A104862 -id:A104862 - cp's OEIS frontend

A104843 Primes from merging of 2 successive digits in decimal expansion of e.

71, 59, 23, 53, 47, 71, 13, 97, 47, 59, 67, 53, 47, 59, 71, 13, 17, 19, 59, 17, 41, 13, 59, 29, 43, 29, 29, 59, 73, 13, 23, 79, 43, 23, 29, 53, 31, 19, 19, 11, 73, 83, 41, 79, 89, 41, 67, 47, 61, 47, 41, 11, 53, 37, 23, 43, 37, 71, 53, 17, 61, 83, 61, 13, 31, 13, 83, 29, 97

Offset: 1

Andrew G. West (WestA(AT)wlu.edu), Mar 27 2005

Scan decimal expansion of e from left to right, recording any 2-digit primes seen. Overlaps are allowed in all the sequences in this family. - N. J. A. Sloane, Feb 05 2012

Leading zeros are not permitted, so each term is 2 digits in length. - Harvey P. Dale, Oct 23 2011

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, e

Cf. A001113, A073246 (the one-digit primes in e), A104844 - A104862.

With[{len=2},FromDigits/@Select[Partition[RealDigits[E,10,1000][[1]], len,1],PrimeQ[FromDigits[#]]&&IntegerLength[FromDigits[#]]==len&]] (* Harvey P. Dale, Oct 23 2011 *)

Changed offset from 0 to 1 by Vincenzo Librandi, Apr 21 2013

A104850 Primes from merging of 9 successive digits in decimal expansion of e.

Original entry on oeis.org

360287471, 526059563, 132328627, 862794349, 573834187, 383418793, 879307021, 154089149, 675092447, 509244761, 234544243, 907774499, 744992069, 551702761, 860626133, 331384583, 297606737, 976067371, 113200709, 704723069, 108657463

Offset: 1

Andrew G. West (WestA(AT)wlu.edu), Mar 27 2005

Leading zeros are not permitted, so each term is 9 digits in length. - Harvey P. Dale, Oct 23 2011

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A104843-A104862.

With[{len=9},FromDigits/@Select[Partition[RealDigits[E,10,1000][[1]], len,1],PrimeQ[ FromDigits[ #]]&&IntegerLength[FromDigits[#]]==len&]] (* Harvey P. Dale, Oct 23 2011 *)

Offset changed from 0 to 1 by Vincenzo Librandi, Apr 21 2013

A104844 Primes from merging of 3 successive digits in decimal expansion of e.

Original entry on oeis.org

271, 281, 523, 353, 977, 757, 709, 967, 277, 353, 547, 457, 571, 821, 251, 919, 193, 599, 181, 359, 563, 307, 349, 907, 233, 829, 251, 101, 157, 383, 307, 149, 499, 167, 509, 761, 853, 107, 907, 449, 499, 761, 613, 331, 313, 449, 673, 113, 709, 709, 127

Offset: 1

Andrew G. West (WestA(AT)wlu.edu), Mar 27 2005

Leading zeros are not permitted, so each term is 3 digits in length. - Harvey P. Dale, Oct 23 2011

Vincenzo Librandi, Table of n, a(n) for n = 1..3000

Cf. A104843-A104862.

With[{len=3},FromDigits/@Select[Partition[RealDigits[E,10,1000][[1]], len,1],PrimeQ[FromDigits[#]]&&IntegerLength[FromDigits[#]]==len&]] (* Harvey P. Dale, Oct 23 2011 *)

Offset changed from 0 to 1 by Vincenzo Librandi, Apr 21 2013

A104847 Primes from merging of 6 successive digits in decimal expansion of e.

Original entry on oeis.org

904523, 360287, 624977, 757247, 995957, 967627, 630353, 759457, 594571, 932003, 904357, 290033, 307381, 381323, 627943, 525101, 738341, 341879, 418793, 884167, 847741, 560297, 606737, 328709, 977209, 720931, 169283, 695369, 644549, 312773

Offset: 1

Andrew G. West (WestA(AT)wlu.edu), Mar 27 2005

Leading zeros are not permitted, so each term is 6 digits in length. - Harvey P. Dale, Oct 23 2011

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A104843-A104862.

With[{len=6},FromDigits/@Select[Partition[RealDigits[E,10,1000][[1]], len,1],PrimeQ[ FromDigits[#]] && IntegerLength[FromDigits[#]]==len&]] (* Harvey P. Dale, Oct 23 2011 *)

Offset changed from 0 to 1 by Vincenzo Librandi, Apr 21 2013

A104845 Primes from merging of 4 successive digits in decimal expansion of e.

