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 1-4 of 4 results.

A208272 Primes containing a digit 2.

Original entry on oeis.org

2, 23, 29, 127, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 421, 521, 523, 727, 821, 823, 827, 829, 929, 1021, 1123, 1129, 1201, 1213, 1217, 1223, 1229, 1231, 1237, 1249, 1259, 1277, 1279, 1283, 1289, 1291, 1297, 1321, 1327
Offset: 1

Views

Author

Jaroslav Krizek, Mar 04 2012

Keywords

Comments

Supersequence of A045708. Subsequence of A011532.
Complement of A208273 with respect to A011532.
Also primes p whose divisors d_k (k = 1, 2; 1 = d_1 < d_2 = p) contain digit equal to number k.
Complement of A208275 with respect to A208274.
Primes with at least one digit equal to 2. - Harvey P. Dale, Aug 29 2012

Links

Crossrefs

Cf. A208273 (composites containing a digit 2), A011532 (numbers containing a digit 2).

Programs

  • Mathematica
    Select[Range[2000], PrimeQ[#] && MemberQ[IntegerDigits[#], 2] &] (* T. D. Noe, Mar 06 2012 *)
Select[Prime[Range[300]],DigitCount[#,10,2]>0&] (* Harvey P. Dale, Aug 29 2012 *)

Formula

a(n) ~ n log n. - Charles R Greathouse IV, Nov 01 2022

A112128 Expansion of phi(q^4) / phi(q) in powers of q where phi() is a Ramanujan theta function.

Original entry on oeis.org

1, -2, 4, -8, 16, -28, 48, -80, 128, -202, 312, -472, 704, -1036, 1504, -2160, 3072, -4324, 6036, -8360, 11488, -15680, 21264, -28656, 38400, -51182, 67864, -89552, 117632, -153836, 200352, -259904, 335872, -432480, 554952, -709728, 904784, -1149916, 1457136
Offset: 0

Views

Author

Michael Somos, Aug 27 2005

Keywords

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Examples

			G.f. = 1 - 2*q + 4*q^2 - 8*q^3 + 16*q^4 - 28*q^5 + 48*q^6 - 80*q^7 + 128*q^8 + ...

Links

Crossrefs

Cf. A093160, A131126, A208724, A208933.

Programs

  • Mathematica
    QP = QPochhammer; s = QP[q]^2*(QP[q^8]^5/QP[q^2]^5/QP[q^16]^2) + O[q]^40; CoefficientList[s, q] (* Jean-François Alcover, Nov 30 2015, adapted from PARI *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q^4] / EllipticTheta[ 3, 0, q], {q, 0, n}]; (* Michael Somos, Dec 11 2016 *)
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^8 + A)^5 / (eta(x^2 + A)^5 * eta(x^16 + A)^2), n))};

Formula

Expansion of (eta(q) / eta(q^16))^2 * (eta(q^8) / eta(q^2))^5 in powers of q.
Euler transform of period 16 sequence [ -2, 3, -2, 3, -2, 3, -2, -2, -2, 3, -2, 3, -2, 3, -2, 0, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = u^2 - (1 - 2*u + 2*u^2) * (1 - 2*v + 2*v^2).
G.f.: (Sum_{k in Z} x^(4*k^2)) / (Sum_{k in Z} x^(k^2)) = theta_3(0, x^4) / theta_3(0, x).
G.f.: Product_{k>0} ((1 + x^(2*k)) * (1 + x^(4*k)))^3 / ((1 + x^k) * (1 + x^(8*k)))^2.
Expansion of continued fraction 1 / (1 + 2*x / (1 - x^2 + (x^1 + x^3)^2 / (1 - x^6 + (x^2 + x^6)^2 / (1 - x^10 + (x^3 + x^9)^2 / ...)))).
G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = 1/2 * g(t) where q = exp(2 Pi i t) and g() is the g.f. for A208724.
(-1)^n * a(n) = A208933(n). a(2*n) = A131126(n). a(2*n + 2) = -2 * A093160(n). - Michael Somos, Dec 11 2016
Convolution inverse of A208274. - Michael Somos, Dec 11 2016
a(n) ~ (-1)^n * exp(sqrt(n)*Pi) / (2^(7/2) * n^(3/4)). - Vaclav Kotesovec, Nov 15 2017

A208604 Expansion of phi(-q) / phi(q^4) in powers of q where phi() is a Ramanujan theta function.

