A099969 -id:A099969 - cp's OEIS frontend

A099970 Write 1/e as a binary fraction; read this from left to right and whenever a 1 appears, note the integer formed by reading leftwards from that 1. Then convert those integers from binary into decimal numbers.

1, 5, 13, 29, 61, 573, 2621, 6717, 23101, 88637, 350781, 875069, 9263677, 26040893, 93149757, 227367485, 2374851133, 10964785725, 28144654909, 165583608381, 440461515325, 990217329213, 3189240584765, 7587287095869, 16383380118077

Offset: 0

N. J. A. Sloane, Nov 13 2004, based on correspondence from Artur Jasinski, Mar 25 2003

			1/e = 0.367879441171442321595523770161460867445811131031767834507... = 0.010111100010110101011000110110001011001110111100110111110001101010111010110111 in binary.

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Cf. A068985, A099969, A099971, A099972, A099973, A099974.

d = 100; l = First[RealDigits[N[1/E, d], 2]]; Do[m = Take[l, n]; k = Length[m]; If[m[[k]] == 1, Print[FromDigits[Reverse[m], 2]]], {n, 1, d}] (* Ryan Propper, Aug 18 2005 *)
Module[{nn=50,e},e=RealDigits[1/E,2, 50][[1]];Table[If[e[[n]]== 0, Nothing,FromDigits[ Reverse[Take[e,n]],2]],{n,nn}]] (* Harvey P. Dale, Sep 17 2020 *)

a(n) = A099969(n)/2. - Michel Marcus, Nov 03 2013

More terms from Ryan Propper, Aug 18 2005

Definition amended by Harvey P. Dale, Sep 17 2020

A099971 Write (sqrt(5)-1)/2 as a binary fraction; read this from left to right and whenever a 1 appears, note the integer formed by reading leftwards from that 1.

Original entry on oeis.org

1, 9, 25, 57, 121, 1145, 3193, 11385, 27769, 60537, 191609, 453753, 978041, 2026617, 10415225, 27192441, 94301305, 228519033, 496954489, 2644438137, 11234372729, 28414241913, 62773980281, 131493457017, 268932410489, 543810317433

Offset: 0

N. J. A. Sloane, Nov 13 2004, based on correspondence from Artur Jasinski, Mar 25 2003

			(sqrt(5)-1)/2 = 0.618033988749894848204586834365638117720309179805762862135... = 0.100111100011011101111001101110010111111101001010011111000001010111110011... in binary.

Cf. A094214, A099969, A099970, A099972, A099973, A099974.

d = 100; l = First[RealDigits[N[(Sqrt[5]-1)/2, d], 2]]; Do[m = Take[l, n]; k = Length[m]; If[m[[k]] == 1, Print[FromDigits[Reverse[m], 2]]], {n, 1, d}] (* Ryan Propper, Aug 18 2005 *)

More terms from Ryan Propper, Aug 18 2005

A099972 Write 1/sqrt(2) as a binary fraction; read this from left to right and whenever a 1 appears, note the integer formed by reading leftwards from that 1.

Original entry on oeis.org

1, 5, 13, 45, 173, 8365, 73901, 204973, 467117, 991405, 5185709, 13574317, 80683181, 214900909, 1288642733, 3436126381, 7731093677, 16321028269, 33500897453, 67860635821, 136580112557, 686335926445, 1785847554221

Offset: 0

N. J. A. Sloane, Nov 13 2004, based on correspondence from Artur Jasinski, Mar 25 2003

			1/sqrt(2) = 0.7071067811865475244008443621048490392848359376885... = 0.10110101000001001111001100110011111110011101111001100100100001000101100101111101100010011011 in binary.

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Cf. A010503, A099969, A099970, A099971, A099973, A099974.

d = 100; l = First[RealDigits[N[1/Sqrt[2], d], 2]]; Do[m = Take[l, n]; k = Length[m]; If[m[[k]] == 1, Print[FromDigits[Reverse[m], 2]]], {n, 1, d}] (* Ryan Propper, Aug 18 2005 *)
Module[{rd=RealDigits[1/Sqrt[2],2,50][[1]],pos},pos=Flatten[Position[rd,1]];Table[ FromDigits[ Reverse[Take[rd,n]],2],{n,pos}]] (* Harvey P. Dale, Jul 29 2013 *)

More terms from Ryan Propper, Aug 18 2005

A099974 Write log(2) as a binary fraction; read this from left to right and whenever a 1 appears, note the integer formed by reading leftwards from that 1.

