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Index to fractions in OEIS

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Index to Sequences of Fractions (or Rational Numbers) in the OEIS

Introduction

  • Sequences of fractions (or rational numbers) appear in the OEIS® as a linked pair of integer sequences, giving the numerators and denominators separately.
  • This page contains a bare-bones list that gives the A-numbers of the numerator and denominator sequences for the most important sequences of rational numbers.
  • The entries are arranged in lexicographic order, according to the following rules (which are essentially the same as those used when sorting the main database into lexicographic order):
    • (i) ignore signs,
    • (ii) ignore any initial terms that are 0, +1 or -1 (the leading term in what's left is the first nontrivial term),
    • (iii) sort the resulting sequences of fractions into lexicographic order.
    • The all-ones sequence (A000012) has been artificially inserted to mark the boundary between sequences whose first nontrivial term is less than 1 from those in which it is greater than 1.
  • The OEIS contains over 3400 entries with keyword "frac", meaning over 1700 sequences of fractions. Many more need to be added here. If you would like to help, please do so!
  • For more information (including how to add further entries) see the Comments at the end of this file.
  • See also the main Index to OEIS.


List of fractions

First nontrivial term <= 1/3

  • 1/192, 1/21504, 1/190080, 27/573440, 127/218880, 34603/40480, 278617/2106, 264156210586399/53329920, ... = A048615/A048616 (Hadamard mass)
  • 1, 1/48, 77/7680, 17017/1105920, 52055003/1061683200, 1509595087/5662310400, 3603403472669/1630745395200, 10151817126357907/391378894848000, ... = A226256/A226257
  • 0, 1/15, 8/105, 5/63, 8/99, 35/429, 16/195, 7/85, 80/969, 11/133, 40/483, 143/1725, 56/675, 65/783, 224/2697, 85/1023, 32/385, ... = A144454/A300295
  • 1, 1/12, 1/288, -139/51840, -571/2488320, 163879/209018880, 5246819/75246796800, -534703531/902961561600, ... = A001163/A001164 (Stirling approximation)
  • 1, -1/12, 7/240, -31/1344, 127/3840, -2555/33792, 1414477/5591040, -57337/49152, 118518239/16711680, ... = A001896/A033469 (Bernoulli(2n,1/2))
  • 0, 1/12, 8/81, 5/48, 8/75, 35/324, 16/147, 7/64, 80/729, 11/100, 40/363, 143/1296, 56/507, 65/588, ... = A144454/A147650
  • 1/10, 3/10, 567/130, 43659/170, 392931/10, ... = A002306/A047817 (Hurwitz numbers H_n))
  • 1/9, 10/81, 100/729, 1000/6561, 10000/59049, ... = A100061/A100062
  • 1/8, 1/12, 11/96, 17/72, 619/960, 709/324, ... = A226258/A226259
  • 1, 1/8, 35/384, 385/3072, 25025/98304, 1616615/2359296, 260275015/113246208, 929553625/100663296, 835668708875/19327352832, ... = A225697/A225698
  • 1/8, 16/45, 25/144, 34/105, 2989/17280, 5248/14175, 1209/5600, 5675/12474, 560593/1935360, 893128/1576575, 11148172711/28740096000, 109420087/156370500, ... = A100647/A100648 (Cotesian C(n,3))
