A016923 -id:A016923 - cp's OEIS frontend

A016929 a(n) = (6*n + 1)^9.

1, 40353607, 10604499373, 322687697779, 3814697265625, 26439622160671, 129961739795077, 502592611936843, 1628413597910449, 4605366583984375, 11694146092834141, 27206534396294947, 58871586708267913, 119851595982618319, 231616946283203125, 427929800129788411

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..2000
Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).

Cf. A016921, A016922, A016923, A016924, A016925, A016926, A016927, A016928.

[(6*n+1)^9: n in [0..25]]; // Vincenzo Librandi, May 04 2011

(6*Range[0,20]+1)^9 (* or *) LinearRecurrence[{10,-45,120,-210,252,-210,120,-45,10,-1},{1,40353607,10604499373,322687697779,3814697265625,26439622160671,129961739795077,502592611936843,1628413597910449,4605366583984375},20] (* Harvey P. Dale, Mar 22 2015 *)

a(n) = 10*a(n-1) - 45*a(n-2) + 120*a(n-3) - 210*a(n-4) + 252*a(n-5) - 210*a(n-6) + 120*a(n-7) - 45*a(n-8) + 10*a(n-9) - a(n-10). - Harvey P. Dale, Mar 22 2015
From Amiram Eldar, Mar 28 2022: (Start)
a(n) = A016921(n)^9 = A016923(n)^3.
Sum_{n>=0} 1/a(n) = 15371*Pi^9/(529079040*sqrt(3)) + 5028751*zeta(9)/10077696. (End)

A016930 a(n) = (6*n + 1)^10.

Original entry on oeis.org

1, 282475249, 137858491849, 6131066257801, 95367431640625, 819628286980801, 4808584372417849, 21611482313284249, 79792266297612001, 253295162119140625, 713342911662882601, 1822837804551761449, 4297625829703557649, 9468276082626847201, 19687440434072265625

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (11,-55,165,-330,462,-462,330,-165,55,-11,1).

Cf. A016921, A016922, A016923, A016924, A016925, A016926, A016927, A016928, A016929.

[(6*n+1)^10: n in [0..25]]; // Vincenzo Librandi, May 04 2011

(6 Range[0, 15] + 1)^10 (* Wesley Ivan Hurt, Jan 15 2022 *)
LinearRecurrence[{11,-55,165,-330,462,-462,330,-165,55,-11,1},{1,282475249,137858491849,6131066257801,95367431640625,819628286980801,4808584372417849,21611482313284249,79792266297612001,253295162119140625,713342911662882601},20] (* Harvey P. Dale, Sep 05 2023 *)

From Amiram Eldar, Mar 28 2022: (Start)
a(n) = A016921(n)^10 = A016922(n)^5 = A016925(n)^2.
Sum_{n>=0} 1/a(n) = PolyGamma(9, 1/6)/21941965946880. (End)

A377560 Decimal expansion of Pi^3/(36sqrt(3)) + 91zeta(3)/216.

Original entry on oeis.org

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

Offset: 1

Stefano Spezia, Nov 01 2024

			1.00368551534795269706323013702486057315272784359...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.6.3, p. 42.

Michael I. Shamos, A catalog of the real numbers, (2007). See p. 28.

Cf. A002117, A002194, A016921, A016923, A091925, A233091, A377557, A377558, A377559.

RealDigits[Pi^3/(36*Sqrt[3])+91*Zeta[3]/216,10,100][[1]]

Equals Sum_{k>=0} 1/(6*k + 1)^3 (see Finch).

Equals -psi''(1/6)/432 (see Shamos).

A016931 a(n) = (6*n + 1)^11.

Original entry on oeis.org

1, 1977326743, 1792160394037, 116490258898219, 2384185791015625, 25408476896404831, 177917621779460413, 929293739471222707, 3909821048582988049, 13931233916552734375, 43513917611435838661, 122130132904968017083, 313726685568359708377, 747993810527520928879

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (12,-66,220,-495,792,-924,792,-495,220,-66,12,-1).

Cf. A016921, A016922, A016923, A016924, A016925, A016926, A016927, A016928, A016929, A016930.

[(6*n+1)^11: n in [0..25]]; // Vincenzo Librandi, May 04 2011

(6*Range[0,20]+1)^11 (* Harvey P. Dale, Aug 23 2017 *)

From Amiram Eldar, Mar 28 2022: (Start)
a(n) = A016921(n)^11.
Sum_{n>=0} 1/a(n) = 1261501*Pi^11/(428554022400*sqrt(3)) + 181308931*zeta(11)/362797056. (End)

A166853 a(n) is the smallest number m such that m^m-n is prime, or zero if there is no such m.

