A081085 -id:A081085 - cp's OEIS frontend

A291280 Primes p such that p does not divide any term of the Apery-like sequence A125143.

2, 5, 7, 11, 13, 19, 29, 41, 47, 61, 67, 71, 73, 89, 97, 101, 103, 131, 137, 139, 149, 157, 163, 167, 173, 179, 181, 193, 197, 199, 211, 223, 227, 229, 233, 241, 251, 257, 263, 269, 281, 283, 293, 317, 331, 347, 349, 359, 367, 373, 379, 383

Offset: 1

N. J. A. Sloane, Aug 21 2017

nonn

Amita Malik and Armin Straub, Mathematica notebook for generating A133370 and A260793, A291275-A291284
Amita Malik and Armin Straub, Lists of all primes up to 10000 in A133370 and A260793, A291275-A291284, together with Mathematica code.
Amita Malik and Armin Straub, Divisibility properties of sporadic Apéry-like numbers, Research in Number Theory, 2016, 2:5

For primes that do not divide the terms of the sequences A000172, A005258, A002893, A081085, A006077, A093388, A125143, A229111, A002895, A290575, A290576, A005259 see A260793, A291275-A291284 and A133370 respectively.

A291281 Primes p such that p does not divide any term of the Apery-like sequence A229111.

Original entry on oeis.org

2, 3, 17, 19, 23, 31, 47, 53, 61, 107, 109, 113, 137, 139, 151, 173, 197, 199, 211, 227, 229, 233, 241, 257, 263, 293, 317, 347, 353, 383, 421, 439, 443, 467, 499, 541, 587, 593, 619, 647, 661, 677, 683, 691, 751, 769, 773, 857, 919

Offset: 1

N. J. A. Sloane, Aug 21 2017

nonn

Amita Malik and Armin Straub, Mathematica notebook for generating A133370 and A260793, A291275-A291284
Amita Malik and Armin Straub, Lists of all primes up to 10000 in A133370 and A260793, A291275-A291284, together with Mathematica code.
Amita Malik and Armin Straub, Divisibility properties of sporadic Apéry-like numbers, Research in Number Theory, 2016, 2:5

For primes that do not divide the terms of the sequences A000172, A005258, A002893, A081085, A006077, A093388, A125143, A229111, A002895, A290575, A290576, A005259 see A260793, A291275-A291284 and A133370 respectively.

A291282 Primes p such that p does not divide any term of the Apery-like sequence A002895.

Original entry on oeis.org

3, 5, 13, 17, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 83, 89, 101, 103, 107, 109, 127, 131, 137, 163, 167, 173, 181, 191, 193, 199, 229, 233, 239, 241, 251, 257, 269, 307, 311, 317, 337, 347, 349, 359, 367, 373, 409, 419, 421, 449, 457

Offset: 1

N. J. A. Sloane, Aug 21 2017

nonn

Amita Malik and Armin Straub, Mathematica notebook for generating A133370 and A260793, A291275-A291284
Amita Malik and Armin Straub, Lists of all primes up to 10000 in A133370 and A260793, A291275-A291284, together with Mathematica code.
Amita Malik and Armin Straub, Divisibility properties of sporadic Apéry-like numbers, Research in Number Theory, 2016, 2:5

For primes that do not divide the terms of the sequences A000172, A005258, A002893, A081085, A006077, A093388, A125143, A229111, A002895, A290575, A290576, A005259 see A260793, A291275-A291284 and A133370 respectively.

A291283 Primes p such that p does not divide any term of the Apery-like sequence A290575.