  • 1, -1/7, 1/49, -1/343, 1/2401, -1/16807, 1/117649, -1/823543, 1/5764801, -1/40353607, 1/282475249, -1/1977326743, ... = A033999/A000420 ( terms of the series (-1)^n/7^n, where n=0..inf, which converges to 7/8; actually you could expand this with a denominator sequence related to powers of x (A001018, A001019, A011557, ...) that when applied to the infinite series (-1)^n/x^n it will converge to x/(x+1) )
  • 0, 1, 0, -1/6, 0, 1/120, 0, -1/5040, 0, 1/362880, 0, -1/39916800, 0, 1/6227020800, 0, -1/1307674368000, ... = A033999/A009445 (sin(x))
  • 1, 0, 1/6, 0, 7/360, 0, 31/15120, 0, 127/604800, 0, 73/3421440, 0, 1414477/653837184000, 0, 8191/37362124800, 0, 16931177/762187345920000, ... = A036280/A036281 (x/sin x)
  • 0, 1, 0, 1/6, 0, 3/40, 0, 5/112, 0, 35/1152, 0, 63/2816, 0, 231/13312, 0, 143/10240, 0, 6435/557056, 0, 12155/1245184, 0, 46189/5505024, 0, ... = A055786/A002595
  • 1, -1/6, 1/120, -1/5040, 1/362880, -1/39916800, 1/6227020800, -1/1307674368000, ... = A033999/A009445 (sin(x))
  • 1/6, 1/90, 1/945, 1/9450, 1/93555, 691/638512875, 2/18243225, 3617/325641566250, 43867/38979295480125, ... = A046988/A002432
  • 1, 1/6, 7/360, 31/15120, 127/604800, 73/3421440, 1414477/653837184000, 8191/37362124800, ... = A036280/A036281 (x/sin x)
  • 1, -1/6, 1/36, -1/216, 1/1296, -1/7776, 1/46656, -1/279936, 1/1679616, -1/10077696, 1/60466176, -1/362797056, ... = A033999/A000400 (terms of the series (-1)^n/6^n, where n=0..inf, which converges to 6/7)
  • 1, 1/6, -1/30, 1/42, -1/30, 5/66, -691/2730, 7/6, -3617/510, 43867/798, -174611/330, ... = A000367/A002445 (Bernoulli numbers B_{2n})
  • 1, 1/6, 3/40, 5/112, 35/1152, 63/2816, 231/13312, 143/10240, 6435/557056, 12155/1245184, 46189/5505024, ... = A055786/A002595
  • 1/6, 3/8, 2/15, 25/144, 9/280, 49/640, -464/14175, 27/2240, -16175/199584, -3237113/87091200, -105387/875875, -1737125143/22353408000, -770720657/5003856000, -25881785/229605376, ... = A100645/A100646 = A002179/A002176 (the latter not being in lowest terms) (Cotesian numbers C(n,2))
  • 1, 1, 1/5, 1/61, 1/1385, 1/50521, 1/2702765, 1/199360981, 1/19391512145, 1/2404879675441, ... = 1/A000364
  • 1, -1/5, 1/25, -1/125, 1/625, -1/3125, 1/15625, -1/78125, 1/390625, -1/1953125, 1/9765625, -1/48828125, ... = A033999/A000351 (terms of the series (-1)^n/5^n, where n=0..inf, which converges to 5/6)
  • 5/24, 5/16, 1105/1152, 565/128, 82825/3072, 19675/96, 1282031525/688128, 80727925/4096, ... = A226260/A226261
  • 1/4, 1/32, 5/1536, 61/184320, 277/8257536, 50521/14863564800, 540553/1569592442880, 199360981/5713316492083200, ... = A046976/A053005
  • 1, 1/4, 1/16, 1/64, 1/256, 1/1024, 1/4096, 1/16384, 1/65536, 1/262144, 1/1048576, 1/4194304, ... = 1/A000302 (1/4^n)
  • 1, -1/4, 1/16, -1/64, 1/256, -1/1024, 1/4096, -1/16384, 1/65536, -1/262144, 1/1048576, -1/4194304, ... = A033999/A000302 (terms of the series (-1)^n/4^n, where n=0..inf, which converges to 4/5)
  • 1, 1/4, 1/9, 1/16, 1/25, 1/36, 1/49, 1/64, 1/81, 1/100, 1/121, 1/144, 1/169, 1/196, 1/225, 1/256, ... = 1/A000290 (1/n^2)
  • 1/4, 5/16, 3/16, 21/64, 13/64, 19/64, 11/64, 7/64, ... = A072800/A??????