Original entry on oeis.org

2, 2, 8, 3, 4, 5, 6, 3, 0, 3, 78, 13, 6, 3, 4, 3, 4, 17, 12, 3, 118, 3, 4, 3, 3

Offset: 1

Farideh Firoozbakht and M. F. Hasler, Nov 27 2009

The sequence with the unknown terms a(n) indicated by -n:
(0's occur for n=9, 49, 81, 121....)
2,2,8,3,4,5,6,3,0,3,78,13,6,3,4,3,4,17,12,3,118,3,4,3,3,
-26,4,-28,4,487,90,9,4,-34,24,5,6,271,28,969,-41,5,-43,7,4,5,32,37,0,621,
20,15,34,7,6,9,4,5,4,7,-61,7,4,5,4,-66,6,63,134,27,10,35,102,31,4,
5,4,569,-79,13,0,15,4,5,-85,7,110,5,4,131,1122,7,4,11,8,7,6,9,4,-100,
22,5,-103,-104,4,5,4,11,12,39,-111,...
If they exist, the first two unknown terms, a(26) and a(28), they are greater than 10000. All other unknown terms a(n), for n<112 are greater than 4000.
If it exists, a(26) > 25000. - Robert Price, Apr 26 2019

			We have a(1)=2 since 1^1-1 is not prime, but 2^2-1 is prime.
a(9)=0 since 2^2-9 is not prime, and if m is an even number greater than 2 then m^m-9=(m^(m/2)-3)*(m^(m/2)+3) is composite. So there is no number m such that m^m-9 is prime. The same applies to any odd square > 25.
We have a(25)=3 since 3^3-25=2 is prime. But 25 is the only known square of the form m^m-2, so a(n)=0 for other odd squares > 25, e.g., n = 49,81,121,....
a(115)=2736 is the largest known term. 2736^2736-115 is a probable prime.

Cf. A087037, A087038, A016754, A016923, A100407, A100408, A166852.

a(n)=0 if n=3^2 or n=(2k+1)^2 > 25, or n = (6k+1)^3 = A016923(k) with k>0.

A016932 a(n) = (6*n + 1)^12.

Original entry on oeis.org

1, 13841287201, 23298085122481, 2213314919066161, 59604644775390625, 787662783788549761, 6582952005840035281, 39959630797262576401, 191581231380566414401, 766217865410400390625, 2654348974297586158321, 8182718904632857144561, 22902048046490258711521

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (13,-78,286,-715,1287,-1716,1716,-1287,715,-286,78,-13,1).

Cf. A016921, A016922, A016923, A016924, A016925, A016926, A016927, A016928, A016929, A016930, A016931.

[(6*n+1)^12: n in [0..20]]; // Vincenzo Librandi, May 04 2011

Table[(6*n + 1)^12, {n, 0, 12}] (* Amiram Eldar, Mar 28 2022 *)

From Amiram Eldar, Mar 28 2022: (Start)
a(n) = A016921(n)^12 = A016922(n)^6 = A016923(n)^4 = A016924(n)^3 = A016926(n)^2.
Sum_{n>=0} 1/a(n) = PolyGamma(11, 1/6)/86890185149644800. (End)

A289134 a(n) = 21n^2 - 33n + 13.

Original entry on oeis.org

1, 31, 103, 217, 373, 571, 811, 1093, 1417, 1783, 2191, 2641, 3133, 3667, 4243, 4861, 5521, 6223, 6967, 7753, 8581, 9451, 10363, 11317, 12313, 13351, 14431, 15553, 16717, 17923, 19171, 20461, 21793, 23167, 24583, 26041, 27541, 29083, 30667, 32293

Offset: 1

Daniel Rockwitz Reynolds, Jun 25 2017

a(n) is the sum of all cells in a cellular-automata-like hexagonal lattice growth from a single active seed, based upon whether each hexagonal unit is active plus how many active neighbors each cell is touching for all active cells in the lattice.

The initial hexagonal seed starts with a single 1 representing it is active and touching no active neighbors. In the next time step, all inactive hexagonal neighboring spaces in the surrounding hexagonal lattice which were touching the active seed via edges become active and all active cells are summed together based on whether they are active plus how many active neighbors they are touching via their edges. This continues for each time step with inactive neighbors touching active neighbors in the previous time step becoming active in the current step followed by the described summing.

Colin Barker, Table of n, a(n) for n = 1..1000
D. R. Reynolds, Geometric graphic of t, a(t) for t=1...4
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A033574 (analog for square tiling, von Neumann neighborhood), A016922 (analog for square tiling, Moore neighborhood), A016923 (analog for cubic 3D tiling, Moore neighborhood), A064762.

hexgro[t_]:=7+4*6+5*6*(t-2)+Sum[i*6*7,{i,t-2}]; Table[hexgro[n],{n,40}]
LinearRecurrence[{3,-3,1},{1,31,103},40] (* Harvey P. Dale, Apr 23 2020 *)

Vec(x*(1 + 28*x + 13*x^2) / (1 - x)^3 + O(x^60)) \\ Colin Barker, Jun 28 2017

G.f.: x*(1 + 28*x + 13*x^2) / (1 - x)^3. - Colin Barker, Jun 28 2017

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>3. - Colin Barker, Jul 29 2017

New Name from Omar E. Pol, Jun 25 2017

cp's OEIS Frontend

A016929 a(n) = (6*n + 1)^9.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

Formula

A016930 a(n) = (6*n + 1)^10.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

Formula

A377560 Decimal expansion of Pi^3/(36*sqrt(3)) + 91*zeta(3)/216.

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Mathematica

Formula

A016931 a(n) = (6*n + 1)^11.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

Formula

A166853 a(n) is the smallest number m such that m^m-n is prime, or zero if there is no such m.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A016932 a(n) = (6*n + 1)^12.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

Formula

A289134 a(n) = 21*n^2 - 33*n + 13.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A377560 Decimal expansion of Pi^3/(36sqrt(3)) + 91zeta(3)/216.

A289134 a(n) = 21n^2 - 33n + 13.