Mark A. Thomas - cp's OEIS frontend

A322171 Expansion of x(3 + 5x + x^2 + x^3)/((1 - x)^2*(1 + x^2)).

3, 11, 17, 19, 23, 31, 37, 39, 43, 51, 57, 59, 63, 71, 77, 79, 83, 91, 97, 99, 103, 111, 117, 119, 123, 131, 137, 139, 143, 151, 157, 159, 163, 171, 177, 179, 183, 191, 197, 199, 203, 211, 217, 219, 223, 231, 237, 239, 243, 251, 257, 259, 263, 271, 277, 279, 283, 291, 297, 299

Offset: 1

Mark A. Thomas, Nov 29 2018

Index entries for linear recurrences with constant coefficients, signature (2,-2,2,-1).

Cf. A228826.

I:=[3,11,17,19]; [n le 4 select I[n] else 2*Self(n-1)-2*Self(n-2)+2*Self(n-3)-Self(n-4): n in [1..60]]; // Vincenzo Librandi, Dec 06 2018

seq(coeff(series(x*(x^3+x^2+5*x+3)/((1-x)^2*(1+x^2)),x,n+1), x, n), n = 1 .. 60); # Muniru A Asiru, Dec 06 2018

CoefficientList[Series[(x^3 + x^2 + 5 x + 3)/((x - 1)^2 (x^2 + 1)), {x, 0, 50}], x] (* or *)
a[n_]:= (1/2) (10 n - (1 + 2 * I) (-I)^n - (1 - 2 I) I^n); Simplify[Array[a, 50]] (* Stefano Spezia, Nov 29 2018 *)
LinearRecurrence[{2, -2, 2, -1}, {3, 11, 17, 19}, 60] (* Vincenzo Librandi, Dec 06 2018 *)

Vec((3 + 5*x + x^2 + x^3)/((1 - x)^2*(1 + x^2)) + O(x^60)) \\ Andrew Howroyd, Nov 29 2018

a(n) = (1/2)*(10*n - (1+2*i)*(-i)^n - (1-2*i)*i^n), where i = sqrt(-1).
a(n) = 5*n - 2*sin(Pi*n/2) - cos(Pi*n/2).
a(n) = 5*n - A228826(n-1). - Andrew Howroyd, Nov 29 2018
G.f.: x*(x^3 + x^2 + 5*x + 3) / ((x - 1)^2 *(x^2 + 1)). - Vincenzo Librandi, Dec 06 2018

A320141 a(n) is the sum of the nearest integer to the imaginary part of the n-th zero of the Riemann zeta function and the n-th prime.

Original entry on oeis.org

16, 24, 30, 37, 44, 51, 58, 62, 71, 79, 84, 93, 100, 104, 112, 120, 129, 133, 143, 148, 152, 162, 168, 176, 186, 193, 198, 203, 208, 214, 231, 236, 244, 250, 261, 265, 273, 282, 288, 296, 303, 309, 321, 324, 330, 334, 349, 363, 368, 372, 379, 386, 391, 402, 410, 419, 427, 430, 438, 444

Offset: 1

Mark A. Thomas, Oct 06 2018

nonn

			14 + 2 = 16, 21 + 3 = 24, 25 + 5 = 30.

Cf. A000040, A002410.

lista(nn) = my(v=apply(round, lfunzeros(lzeta, nn))); vector(#v, n, v[n] + prime(n)); \\

a(n) = A002410(n) + A000040(n).

A319229 a(n) is equal to the difference between the nearest integer to the imaginary part of the n-th zero of the Riemann zeta function and the n-th prime.

Original entry on oeis.org

12, 18, 20, 23, 22, 25, 24, 24, 25, 21, 22, 19, 18, 18, 18, 14, 11, 11, 9, 6, 6, 4, 2, -2, -8, -9, -8, -11, -10, -12, -23, -26, -30, -28, -37, -37, -41, -44, -46, -50, -55, -53, -61, -62, -64, -64, -73, -83, -86, -86, -87, -92, -91, -100, -104, -107, -111, -112, -116, -118

Offset: 1

Mark A. Thomas, Sep 14 2018

sign

			14 - 2 = 12, 21 - 3 = 18, 25 - 5 = 20.