  • 1/4, 2/5, 1/5, 1/2, 1/10, 1/10, 2/11, 1/10, 1/4, 1/4, 1/10, 4/13, 1/10, ... = A291093/ A291094
  • 1/4, 26/65, 19/95, 49/98, 11/110, 12/120, 22/121, 13/130, 33/132, 34/136, ... = A291093/A291094
  • 1/4, 5/12, 13/24, 47/72, 271/360, 301/360, 2287/2520, 491/504, 1045/1008, 367/336, 1919/1680, 1999/1680, 22829/18480, ... = A250133/A296358
  • 1, 1/4, 3/2, 3/16, 15/8, 5/32, 35/16, 35/256, 315/128, 63/512, 693/256, 231/2048, ... = A163590/A240988 ((((n-1)!!)/(n!! * 2^((1 + (-1)^n)/2)))^((-1)^n), starting at n = 1)
  • -691/2730, 691/2730, 7/6, -7/6, -3617/510, 3617/510, ... = A051716/A051717 (Bernoulli twin numbers)
  • 1, 0, -1/3, 0, -1/45, 0, -2/945, 0, -1/4725, 0, -2/93555, 0, -1382/638512875, 0, -4/18243225, 0, -3617/162820783125, 0, ... = A002431/A036278 (cot(x))
  • 0, 1, 0, -1/3, 0, 2/15, 0, -17/315, 0, 62/2835, 0, -1382/155925, 0, 21844/6081075, 0, -929569/638512875, 0, 6404582/10854718875, 0, -443861162/1856156927625, ... = A002430/A036279 (tan(x))
  • 1, -1/3, -1/45, -2/945, -1/4725, -2/93555, -1382/638512875, -4/18243225, -3617/162820783125, -87734/38979295480125, -349222/1531329465290625, ... = A002431/A036278 (cot(x))
  • 1, 1/3, 1/9, 1/27, 1/81, 1/243, 1/729, 1/2187, 1/6561, 1/19683, 1/59049, 1/177147, 1/531441, ... = 1/A000244 (1/3^n)
  • 1, -1/3, 1/9, -1/27, 1/81, -1/243, 1/729, -1/2187, 1/6561, -1/19683, 1/59049, -1/177147, 1/531441, ... = A033999/A000244 (terms of the series (-1)^n/3^n, where n=0..inf, which converges to 3/4)
  • 1, 1/3, 2/15, 17/315, 62/2835, 1382/155925, 21844/6081075, 929569/638512875, 6404582/10854718875, ... = A002430/A036279 (tan(x))
  • 1, 1/3, 1/6, 1/10, 1/15, 1/21, 1/28, 1/36, 1/45, 1/55, 1/66, 1/78, ... = 1/A000217 (1/triangular numbers)
  • 1, -1/3, 1/5, -1/7, 1/9, -1/11, 1/13, -1/15, 1/17, -1/19, 1/21, -1/23, 1/25, -1/27, 1/29, -1/31, 1/33, -1/35, ... = A033999/A005408 ((+-1)/(2n+1))
  • 1, 1/3, 1/5, 1/7, 1/9, 1/11, 1/13, 1/15, 1/17, 1/19, 1/21, 1/23, 1/25, 1/27, 1/29, 1/31, 1/33, ... = 1/A005408 (1/(2n+1))
  • 1, -1/3, 7/15, -31/21, 127/15, -2555/33, 1414477/1365, -57337/3, 118518239/255, -5749691557/399, 91546277357/165, -1792042792463/69, 1982765468311237/1365, -286994504449393/3, 3187598676787461083/435, ... = A001896/A001897 (cosecant numbers)

First nontrivial term in range (1/3, 1/2]

  • 1, 0, -1/2, 0, 1/24, 0, -1/720, 0, 1/40320, 0, -1/3628800, 0, 1/479001600, 0, -1/87178291200, ... = A033999/A010050 (cos(x))
  • 1, 0, 1/2, 0, 5/24, 0, 61/720, 0, 277/8064, 0, 50521/3628800, 0, 540553/95800320, 0, 199360981/87178291200, ... = A046976/A046977 (sec(x))
  • 1, -1/2, 1/24, -1/720, 1/40320, -1/3628800, 1/479001600, -1/87178291200, ... = A033999/A010050 (cos(x))
  • 1, -1/2, 1/12, 0, -1/720, 0, 1/30240, 0, -1/1209600, 0, 1/47900160, 0, -691/1307674368000, 0, 1/74724249600, 0, -3617/10670622842880000, 0, ... = A120082/A227829
  • 0, 1/2, -1/12, 1/24, -19/720, 3/160, -863/60480, 275/24192, -33953/3628800, 8183/1036800, -3250433/479001600, 4671/788480, -13695779093/2615348736000, 2224234463/475517952000, ... = A002206/A002207 (logarithmic numbers)
  • 1, 1/2, -1/8, 1/16, -5/128, 7/256, -21/1024, 33/2048, -429/32768, 715/65536, ... = A002596/A046161 (sqrt(1+x))
  • 1, -1/2, 1/6, 0, -1/30, 0, 1/42, 0, -1/30, 0, 5/66, 0, -691/2730, 0, 7/6, 0, -3617/510, 0, 43867/798, 0, -174611/330, ... = A027641/A027642 (Bernoulli numbers B_n)
  • 1, 1, 1/2, 1/6, 1/24, 1/120, 1/720, 1/5040, 1/40320, 1/362880, 1/3628800, 1/39916800, ... = 1/A000142 (1/n!)
