cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 10 results.

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.

Original entry on oeis.org

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

Views

Author

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

Keywords

Examples

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

Links

Crossrefs

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

Programs

  • Mathematica
    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 *)

Formula

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

Extensions

More terms from Ryan Propper, Aug 18 2005
Definition amended by Harvey P. Dale, Sep 17 2020

A099969 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.

Original entry on oeis.org

2, 10, 26, 58, 122, 1146, 5242, 13434, 46202, 177274, 701562, 1750138, 18527354, 52081786, 186299514, 454734970, 4749702266, 21929571450, 56289309818, 331167216762, 880923030650, 1980434658426, 6378481169530, 15174574191738
Offset: 0

Views

Author

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

Keywords

Examples

			1/e = 0.367879441171442321595523770161460867445811131031767834507... = 0.010111100010110101011000110110001011001110111100110111110001101010111010110111 in binary.
From the binary expansion we get 10 = 2, 1010 = 10, 11010 = 26, 111010 = 58, 1111010 = 122, etc.

Crossrefs

Cf. A068985, A099970, A099971, A099972, A099973, A099974.

Programs

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

Extensions

More terms from Ryan Propper, Aug 18 2005

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

Views

Author

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

Keywords

Examples

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

Crossrefs

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

Programs

  • Mathematica
    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 *)

Extensions

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

Views

Author

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

Keywords

Examples

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

Crossrefs

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

Programs

  • Mathematica
    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 *)

Extensions

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

Views

Author

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

Keywords

Examples

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

Crossrefs

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

Programs

  • Mathematica
    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 *)

Extensions

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

Views

Author

Artur Jasinski, Jan 23 2006

Keywords

Comments

Next term 205688069665150755269371147819668813122842057000180977011589271 + 2^1752 is too large to include here.
Those powers of two are A073924.

Examples

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

Crossrefs

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

Extensions

Edited by Don Reble, Jan 25 2006

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

Views

Author

Artur Jasinski, Jan 25 2006

Keywords

Comments

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].

Crossrefs

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

Extensions

Naming a sequence after oneself is deprecated. - N. J. A. Sloane.
Corrected and extended by R. J. Mathar, Aug 31 2007

A113914 (1,2,3) Jasinski-like positive power sequence.

Original entry on oeis.org

1, 5, 13, 29, 61, 131, 271, 569, 1381, 2789, 5581, 11171, 22369, 44741, 89491, 185543, 373273, 766229, 1532701, 3065411, 6130849, 12261701, 24700549, 49401101, 98802211, 202387391, 409557751, 819116231, 1638232471, 3276464969
Offset: 1

Views

Author

Jonathan Vos Post, Jan 29 2006

Keywords

Comments

In general, the (b,c,d) Jasinski-like positive power sequence is defined as follows: a(1) = b, a(n+1) = the least prime p such that p = c*a(n) + d^k for positive integer k. The (b,c,d) Jasinski-like nonnegative power sequence is defined: a(1) = b, a(n+1) = the least prime p such that p = c*a(n) + d^k for integer k. In this notation, A113824 is the (1,2,2) Jasinski-like nonnegative power sequence. The first differences of such sequences are powers of d, with no closed-form known upper bound.

Examples

			a(1) = 1 by definition.
a(2) = 2*1 + 3^1 = 5.
a(3) = 2*5 + 3^1 = 13.
a(4) = 2*13 + 3^1 = 29.
a(5) = 2*29 + 3^1 = 61.
a(6) = 2*61 + 3^2 = 271.
a(7) = 2*271 + 3^2 = 569.
a(32) = 2*6553461379 + 3^49 = 239299329230630636512841. Here 49 is a record value for the exponent.

Crossrefs

Cf. A073924, A080355, A080567, A099969, A099970, A099971, A099972, A113824.

Formula

a(1) = 1, a(n+1) = the least prime p such that p = 2*a(n) + 3^k for integer k>0.

A113927 a(1)=1, and recursively a(n+1) is the smallest prime p of the form p = 2*a(n) + 5^k for some k>0.

Original entry on oeis.org

1, 7, 19, 43, 211, 547, 4219, 8443, 17011, 34147, 71419, 142963, 1220989051, 3662681227, 19080811690579, 38161623381163, 76324467465451, 152648936884027, 305299094471179, 4656613483675581520483
Offset: 1

Views

Author

Jonathan Vos Post, Jan 30 2006

Keywords

Comments

Note that last digits cycle 7, 9, 3, 1; 7, 9, 3, 1. Note that the exponent k of 5^k is always odd. This follows from taking this sequence mod 6.
Since the first prime value a(2) = 7 == 1 mod 6, all values a(n) thereafter are primes of the form 6*d+1. Hence a(n+1) = [2*(6*d+1) + 5^2] mod 6 == 12*d + 2 + 1 == 3 mod 6 and would be divisible by 3; a(n+1) = [2*(6*d+1) + 5^4] mod 6 == 12*d + 2 + 1 == 3 mod 6 and would be divisible by 3; and so for all even exponents.
In general, the (b,c,d) Jasinski-like positive power sequence is defined as follows: a(1) = b, a(n+1) = the least prime p such that p = c*a(n) + d^k for positive integer k. The (b,c,d) Jasinski-like nonnegative power sequence is defined: a(1) = b, a(n+1) = the least prime p such that p = c*a(n) + d^k for integer k. In this notation, A113824 is the (1,2,2) Jasinski-like nonnegative power sequence. A113914 is the (1,2,3) Jasinski-like positive power sequence, and this here the (1,2,5) Jasinski-like power sequence.

Examples

			a(1) = 1 by definition.
a(2) = 2*1 + 5^1 = 7.
a(3) = 2*7 + 5^1 = 19.
a(4) = 2*19 + 5^1 = 43.
a(5) = 2*43 + 5^3 = 211.
a(6) = 2*211 + 5^3 = 547.
a(7) = 2*547 + 5^5 = 4219.
a(13) = 2*142963 + 5^13 = 1220989051.
a(20) = 2*305299094471179 + 5^31 = 4656613483675581520483, where 31 is a record exponent.
a(22) = 2*9313226967351163119091 + 5^45 = 28421709449030461369547296941307 and 45 is the new record exponent.

Crossrefs

Cf. A073924, A080355, A080567, A099969, A099970, A099971, A099972, A113824, A113914.

A151891 a(n) = Pi(A113824(n+1)).

Original entry on oeis.org

2, 4, 9, 36, 6561, 252252704150178, 1650016588712720468
Offset: 1

Views

Author

Artur Jasinski, Apr 04 2008

Keywords

Links

Crossrefs

Cf. A073924, A080355, A080567, A099969, A099970, A099971, A099972, A113824.

Extensions

a(6)-a(7) using Kim Walisch's primecount, from Amiram Eldar, Mar 13 2020
Showing 1-10 of 10 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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