Cf. A000040, A002410.

lista(nn) = my(v=apply(round, lfunzeros(lzeta, nn))); vector(#v, n, v[n] - prime(n)); \\ Michel Marcus, Sep 14 2018

a(n) = A002410(n) - prime(n).

A198343 Divisors of 196560.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 18, 20, 21, 24, 26, 27, 28, 30, 35, 36, 39, 40, 42, 45, 48, 52, 54, 56, 60, 63, 65, 70, 72, 78, 80, 84, 90, 91, 104, 105, 108, 112, 117, 120, 126, 130, 135, 140, 144, 156, 168, 180, 182, 189, 195, 208, 210, 216, 234, 240, 252, 260, 270, 273, 280, 312, 315, 336, 351, 360, 364, 378, 390, 420, 432, 455

Offset: 1

N. J. A. Sloane, Oct 23 2011, following a suggestion from Mark A. Thomas

196560 is the kissing number of the Leech lattice (cf. A008408). It is a famous number in the "Moonshine" investigations.

N. J. A. Sloane, Table of n, a(n) for n = 1..160
Eiichi Bannai and N. J. A. Sloane, Uniqueness of certain spherical codes, Canad. J. Math. 33 (1981), no. 2, 437-449.
J. H. Conway and S. P. Norton, Monstrous Moonshine, Bull. Lond. Math. Soc. 11 (1979), no. 3, 308-339.
A. M. Odlyzko and N. J. A. Sloane, New bounds on the number of unit spheres that can touch a unit sphere in n dimensions, J. Combin. Theory Ser. A 26 (1979), no. 2, 210-214.
J. G. Thompson, Some numerology between the Fischer-Griess Monster and the elliptic modular function, Bull. London Math. Soc., 11 (1979), no. 3, 352-353.
Index entries for sequences related to divisors of numbers

Cf. A008408.

Divisors[196560] (* Paolo Xausa, Jul 01 2024 *)

divisors(196560) \\ Charles R Greathouse IV, Feb 21 2013

A181164 Decimal expansion of exp(5*Pi)/8.

Original entry on oeis.org

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

Offset: 6

Mark A. Thomas, Oct 07 2010

This real number is close to the prime number 829453.

			829452.999917641779158283008387388115229513366768873400405895867044280150...

G. C. Greubel, Table of n, a(n) for n = 6..10000

using Nemo
RR = RealField(366)
t = RR(5) * const_pi(RR)
exp(t)/RR(8) |> println # Peter Luschny, Mar 13 2018

R:= RealField(); Exp(5*Pi(R))/8; // G. C. Greubel, Feb 14 2018

RealDigits[Exp[5*Pi]/8, 10, 100][[1]] (* G. C. Greubel, Feb 14 2018 *)

exp(5*Pi)/8 \\ G. C. Greubel, Feb 14 2018

Equals exp(5*Pi)/8.

A181165 Decimal expansion of exp(Pi*sqrt43)/24.

Original entry on oeis.org

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

Offset: 8

Mark A. Thomas, Oct 07 2010

This real number is close to the prime number 36864031.

			exp(Pi*sqrt(43))/24 = 36864030.9999907277514544442473942532743907019... This is almost the prime 36864031.

G. C. Greubel, Table of n, a(n) for n = 8..10007

R:= RealField(); Exp(Pi(R)*Sqrt(43)); // G. C. Greubel, Feb 14 2018

RealDigits[E^(Pi Sqrt[43])/24,10,120][[1]] (* Harvey P. Dale, Oct 17 2011 *)

exp(Pi*sqrt(43))/24 \\ G. C. Greubel, Feb 14 2018

Equals exp(Pi*sqrt(43))/24.

A181166 Decimal expansion of exp(Pi*sqrt(67))/24.

Original entry on oeis.org

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

Offset: 10

Mark A. Thomas, Oct 07 2010

This real number is close to the prime number 6133248031.

			exp(Pi*sqrt(67))/24 = 6133248030.999999944268926021117885888024109521... This is almost the prime 6133248031.

G. C. Greubel, Table of n, a(n) for n = 10..10000

RealField(); Exp(Pi(R)*Sqrt(67))/24; // G. C. Greubel, Feb 14 2018

E^(Pi Sqrt[67])/24
RealDigits[Exp[Pi Sqrt[67]]/24, 10, 100][[1]] (* G. C. Greubel, Feb 14 2018 *)

exp(Pi*sqrt(67))/24 \\ G. C. Greubel, Feb 14 2018

Equals exp(Pi*sqrt(67))/24.