  • 0/1, 1/1, 1/2, 1/6, 1/12, 7/60, 1/20, 11/420, 13/840, 11/2520, 11/2520, 23/27720, 23/27720, 607/360360, ... = A232111/A232112
  • 0, 1/2, 1/6, 1/8, 7/90, 19/288, 41/840, 751/17280, 989/28350, 2857/89600, 16067/598752, 434293/17418240, 1364651/63063000, 8181904909/402361344000, ... = A100640/A100641 = A002177/A002176 (the latter is not in lowest terms)
  • 1, 1/2, -1/6, 1/4, -19/30, 9/4, -863/84, 1375/24, -33953/90,... = A006232/A006233 (Cauchy numbers)
  • 0, 1, 1/2, 1/6, 5/12, 13/60, 1/20, 27/140, 19/280, 451/2520, 199/2520, 4709/27720. ... = A231692/A231693 (fractional Recaman)
  • 1, -1/2, 1/5, -1/20, -1/350, 1/140, 1/1050, -1/300, -37/57750, 111/38500, 177/250250, -177/45500, -2753/2388750, 2753/367500, ... = A003163/A003164 (Van der Pol numbers)
  • 1, 1, 1/2, 1/5, 1/14, 1/42, 1/132, 1/429, 1/1430, 1/4862, 1/16796, 1/58786, 1/208012, 1/742900, 1/2674440, ... = 1/A000108 (1/Catalan)
  • 1, 1, 1, 1/2, 1/5, 1/16, 1/61, 1/272, 1/1385, 1/7936, 1/50521, 1/353792, 1/2702765, 1/22368256, ... = 1/A000111 (1/Euler)
  • 1, 1/2, 5/24, 61/720, 277/8064, 50521/3628800, 540553/95800320, 199360981/87178291200, ... = A046976/A046977 (sec(x))
  • 1, 1, 1/2, 1/4, 1/8, 0, 1/16, 3/64, -29/128, 25/128, 263/256, -1481/512, -5493/1024, 80505/2048, ... = A030274/A030275
  • 1, 1/2, 1/4, 1/8, 1/16, 1/32, 1/64, 1/128, 1/256, 1/512, 1/1024, 1/2048, 1/4096, 1/8192, 1/16384, ... = 1/A000079 (1/2^n)
  • 1, -1/2, 1/4, -1/8, 1/16, -1/32, 1/64, -1/128, 1/256, -1/512, 1/1024, -1/2048, 1/4096, -1/8192, 1/16384, ... = A033999/A000079 (terms of the series (-1)^n/2^n, where n=0..inf, which converges to 2/3)
  • 1/2, 1/4, 1/6, 1/8, 1/10, 1/12, 1/14, 1/16, 1/18, 1/20, 1/22, 1/24, 1/26, 1/28, 1/30, 1/32, 1/34, 1/36, 1/38, ... = 1/A005843 (1/(2n))
  • 0, 1/2, 1/4, 3/4, 1/8, 5/8, 3/8, 7/8, 1/16, 9/16, 5/16, 13/16, 3/16, 11/16, ... = A030101/A062383 (van der Corput)
  • 1, -1/2, -1/3, -1/6, -1/30, 1/30, 1/42, -1/42, -1/30, 1/30, 5/66, -5/66, -691/2730, 691/2730, 7/6, -7/6, ... = A051716/A051717
  • 1, 1, 1/2, 1/3, 1/6, 7/60, 19/360, 3/70, 5/336, 13/756, 199/75600, 1663/207900, -10819/9979200, 117119/25945920, ... = A226932/A226933
  • 1, 1, 1/2, 1/3, 5/24, 2/15, 61/720, 17/315, 277/8064, 62/2835, 50521/3628800, 1382/155925, 540553/95800320, ... = A099612/A099617
  • 1/2, 1/3, 1/5, 1/7, 1/11, 1/13, 1/17, 1/19, 1/23, 1/29, 1/31, 1/37, 1/41, 1/43, 1/47, 1/53, 1/59, 1/61, ... = 1/A000040
  • 1, 1/2, 1/3, 1/4, 1/5, 1/6, 1/7, 1/8, 1/9, 1/10, 1/11, 1/12, 1/13, 1/14, 1/15, 1/16 ... = 1/A000027