Original entry on oeis.org

1, 5, 13, 141, 653, 1677, 3725, 20109, 544397, 2641549, 6835853, 15224461, 32001677, 65556109, 132664973, 266882701, 803753613, 1877495437, 4024979085, 8319946381, 16909880973, 51269619341, 601025433229, 1700537061005

Offset: 0

N. J. A. Sloane, Nov 13 2004, based on correspondence from Artur Jasinski, Mar 25 2003

			log(2) = 0.69314718055994530941723212145817656807550013436025525412... = 0.1011000101110010000101111111011111010001110011110111100110101011110010011110... in binary.

Cf. A002162, A099969, A099970, A099971, A099972, A099973.

d = 100; l = First[RealDigits[N[Log[2], d], 2]]; Do[m = Take[l, n]; k = Length[m]; If[m[[k]] == 1, Print[FromDigits[Reverse[m], 2]]], {n, 1, d}] (* Ryan Propper, Aug 17 2005 *)
Module[{nn=50,l2},l2=RealDigits[Log[2],2,nn][[1]];Table[FromDigits[ Reverse[ Take[ l2,n]],2],{n,nn}]]//Union (* Harvey P. Dale, Mar 29 2016 *)

More terms from Ryan Propper, Aug 17 2005

A099973 Write Euler's constant gamma as a binary fraction; read this from left to right and whenever a 1 appears, note the integer formed by reading leftwards from that 1.

Original entry on oeis.org

1, 9, 73, 201, 457, 969, 9161, 140233, 402377, 2499529, 6693833, 15082441, 31859657, 65414089, 132522953, 1206264777, 3353748425, 11943683017, 29123552201, 63483290569, 132202767305, 269641720777, 819397534665

Offset: 0

N. J. A. Sloane, Nov 13 2004, based on correspondence from Artur Jasinski, Mar 25 2003

			gamma = 0.577215664901532860606512090082402431042159335939923598805... = 0.1001001111000100011001111110001101111101101100001100011110100100110100011011... in binary.

Cf. A001620, A099969, A099970, A099971, A099972, A099974.

d = 100; l = First[RealDigits[N[EulerGamma, d], 2]]; Do[m = Take[l, n]; k = Length[m]; If[m[[k]] == 1, Print[FromDigits[Reverse[m], 2]]], {n, 1, d}] (* Ryan Propper, Aug 18 2005 *)

More terms from Ryan Propper, Aug 18 2005

A113824 a(1)=1; a(n+1) = the least prime greater than 2*a(n) which is a(n) plus a power of two.

Original entry on oeis.org

1, 3, 7, 23, 151, 65687, 9007199254806679, 73795983494093013143, 205688069665150755269371147819668813122842057000180977011589271

Offset: 1

Artur Jasinski, Jan 23 2006

nonn

Next term 205688069665150755269371147819668813122842057000180977011589271 + 2^1752 is too large to include here.

Those powers of two are A073924.

			151 is there because 23 + 2^7 = 151 is prime.

Cf. A073924, A080355, A080567, A099969, A099970, A099971, A099972.

Edited by Don Reble, Jan 25 2006

A113835 a(n) = a(n-1) + 2^(A007494(n-1)).

Original entry on oeis.org

1, 5, 13, 45, 109, 365, 877, 2925, 7021, 23405, 56173, 187245, 449389, 1497965, 3595117, 11983725, 28760941, 95869805, 230087533, 766958445, 1840700269, 6135667565, 14725602157, 49085340525, 117804817261, 392682724205

Offset: 1

Artur Jasinski, Jan 27 2006

nonn

Cf. A099974, A112627, A080355, A080567, A099969, A099970, A099971.

Partial sums of A094015.

Empirical g.f.: x*(4*x+1) / ((x-1)*(8*x^2-1)). - Colin Barker, Sep 01 2013

Edited with better definition and offset corrected by Omar E. Pol, Jan 08 2009

A113829 a(n) = a(n-1) + 2^(k(n)), where k(n) is the n-th term of the sequence of numbers that are congruent to {0,3,4,5,7,8} mod 12.

Original entry on oeis.org

1, 9, 25, 57, 185, 441, 4537, 37305, 102841, 233913, 758201, 1806777, 18583993, 152801721, 421237177, 958108089, 3105591737, 7400559033, 76120035769, 625875849657, 1725387477433, 3924410732985, 12720503755193, 30312689799609

Offset: 1

Artur Jasinski, Jan 27 2006

Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,4096,-4096).

Cf. A099974, A112627, A080355, A080567, A099969, A099970, A099971, A154571.