Original entry on oeis.org

3, 7, 13, 19, 23, 29, 31, 37, 43, 47, 61, 67, 73, 83, 89, 101, 103, 107, 109, 113, 127, 131, 139, 149, 157, 167, 179, 191, 193, 197, 223, 227, 229, 241, 251, 257, 271, 281, 293, 307, 311, 313, 353, 367, 373, 379, 389, 401, 419, 431, 433

Offset: 1

N. J. A. Sloane, Aug 21 2017

nonn

Amita Malik and Armin Straub, Mathematica notebook for generating A133370 and A260793, A291275-A291284
Amita Malik and Armin Straub, Lists of all primes up to 10000 in A133370 and A260793, A291275-A291284, together with Mathematica code.
Amita Malik and Armin Straub, Divisibility properties of sporadic Apéry-like numbers, Research in Number Theory, 2016, 2:5

For primes that do not divide the terms of the sequences A000172, A005258, A002893, A081085, A006077, A093388, A125143, A229111, A002895, A290575, A290576, A005259 see A260793, A291275-A291284 and A133370 respectively.

A294454 a(n) = n! * [x^n] exp(2nx)BesselI(0,2x)^n.

Original entry on oeis.org

1, 2, 20, 324, 7336, 213500, 7593744, 319195800, 15481238224, 850968357228, 52279073479120, 3549850939488392, 263999303861731200, 21340730504572110008, 1863120652816098506432, 174706136370865217610000, 17512175948995988236164000, 1868638289932305589084614220, 211478046685658614366937497296

Offset: 0

Ilya Gutkovskiy, Nov 23 2017

nonn

The n-th term of the n-fold exponential convolution of A000984 with themselves.

N. J. A. Sloane, Transforms

Cf. A000984, A081085, A091527, A293491.

Table[n! SeriesCoefficient[Exp[2 n x] BesselI[0, 2 x]^n, {x, 0, n}], {n, 0, 18}]

a(n) ~ c * d^n * n! / sqrt(n), where d = 6.46710510392662827829435747085578126903789467159876086... and c = 0.36028050364743885143298970162021762094091934461095... - Vaclav Kotesovec, May 04 2024

A327834 Expansion of 1 / AGM(1, 1 - 8*x)^2 in powers of x.

Original entry on oeis.org

1, 8, 56, 384, 2648, 18496, 131008, 940032, 6821848, 49985984, 369258560, 2746629120, 20549693888, 154518118912, 1166873394688, 8844937101312, 67265481552856, 513038965707968, 3923108472072512, 30068733313938432, 230943237733355840, 1777114026405752320

Offset: 0

Vaclav Kotesovec, Sep 27 2019

nonn

AGM(x,y) = AGM((x+y)/2,sqrt(x*y)) is the arithmetic-geometric mean.

Eric Weisstein's World of Mathematics, Arithmetic-Geometric Mean

Cf. A081085, A089603.

CoefficientList[Series[(2*EllipticK[1/(1 - 1/(4*x))^2]/(Pi*(1 - 4*x)))^2, {x, 0, 25}], x]
CoefficientList[Series[Hypergeometric2F1[1/2, 1/2, 1, 16*x*(1 - 4*x)]^2, {x, 0, 25}], x]

Recurrence: n^3*a(n) = 4*(2*n - 1)*(3*n^2 - 3*n + 2)*a(n-1) - 16*(n-1)*(13*n^2 - 26*n + 20)*a(n-2) + 128*(2*n - 3)*(3*n^2 - 9*n + 8)*a(n-3) - 1024*(n-2)^3*a(n-4).

a(n) ~ 2^(3*n + 3) * (log(4*n) + gamma) / (Pi^2 * n), where gamma is the Euler-Mascheroni constant A001620.

cp's OEIS Frontend

A291280 Primes p such that p does not divide any term of the Apery-like sequence A125143.

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A291281 Primes p such that p does not divide any term of the Apery-like sequence A229111.

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A291282 Primes p such that p does not divide any term of the Apery-like sequence A002895.

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A291283 Primes p such that p does not divide any term of the Apery-like sequence A290575.

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A294454 a(n) = n! * [x^n] exp(2*n*x)*BesselI(0,2*x)^n.

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A327834 Expansion of 1 / AGM(1, 1 - 8*x)^2 in powers of x.

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Programs

Mathematica

Formula

A294454 a(n) = n! * [x^n] exp(2nx)BesselI(0,2x)^n.