  • 1, -1/2, 1/3, -1/4, 1/5, -1/6, 1/7, -1/8, 1/9, -1/10, 1/11, -1/12, 1/13, -1/14, 1/15, -1/16 ... = A033999/A000027 (terms of the Maclaurin series for ln(1+x), where x = 1; so, the series (-1)^n/(n+1), where n=0..inf, converges to ln(2))
  • 1, 1/2, 1/3, 1/3, 3/10, 3/10, 29/105, 29/105, 17/70, 17/70, 193/1155, 193/1155, -123/1430, -123/1430, -2687/2145, -2687/2145, -202863/24310, -202863/24310, -307072861/4849845, ... = A100651/A100652
  • 1, 0, 1/2, 1/3, 3/8, 11/30, 53/144, 103/280, 2119/5760, 16687/45360, 16481/44800, 1468457/3991680, 16019531/43545600, 63633137/172972800, 2467007773/6706022400, ... = A053557/A053556
  • 1, 1/2, 1/3, 2/3, 1/5, 2/5, 3/5, 1/7, 2/7, 3/7, 5/7, 1/8, 1/4, 3/8, 5/8, 7/8, 1/11, 2/11, ... = A066657/A066658
  • 0, 1, 1/2, 1/3, 2/3, 1/4, 2/5, 3/5, 3/4, 1/5, 2/7, 3/8, 3/7, 4/7, 5/8, 5/7, 4/5, ... = A007305/A007306
  • 1/2, 1/3, 2/3, 1/4, 3/4, 2/5, 3/5, 1/5, 4/5, 3/7, 4/7, 2/7, 5/7, 3/8, 5/8, 1/6, 5/6, 4/9, 5/9, 3/10, 7/10, ... = A020651/A086592
  • 1, 1/2, 3/8, 5/16, 35/128, 63/256, 231/1024, 429/2048, 6435/32768, 12155/65536, 46189/262144, ... = A001790/A046161
  • 1/2, 2/5, 5/12, 7/17, 33/80, 106/257, 563/1365, 1232/2987, 1795/4352, 8412/20395, ... = A085394/A085395
  • 1/2, 5/12, 3/8, 251/720, 233/720, ... = erroneous version of A002208/A002209
  • 1, 1/2, 5/12, 3/8, 251/720, 95/288, 19087/60480, 5257/17280, 1070017/3628800, 25713/89600, 26842253/95800320, 4777223/17418240, 703604254357/2615348736000, 106364763817/402361344000, ... = A002208/A002209
  • 1, -1/2, 11/24, -7/16, 2447/5760, -959/2304, 238043/580608, -67223/165888, 559440199/1393459200, -123377159/309657600, ... = A055505/A055535
  • 1, 1/2, 1/2, 1/6, 1/3, 1/2, 1/6, 1/4, 1/4, 1/3, 2/15, 1/6, 1/5, 1/4, 1/4, ... = A344008/A344007
  • 0, 1/2, 1/2, 1/6, 2/3, 1/6, 1/8, 3/8, 3/8, 1/8, 7/90, 16/45, 2/15, 16/45, 7/90, 19/288, 25/96, 25/144, 25/144, 25/96, 19/288, 41/840, 9/35, 9/280, 34/105, 9/280, 9/35, 41/840, ... = A100640/A100641 = A100642/A002176 (the latter is not in lowest terms)
  • 1; 1/2, 1/2; 1/3, 1/6, 1/3; 1/4, 1/12, 1/12, 1/4; 1/5, 1/20, 1/30, 1/20, 1/5; ... = 1/A003506 (Leibnitz's Harmonic Numbers)
  • 1, 1/2, 1/2, 1/3, 2/3, 1/3, 2/3, 1/4, 3/4, 2/5, 3/5, 1/4, 3/4, 2/5, 3/5, 1/5, 4/5, 3/7, 4/7, 2/7, 5/7, 3/8, 5/8, ... = A093873/A093875
  • 1, 1/2, 1/2, 11/24, 13/30, 77/180, 29/70, 459/1120, ... = A226242/A226243 (Sultan's dowry)
  • 1, 1/2, 1/2, 2/3, 25/24, 9/5, 2401/720, 2048/315, 59049/4480, 15625/567, 214358881/3628800, ... = A036502/A036503 (n^(n-2)/n!)