Original entry on oeis.org

1, -2, 0, 0, 0, 4, 0, 0, 0, -10, 0, 0, 0, 20, 0, 0, 0, -36, 0, 0, 0, 64, 0, 0, 0, -110, 0, 0, 0, 180, 0, 0, 0, -288, 0, 0, 0, 452, 0, 0, 0, -692, 0, 0, 0, 1044, 0, 0, 0, -1554, 0, 0, 0, 2276, 0, 0, 0, -3296, 0, 0, 0, 4724, 0, 0, 0, -6696, 0, 0, 0, 9408, 0, 0
Offset: 0

Views

Author

Michael Somos, Feb 29 2012

Keywords

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Examples

			G.f. = 1 - 2*q + 4*q^5 - 10*q^9 + 20*q^13 - 36*q^17 + 64*q^21 - 110*q^25 + ...

Links

Crossrefs

Cf. A079006, A208274, A208933.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q] / EllipticTheta[ 3, 0, q^4], {q, 0, n}]; (* Michael Somos, Dec 11 2016 *)
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^4 + A)^2 * eta(x^16 + A)^2 / (eta(x^2 + A) * eta(x^8 + A)^5), n))};

Formula

Expansion of eta(q)^2 * eta(q^4)^2 * eta(q^16)^2 / (eta(q^2) * eta(q^8)^5) in powers of q.
Euler transform of period 16 sequence [ -2, -1, -2, -3, -2, -1, -2, 2, -2, -1, -2, -3, -2, -1, -2, 0, ...].
G.f.: (Sum_{k in Z} (-1)^k * x^k^2) / (Sum_{k in Z} x^(4 * k^2)).
a(4*n) = 0 unless n=0. a(4*n + 2) = a(4*n + 3) = 0. a(4*n + 1) = -2 * A079006(n).
a(n) = (-1)^n * A208274(n). Convolution inverse of A208933. - Michael Somos, Dec 11 2016
G.f.: Product_{k>0} (1 + x^(8*k)) / ((1 + x^k)^2 * (1 + x^(2*k)) * (1 + x^(4*k))^3). - Michael Somos, Dec 11 2016

A216060 Expansion of (phi(q) / phi(q^4))^2 in powers of q where phi() is a Ramanujan theta function.

Original entry on oeis.org

1, 4, 4, 0, 0, -8, -16, 0, 0, 20, 56, 0, 0, -40, -160, 0, 0, 72, 404, 0, 0, -128, -944, 0, 0, 220, 2072, 0, 0, -360, -4320, 0, 0, 576, 8648, 0, 0, -904, -16720, 0, 0, 1384, 31360, 0, 0, -2088, -57312, 0, 0, 3108, 102364, 0, 0, -4552, -179104, 0, 0, 6592
Offset: 0

Views

Author

Michael Somos, Aug 31 2012

Keywords

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Examples

			1 + 4*q + 4*q^2 - 8*q^5 - 16*q^6 + 20*q^9 + 56*q^10 - 40*q^13 - 160*q^14 + ...

Links

Crossrefs

Cf. A001938, A079006, A208274.

Programs

  • Mathematica
    a[n_]:= SeriesCoefficient[(EllipticTheta[3, 0, q]/EllipticTheta[3, 0, q^4])^2, {q, 0, n}]; Table[a[n], {n, 0, 50}] (* G. C. Greubel, Dec 04 2017 *)
  • PARI
    {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^5 * eta(x^16 + A)^2 / (eta(x + A)^2 * eta(x^8 + A)^5))^2, n))}

Formula

Expansion of (eta(q^2)^5 * eta(q^16)^2 / (eta(q)^2 * eta(q^8)^5))^2 in powers of q.
Euler transform of period 16 sequence [ 4, -6, 4, -6, 4, -6, 4, 4, 4, -6, 4, -6, 4, -6, 4, 0, ...].
a(4*n) = 0 unless n=0. a(4*n + 3) = 0. a(4*n + 1) = 4 * A079006(n). a(4*n + 2) = 4 * A001938(n).
Convolution square of A208274.
Empirical: Sum{n>=0} a(n)/exp(Pi*n) = 40 + 28*sqrt(2) - 8*sqrt(48+34*sqrt(2)). - Simon Plouffe, Mar 02 2021
Showing 1-4 of 4 results.

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