Original entry on oeis.org

4523, 8747, 7757, 7247, 5749, 6967, 6277, 3547, 4759, 3821, 6427, 4663, 3919, 2003, 1741, 9043, 4357, 8627, 4349, 6323, 8807, 5101, 1019, 1901, 9011, 1879, 1499, 4993, 2447, 7741, 7411, 8537, 4243, 2437, 3907, 2069, 9551, 7027, 6133, 3313, 4583, 4493

Offset: 1

Andrew G. West (WestA(AT)wlu.edu), Mar 27 2005

Leading zeros are not permitted, so each term is 4 digits in length. - Harvey P. Dale, Oct 23 2011

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A104843-A104862.

With[{len=4},FromDigits/@Select[Partition[RealDigits[E,10,1000][[1]], len,1],PrimeQ[FromDigits[#]]&&IntegerLength[FromDigits[#]]==len&]] (* Harvey P. Dale, Oct 23 2011 *)

Offset changed from 0 to 1 by Vincenzo Librandi, Apr 21 2013

A104846 Primes from merging of 5 successive digits in decimal expansion of e.

Original entry on oeis.org

74713, 62497, 24977, 24709, 47093, 95957, 49669, 27427, 46639, 32003, 59921, 21817, 35729, 63073, 28627, 27943, 94349, 33829, 98807, 57383, 41879, 18793, 91499, 68477, 47741, 37423, 42437, 24371, 10753, 17027, 61331, 13313, 93287

Offset: 1

Andrew G. West (WestA(AT)wlu.edu), Mar 27 2005

Leading zeros are not permitted, so each term is 5 digits in length. - Harvey P. Dale, Oct 23 2011

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A104843-A104862.

With[{len=5},FromDigits/@Select[Partition[RealDigits[E,10,1000][[1]], len,1],PrimeQ[FromDigits[#]]&&IntegerLength[FromDigits[#]]==len&]] (* Harvey P. Dale, Oct 23 2011 *)

Offset changed from 0 to 1 by Vincenzo Librandi, Apr 20 2013

A104848 Primes from merging of 7 successive digits in decimal expansion of e.

Original entry on oeis.org

2718281, 6028747, 2497757, 5354759, 4759457, 7594571, 5945713, 7138217, 3059921, 9921817, 6059563, 6307381, 3073813, 9525101, 5251019, 4089149, 8914993, 9348841, 4167509, 7774499, 8606261, 2613313, 5830007, 1274437

Offset: 1

Andrew G. West (WestA(AT)wlu.edu), Mar 27 2005

Leading zeros are not permitted, so each term is 7 digits in length. - Harvey P. Dale, Oct 23 2011

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A104843-A104862.

With[{len=7},FromDigits/@Select[Partition[RealDigits[E,10,1000][[1]], len,1],PrimeQ[FromDigits[#]]&&IntegerLength[FromDigits[#]]==len&]] (* Harvey P. Dale, Oct 23 2011 *)

Offset changed from 0 to 1 by Vincenzo Librandi, Apr 19 2013

A104849 Primes from merging of 8 successive digits in decimal expansion of e.

Original entry on oeis.org

72407663, 40766303, 54759457, 57138217, 20030599, 98807531, 15738341, 83418793, 34884167, 84167509, 22648001, 10753907, 20695517, 38606261, 82656029, 29760673, 13200709, 27443747, 74704723, 69772093, 92836819

Offset: 1

Andrew G. West (WestA(AT)wlu.edu), Mar 27 2005

Leading zeros are not permitted, so each term is 8 digits in length. - Harvey P. Dale, Oct 23 2011

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A104843-A104862.

With[{len=8},FromDigits/@Select[Partition[RealDigits[E,10,1000][[1]], len,1],PrimeQ[FromDigits[#]]&&IntegerLength[FromDigits[#]]==len&]] (* Harvey P. Dale, Oct 23 2011 *)

Offset changed from 0 to 1 by Vincenzo Librandi, Apr 21 2013

A014292 a(n) = 2*a(n-1) - a(n-2) - a(n-4) with a(0) = a(1) = 0, a(2) = 1, a(3) = 2.