  • -1/2, -1/2, -19/8, -593/32, -23877/128, -4496245/2048, ... = A226584/A226585 (Hubbard model)
  • 1/2, 7/12, 37/60, 533/840, 1627/2520, 18107/27720, 237371/360360, 95549/144144, 1632341/2450448, 155685007/232792560, 156188887/232792560, 3602044091/5354228880, ... = A082687/A082688
  • 1/2, 2/3, 3/8, 16/45, 25/96, 9/35, 3577/17280, 2944/14175, 15741/89600, 26575/149688, 4495513/29030400, 12504/79625, 56280729661/402361344000, 44436679/312741000, ... = A100643/A100644 = A002178/A002176 (the latter not being in lowest terms)
  • 0, 1, 1/2, 2/3, 5/8, 19/30, 91/144, 177/280, 3641/5760, 28673/45360, 28319/44800, 2523223/3991680, 27526069/43545600, 109339663/172972800, 4239014627/6706022400, ... = A103816/A053556
  • 0, 1/2, 2/3, 3/4, 4/5, 5/6, 6/7, 7/8, 8/9, 9/10, 10/11, 11/12, 12/13, 13/14, 14/15, 15/16, 16/17, 17/18, ... = A001477/A000027
  • 1/2, 2/3, 4/5, 8/9, 16/17, 32/33, 64/65, 128/129, 256/257, 512/513, 1024/1025, 2048/2049, 4096/4097, ... = A000079/A000051
  • 1, 1, -1/2, 2/3, -9/8, 32/15, -625/144, 324/35, -117649/5760, 131072/2835, -4782969/44800, ... = A264234/A264235 (exp(LambertW))
  • 0, 1/2, 3/4, 7/8, 99/112, 9307/10528, ... = A129660/A129661
  • 0, 1/2, 3/4, 7/8, 15/16, 31/32, 63/64, 127/128, 255/256, 511/512, 1023/1024, 2047/2048, 4095/4096, 8191/8192, 16383/16384, ... = A000225/A000079
  • 1/2, 5/6, 1/2, 31/30, 1/2, 41/42, 1/2, 31/30, 1/2, 61/66, 1/2, 3421/2730, 1/2, 5/6, 1/2, 557/510, ... = A027759/A027760
  • 1, 1/2, 5/6, 7/12, 47/60, 37/60, 319/420, 533/840, 1879/2520, 1627/2520, 20417/27720, 18107/27720, ... = A058313/A058312
  • 0, 1/2, 5/6, 5/6, 31/30, 31/30, 247/210, 247/210, 247/210, 247/210, 2927/2310, 2927/2310, 40361/30030, 40361/30030, 40361/30030, ... = A106830/A034386
  • 1/2, 5/6, 31/30, 247/210, 2927/2310, 40361/30030, 716167/510510, 14117683/9699690, 334406399/223092870, 9920878441/6469693230, ... = A024451/A002110
  • 1, 1/2, 5/6, 9/4, 251/30, 475/12, 19087/84, 36799/24, 1070017/90, ... = A002657/A002790 (Cauchy numbers)
  • 0, 1, 0, 1/2, 1, 0, 1/3, 1/2, 2/3, 1, 0, 1/4, 1/3, 2/5, 1/2, 3/5, 2/3, 3/4, 1, 0, 1/5, 1/4, 2/7, 1/3, 3/8, 2/5, 3/7, 1/2, ... = A049455/A049456
  • 0, 0, 1, 0, 1/2, 1, 0, 1/3, 1/2, 2/3, 1, 0, 1/4, 1/3, 1/2, 2/3, 3/4, 1, 0, 1/5, 1/4, 1/3, 2/5, 1/2, 3/5, 2/3, 3/4, 4/5, ... = A006842/A006843
  • 1, -1/2, 1, 0, -1, 1, 1/4, 0, -3/2, 1, 0, 1, 0, -2, 1, -1/2, 0, 5/2, 0, -5/2, 1, 0, -3, 0, 5, 0, -3, 1, 17/8, 0, -21/2, 0, 35/4, 0, -7/2, 1, 0, 17, 0, -28, 0, 14, 0, -4, 1, ... = A060096/A060097
  • 1, -1/2, 1, 1/6, -1, 1, 0, 1/2, -3/2, 1, -1/30, 0, 1, -2, 1, 0, -1/6, 0, 5/3, -5/2, 1, 1/42, 0, -1/2, 0, 5/2, -3, 1, ... = A053382/A053383 (refelected) (Bernoulli polynomials)
  • 0, 1/2, 1, 1/3, 2/3, 1/2, 2, 3/4, 3/5, 1/5, 1/2, 1, 1, 2, 5, 1, 4/7, 1/8, 2/7, 3/7, 3/8, 4/7, 1, ... = A007305/A047679 (another version of Stern-Brocot)
