = 0.3655409... (A291050). - Amiram Eldar, Nov 17 2022
Dirichlet g.f.: zeta(s)*zeta(s-1)*(1-2^(1-s)-3^(-s)+2^(1-s)*3^(-s)). - Amiram Eldar, Jan 03 2023

A229852 3*h^2, where h is an odd integer not divisible by 3.

Original entry on oeis.org

3, 75, 147, 363, 507, 867, 1083, 1587, 1875, 2523, 2883, 3675, 4107, 5043, 5547, 6627, 7203, 8427, 9075, 10443, 11163, 12675, 13467, 15123, 15987, 17787, 18723, 20667, 21675, 23763, 24843, 27075, 28227, 30603, 31827, 34347, 35643, 38307, 39675, 42483, 43923
Offset: 1

Views

Author

Arkadiusz Wesolowski, Oct 01 2013

Keywords

Comments

If p = a(n)*2^k + 1 divides a composite Fermat number 2^(2^m) + 1 and p is a prime, then k is odd.
More precisely, k == 1 (mod 4) if h == +/- 1 (mod 5) and k == 3 (mod 4) if h == +/- 2 (mod 5) (Krizek, Luca and Somer).

References

  • M. Krizek, F. Luca, L. Somer, 17 Lectures on Fermat Numbers: From Number Theory to Geometry, CMS Books in Mathematics, vol. 9, Springer-Verlag, New York, 2001, pp. 63-65.

Links

Crossrefs

Cf. A000215, A204620, A291050.

Programs

  • Magma
    [3*h^2 : h in [1..121 by 2] | not IsZero(h mod 3)];
  • Mathematica
    3*Select[Range[1, 121, 2], Mod[#, 3] > 0 &]^2 (* Amiram Eldar, Jan 02 2021 *)
  • PARI
    forstep(h=1, 121, 2, if(!(h%3==0), print1(3*h^2, ", ")));
  • PARI
    Vec(3*x*(1+24*x+22*x^2+24*x^3+x^4) / ((1-x)^3*(1+x)^2) + O(x^100)) \\ Colin Barker, Jan 26 2016

Formula

G.f.: 3*x*(1+24*x+22*x^2+24*x^3+x^4) / ((1-x)^3*(1+x)^2).
a(n) = 3*A104777(n).
From Colin Barker, Jan 26 2016: (Start)
a(n) = 3*(18*n^2+6*(-1)^n*n-18*n-3*(-1)^n+5)/2.
a(n) = 27*n^2-18*n+3 for n even.
a(n) = 27*n^2-36*n+12 for n odd.
(End)
Sum_{n>=1} 1/a(n) = Pi^2/27 (A291050). - Amiram Eldar, Jan 02 2021

A152018 Denominator of 1/n^2-1/(3n)^2 or of 8/(9n^2).

Original entry on oeis.org

9, 9, 81, 18, 225, 81, 441, 72, 729, 225, 1089, 162, 1521, 441, 2025, 288, 2601, 729, 3249, 450, 3969, 1089, 4761, 648, 5625, 1521, 6561, 882, 7569, 2025, 8649, 1152, 9801, 2601, 11025, 1458, 12321, 3249, 13689, 1800, 15129, 3969, 16641, 2178, 18225
Offset: 1

Views

Author

Paul Curtz, Nov 20 2008

Keywords

Comments

The associated terms of the n-th main series of the Hydrogen energy spectrum are A000290(3), A061038(6), A061040(9), A061042(12), A061044(15), A061046(18), A061048(21), A061050(24), etc.
All numbers are multiples of 9.

Links

Crossrefs

Cf. A143025 with a similar principle of construction.
Cf. A291050.

Programs

  • Mathematica
    Denominator/@(8/(9Range[50]^2))  (* Harvey P. Dale, Mar 15 2011 *)

Formula

Sum_{n>=1} 1/a(n) = Pi^2/27 (A291050). - Amiram Eldar, Sep 14 2022

Extensions

Stratified definition, corrected indices, extended, R. J. Mathar, Dec 10 2008
Showing 1-3 of 3 results.

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