Original entry on oeis.org

0, 0, 1, 2, 3, 4, 4, 2, -3, -12, -25, -40, -52, -52, -27, 38, 155, 324, 520, 678, 681, 360, -481, -2000, -4200, -6760, -8839, -8918, -4797, 6084, 25804, 54442, 87877, 115228, 116775, 63880, -76892, -332892, -705667, -1142322

Offset: 0

N. J. A. Sloane

sign

Imaginary part of the sequence of complex numbers defined by c(0) = 1, c(1) = 1, for n>1 c(n) = c(n-1) + i*c(n-2). - Gerald McGarvey, Apr 24 2005
a(n) = sqrt(3)*y where (x,y,y,y) is the quaternion b(n) of the sequence b of quaternions defined by b(0)=1,b(1)=1, b(n) = b(n-1) + b(n-2)*(0,s,s,s) where s = 1/sqrt(3). - Gerald McGarvey, Apr 25 2005
For n>=1, a(n) is equal to -1 times the imaginary part of the determinant of the n X n matrix with the sqrt(i)'s along the superdiagonal and the subdiagonal (i is the imaginary unit), 1's along the main diagonal, and 0's everywhere else (see Mathematica code below). - John M. Campbell, Jun 04 2011

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,-1).

Cf. A104862, A104862.

a:=[0,0,1,2];; for n in [5..40] do a[n]:=2*a[n-1]-a[n-2]-a[n-4]; od; a; # G. C. Greubel, Jun 12 2019

R:=PowerSeriesRing(Integers(), 40); [0,0] cat Coefficients(R!( x^2/(1-2*x+x^2+x^4) )); // G. C. Greubel, Jun 12 2019

Table[-Im[Det[Array[KroneckerDelta[#1 + 1, #2]*Sqrt[I] &, {n, n}] + Array[KroneckerDelta[#1 - 1, #2]*Sqrt[I] &, {n, n}] + IdentityMatrix[n]]], {n, 1, 40}] (* John M. Campbell, Jun 04 2011 *)
LinearRecurrence[{2,-1,0,-1}, {0,0,1,2}, 40] (* G. C. Greubel, Jun 12 2019 *)

my(x='x+O('x^40)); concat([0,0], Vec(x^2/(1-2*x+x^2+x^4))) \\ G. C. Greubel, Jun 12 2019

(x^2/(1-2*x+x^2+x^4)).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, Jun 12 2019

a(n) = Sum_{k=0..floor((n+2)/2)} binomial(n-k+2, k)*sin(Pi*k/2). - Paul Barry, Apr 25 2005

G.f.: x^2/(1 - 2*x + x^2 + x^4). - R. J. Mathar, Oct 22 2008

A106201 Expansion of Re(x/(1-x-2ix^2)), i=sqrt(-1).

Original entry on oeis.org

0, 1, 1, 1, 1, -3, -11, -23, -39, -43, -3, 129, 417, 877, 1349, 1305, -407, -5627, -16243, -32079, -46287, -37987, 35285, 236873, 623609, 1162293, 1559837, 1009889, -2034495, -9728051, -23660955, -41633415, -51467895, -22390171, 101331373, 391586577, 887713361, 1473400829, 1653762805, 267778473, -4669059303, -15499500395

Offset: 0

Paul Barry, Apr 25 2005

Imaginary part is A106202.

Index entries for linear recurrences with constant coefficients, signature (2,-1,0,-4)

Cf. A104862, A014292.

LinearRecurrence[{2,-1,0,-4},{0,1,1,1},50] (* Harvey P. Dale, May 22 2020 *)

G.f.: x*(1-x)/(1-2*x+x^2+4*x^4).

a(n) = sum(k=0..floor((n-1)/2), C(n-k-1, k)*2^k*cos(pi*k/2) ).

cp's OEIS Frontend

A104843 Primes from merging of 2 successive digits in decimal expansion of e.

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A104850 Primes from merging of 9 successive digits in decimal expansion of e.

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A104844 Primes from merging of 3 successive digits in decimal expansion of e.

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A104847 Primes from merging of 6 successive digits in decimal expansion of e.

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A104845 Primes from merging of 4 successive digits in decimal expansion of e.

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A104846 Primes from merging of 5 successive digits in decimal expansion of e.

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A104848 Primes from merging of 7 successive digits in decimal expansion of e.

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A104849 Primes from merging of 8 successive digits in decimal expansion of e.

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A106201 Expansion of Re(x/(1-x-2ix^2)), i=sqrt(-1).