  • 0, 1/2, 1, 2/3, 3/2, 1/3, 2, 3/4, 5/3, 2/5, 5/2, 3/5, 4/3, 1/4, 3, 4/5, 7/4, 3/7, ... = A174981(n)/A002487(n+2) (L-tree)
  • 1, 1, -1/2, 1, -1, 1/6, 1, -3/2, 1/2, 0, 1, -2, 1, 0, -1/30, 1, -5/2, 5/3, 0, -1/6, 0, 1, -3, 5/2, 0, -1/2, 0, 1/42, ... = A053382/A053383 (Bernoulli polynomials)
  • 1/2, 1, 1, 1, 13/10, 1, 1, 3/2, 1, 1, 16/11, 1, 31/26, 10/7, 1, 1, 24/17, 4/3, 1, 7/5, 1, 1, 39/23, 1, 57/50, 18/13, 1, 9/7, 40/29, 1, 1, ... = A225699/A225700
  • 1, 1, 1/2, 1, -5/4, 27/4, -277/8, 895/4, -27655/16, 248185/16, -5052519/32, 28731489/16, -1444496477/64, 19885473347/64, ... = A300621/A300622
  • 1, 1/2, 11/8, 37/16, 563/128, 1695/256, 12255/1024, 36333/2048, 972867/32768, ... = A232976/A046161
  • 0, 1/2, 3/2, 9/4, 3, 7/2, 4, 14/3, 61/12, 16/3, 35/6, 19/3,... = A099319/A099320 (approx to pi(n))
  • 1/2, 2, 1/4, 4/5, 5/2, 2/5, 5/4, 4, 1/6, 2/3, 9/4, 4/13, 13/14, 14/5, 5/12, 4/3, ... = A191379(n)/A191379(n+1)
  • 1, 1/2, 2, 1/3, 2/3, 3, 1/4, 3/2, 3/4, 4, 1/5, 2/5, 3/5, 4/5, ... = A226314/A054531 (enumerating rationals)
  • 0, 1, 1/2, 2, 1/3, 3/2, 2/3, 3, 1/4, 4/3, 3/5, 5/2, 2/5, 5/3, 3/4, 4, 1/5, 5/4, ... = A002487(n)/A002487(n+1) (Stern-Brocot)
  • 1, 1/2, 2, 1/3, 3, 1/4, 2/3, 3/2, 4, 1/5, 5, 1/6, 2/5, 3/4, 4/3, 5/2, 6, ... = A038566/A020653
  • 1, 1/2, 2, 1/3, 3, 2/3, 3/2, 1/4, 4, 3/4, 4/3, ... = A038568/A038569 (rationals <-> integers)
  • 1, 1/2, 2, 2/3, 1/3, 3, 3/2, 3/5, 3/4, 1/4, 2/5, 5/2, 4, 4/3, 5/3, ... = A162909/A162910 (bird tree)
  • 1, 1/2, 2, 2/3, 3, 1/3, 3/2, 3/5, 5/2, 1/4, 4/3, 3/4, 4, 2/5, 5/3, ... = A162911/A162912 (drib tree)
  • 0, 1/2, 7/3, 59/12, 64/5, 581/30, 649/21, 2287/56, 1001/18, 2443/30, 5669/55, 17915/132, 6665/39, 36637/182, 3529/15, 22413/80, 22813/68, ... = A301275/A301276 (variance of primes)

First nontrivial term in range (1/2, 1]

  • 1, -2/3, 7/15, -12/35, 83/315, -146/693, 523/3003, -952/6435, 14051/109395, -26206/230945, 32867/323323, -62260/676039, 1423159/16900975, -2723234/35102025, 10461043/145422675, -20155888/300540195, 623034403/9917826435, ... = A034430/A001147
  • 1, 2/3, 4/7, 8/15, 16/31, 32/63, 64/127, 128/255, 256/511, 512/1023, 1024/2047, 2048/4095, 4096/8191, 8192/16383, 16384/32767, ... = A000079(n)/A000225(n-1)
  • 2/3, 3/5, 5/7, 7/11, 11/13, 13/17, 17/19, 19/23, 23/29, 29/31, 31/37, 37/41, 41/43, 43/47, 47/53, 53/59, 59/61, 61/67, 67/71, ... = A000040(n)/A000040(n+1) (prime(n)/prime(n+1))
  • 1, 3/4, 1/2, 3/8, 19/40, 53/120, 103/168, 4111/6720, 78293/120960, ... = A040173/A040174 (prob. of generating S_n)
  • 1, -1, 5/6, -1/2, 1/10, 1/6, -5/42, -1/6, 7/30, 3/10, -15/22, -5/6, 7601/2730, 691/210, -91/6, -35/2, 3617/34, 3617/30, -745739/798, -43867/42, ... = A100615/A100616.