LinearRecurrence[{1,0,0,0,0,4096,-4096},{1,9,25,57,185,441,4537},30] (* Harvey P. Dale, Aug 04 2018 *)

Vec((-4096*x^6+4096*x^5+256*x^4+128*x^3+32*x^2+16*x+9)/(4096*x^7 - 4096*x^6-x+1)+O(x^99)) \\ Charles R Greathouse IV, Apr 05 2012

G.f.: (9+16*x+32*x^2+128*x^3+256*x^4+4096*x^5-4096*x^6)/(1-x-4096*x^6+4096*x^7). - Charles R Greathouse IV, Apr 05 2012

Better definition, corrected offset and edited by Omar E. Pol, Jan 08 2009

A113841 a(n) = a(n-1) + 2^A047240(n) for n>1, a(1)=1.

Original entry on oeis.org

1, 3, 7, 71, 199, 455, 4551, 12743, 29127, 291271, 815559, 1864135, 18641351, 52195783, 119304647, 1193046471, 3340530119, 7635497415, 76354974151, 213793927623, 488671834567, 4886718345671, 13682811367879, 31274997412295

Offset: 1

Artur Jasinski, Jan 27 2006

Vincenzo Librandi, Table of n, a(n) for n = 1..300
Vladimir Pletser, Congruence conditions on the number of terms in sums of consecutive squared integers equal to squared integers, arXiv:1409.7969 [math.NT], 2014.
Index entries for linear recurrences with constant coefficients, signature (1, 0, 64, -64).

Cf. A099974, A112627, A080355, A080567, A099969, A099970, A099971.

CoefficientList[Series[(1 + 2 x + 4 x^2) / ((-1 + x) (-1 + 4 x) (1 + 4 x + 16 x^2)), {x, 0, 30}], x] (* Vincenzo Librandi, May 19 2013 *)
LinearRecurrence[{1,0,64,-64},{1,3,7,71},30] (* Harvey P. Dale, Nov 18 2013 *)

G.f.: x*(1+2*x+4*x^2)/((-1+x)*(-1+4*x)*(1+4*x+16*x^2)). - Vaclav Kotesovec, Nov 28 2012

a(1)=1, a(2)=3, a(3)=7, a(4)=71, a(n)=a(n-1)+64*a(n-3)-64*a(n-4). - Harvey P. Dale, Nov 18 2013

Edited with better definition and offset corrected by Omar E. Pol, Jan 08 2009

A113860 Start with the binary representation of the Catalan constant (A104338, A006752) = 0.91596559... = sum_{i=1..infinity} b(i)/2^i, where b(i)=1,1,1,0,1,0,1,0,0,1,1,1,1.... Then a(n-1)=sum_{i=1..k: sum_{ j = 1..k} b(j)=n} b(i) * 2^(i-1). In words: scan the binary digits of the number, halt at each nonzero binary digit, add a power of 2 corresponding to the place of this digit after the comma, assign current partial sum to a(n), increment n.

Original entry on oeis.org

1, 3, 7, 23, 87, 599, 1623, 3671, 7767, 15959, 81495, 343639, 867927, 1916503, 18693719, 152911447, 421346903, 958217815, 2031959639, 4179443287, 12769377879, 1112281005655, 9908374027863, 27500560072279, 97869304249943

Offset: 0

Artur Jasinski, Jan 25 2006

An instance of a Jasinski Integer Sequence using the convention JIS[number,counting system] as defined for example in A080355. This is JIS [Catalan constant,binary]=JIS[0.9159655941772190150546..,2].

Cf. A080355, A080567, A099969, A099970, A099971, A099972.

Naming a sequence after oneself is deprecated. - N. J. A. Sloane.

Corrected and extended by R. J. Mathar, Aug 31 2007

cp's OEIS Frontend

A099970 Write 1/e as a binary fraction; read this from left to right and whenever a 1 appears, note the integer formed by reading leftwards from that 1. Then convert those integers from binary into decimal numbers.

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A099971 Write (sqrt(5)-1)/2 as a binary fraction; read this from left to right and whenever a 1 appears, note the integer formed by reading leftwards from that 1.

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A099972 Write 1/sqrt(2) as a binary fraction; read this from left to right and whenever a 1 appears, note the integer formed by reading leftwards from that 1.

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A099974 Write log(2) as a binary fraction; read this from left to right and whenever a 1 appears, note the integer formed by reading leftwards from that 1.

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A099973 Write Euler's constant gamma as a binary fraction; read this from left to right and whenever a 1 appears, note the integer formed by reading leftwards from that 1.

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A113824 a(1)=1; a(n+1) = the least prime greater than 2*a(n) which is a(n) plus a power of two.

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A113835 a(n) = a(n-1) + 2^(A007494(n-1)).

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A113829 a(n) = a(n-1) + 2^(k(n)), where k(n) is the n-th term of the sequence of numbers that are congruent to {0,3,4,5,7,8} mod 12.

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A113841 a(n) = a(n-1) + 2^A047240(n) for n>1, a(1)=1.

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