A181163 Decimal expansion of A169624/8.

Original entry on oeis.org

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

Offset: 10

Mark A. Thomas, Oct 07 2010

This real number is close to the prime number 3073907219.

			e^(Pi*sqrt(58))/8 = 3073907218.999999977776655183697024044408226528451271... This is almost the prime 3073907219.

G. C. Greubel, Table of n, a(n) for n = 10..10000

R:= RealField(); Exp(Pi(R)*Sqrt(58))/8; // G. C. Greubel, Feb 14 2018

E^(Pi Sqrt[58])/8
RealDigits[Exp[Pi*Sqrt[58]]/8, 10, 100][[1]] (* G. C. Greubel, Feb 14 2018 *)

exp(Pi*sqrt(58))/8 \\ G. C. Greubel, Feb 14 2018

Equals exp(Pi*sqrt(58))/8.

A181425 Decimal expansion of e^(Pi*sqrt(22))/8.

Original entry on oeis.org

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

Offset: 6

Mark A. Thomas, Oct 18 2010

This real number is close to the prime number 313619.

			e^(Pi*sqrt(22))/8 = 313618.99978217805839569114926521089530955134185852580...
This is almost the prime 313619.

RealDigits[E^(Pi Sqrt[22])/8,10,120][[1]] (* Harvey P. Dale, Apr 22 2018 *)

exp(Pi*sqrt(22))/8 \\ Charles R Greathouse IV, Mar 26 2012

e^(Pi*sqrt(22))/8.

Prior Mathematica program replaced by Harvey P. Dale, Apr 22 2018

A181045 Decimal expansion of A060295/24.

Original entry on oeis.org

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

Offset: 17

Mark A. Thomas, Sep 30 2010

This real number is close to the prime number 10939058860032031. Also, the only (single) integer values placed in the denominator that will generate 'near-integers' from this relation are the divisors of 24: 1, 2, 3, 4, 6, 8, 12, 24 (cf. A018253). A total of 64 'near-integers' can be obtained from generating powers (1-8) of A060295 and dividing each by one of the divisors of 24. Example: The last (64th) 'near-integer' is A060295^8 = 2.25698985492608864738884...99926422461218840012234... *10^139 (which is split by ... for brevity), the digits of which close to the decimal point are ...218840.012234... . While this does not quite look like a 'near-integer' this is where the pattern of 0's and 9's in the decimal tail cease in the case. See A166532.

			A060295/24 = 10939058860032030.999999999999968753024883257737036639... This is almost the prime 10939058860032031.

G. C. Greubel, Table of n, a(n) for n = 17..10000
Math Overflow, Questions [From _Mark A. Thomas_, Oct 02 2010]
M. A. Thomas, Math Ontological Basis of Quasi Fine-Tuning in Ghc Cosmologies, HAL preprint Id: hal-01232022, 2015.

Cf. A166528, A166529, A166530, A166531.

Magma
```
R:= RealField(); Exp(Pi*Sqrt(163))/24;
```

E^(Pi Sqrt[163])/24
RealDigits[Exp[Pi Sqrt[163]]/24, 10, 100][[1]] (* G. C. Greubel, Feb 14 2018 *)

exp(Pi*sqrt(163))/24 \\ G. C. Greubel, Feb 14 2018

Equals exp(Pi * sqrt(163))/24.

cp's OEIS Frontend

User: Mark A. Thomas

A322171 Expansion of x*(3 + 5*x + x^2 + x^3)/((1 - x)^2*(1 + x^2)).

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A320141 a(n) is the sum of the nearest integer to the imaginary part of the n-th zero of the Riemann zeta function and the n-th prime.

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A319229 a(n) is equal to the difference between the nearest integer to the imaginary part of the n-th zero of the Riemann zeta function and the n-th prime.

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A198343 Divisors of 196560.

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A181164 Decimal expansion of exp(5*Pi)/8.

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Julia

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A181165 Decimal expansion of exp(Pi*sqrt43)/24.

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A181166 Decimal expansion of exp(Pi*sqrt(67))/24.

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A181163 Decimal expansion of A169624/8.

A322171 Expansion of x(3 + 5x + x^2 + x^3)/((1 - x)^2*(1 + x^2)).