  • 0, 1, 1, 1, 1, 24/25, 1, 50/49, 1, 1, 24/25, 120/121, 1, 170/169, 50/49, 24/25, 1, 288/289, 1, 362/361, 24/25, 50/49, 120/121, 528/529, 1, 601/625, ... = A232988/A232989
  • 0, 1, 0, 1, 7/8, 8/9, 15/17, 23/26, 38/43, 61/69, 343/388, 404/457, 747/845, 7127/8062, 29255/33093, ... = A129658/A129659
  • 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ... = A000012 = 1/A000012

First nontrivial term in range (1, 3/2]

  • 1, 33/32, 8051/7776, 257875/248832, ... = A099828/A069052
  • 1, 17/16, 82/81, 273/256, 626/625, 697/648, 2402/2401, 4369/4096, 6643/6561, 5321/5000, 14642/14641, 3731/3456, 28562/28561, 20417/19208, 51332/50625, ... = A017671/A017672
  • 1, 9/8, 28/27, 73/64, 126/125, 7/6, 344/343, 585/512, 757/729, 567/500, 1332/1331, 511/432, 2198/2197, 387/343, 392/375, 4681/4096, 4914/4913, 757/648, ... = A017669/A017670
  • 1, 5/4, 10/9, 21/16, 26/25, 25/18, 50/49, 85/64, 91/81, 13/10, 122/121, 35/24, 170/169, 125/98, 52/45, 341/256, 290/289, 455/324, 362/361, 273/200, 500/441, ... = A017667/A017668
  • 1, 5/4, 17/12, 37/24, 119/72, 631/360, 661/360, 4807/2520, 995/504, 2053/1008, 703/336, 3599/1680, ... = A282511/A282512
  • 1, 3/2, 4/3, 4/3, 41/30, 41/30, 47/35, 47/35, 289/210, 289/210, 1502/1155, 1502/1155, 15551/10010, 15551/10010, 5809/15015, 5809/15015, 3818123/510510, 3818123/510510, ... = A100649/A100650
  • 1, 1, 3/2, 4/3, 17/12, 13/10, 77/60, 128/105, 167/140, 73/63, 319/280, 1294/1155, 30697/27720, 28211/25740, 8707/8008, 9728/9009, 17587/16380, 18181/17017, ... = A100560/A100561
  • 1, 3/2, 4/3, 7/4, 6/5, 2, 8/7, 15/8, 13/9, 9/5, 12/11, 7/3, 14/13, 12/7, 8/5, 31/16, 18/17, 13/6, 20/19, 21/10, 32/21, 18/11, 24/23, 5/2, 31/25, ... = A017665/A017666 (sum of reciprocals of divisors of n)
  • 1, 1, 3/2, 4/3, 25/12, 6/5, 49/20, 176/105, 621/280, 100/63, 7381/2520, 552/385, 86021/27720, 11662/6435, 18075/8008, 91072/45045, ... = A093600/A069220
  • 1, 3/2, 7/5, 17/12, 41/29, 99/70, 239/169, 577/408, 1393/985, 3363/2378, 8119/5741, 19601/13860, 47321/33461, 114243/80782, ... = A001333/A000129 (sqrt(2))
  • 3/2, 5/3, 7/5, 11/7, 13/11, 17/13, 19/17, 23/19, 29/23, 31/29, 37/31, 41/37, 43/41, 47/43, 53/47, 59/53, 61/59, 67/61, 71/67, ... = A000040(n+1)/A000040(n) (prime(n+1)/prime(n))
  • 1, 3/2, 7/4, 15/8, 31/16, 63/32, 127/64, 255/128, 511/256, 1023/512, 2047/1024, 4095/2048, 8191/4096, 16383/8192, 32767/16384, ... = A000225(n+1)/A000079
  • 1, 3/2, 11/6, 25/12, 137/60, 49/20, 363/140, 761/280, 7129/2520, 7381/2520, 83711/27720, 86021/27720, ... = A001008/A002805 (harmonic numbers H_n)
  • 1, 3/2, 2, 13/6, 12/5, 49/20, 93/37, 5/2, 127/51, ... = A095132/A095131
  • 1, 3/2, 21/10, 861/310, 1275141/363010, 2551762438701/594665194510, ... = A079268/A079269
  • 1, 3/2, 21/8, 77/16, 1155/128, 4389/256, 33649/1024, 129789/2048, 4023459/32768 ... = A067002/A046161
  • 0, 1, -1, 3/2, -8/3, 125/24, -54/5, 16807/720, -16384/315, 531441/4480, -156250/567, 2357947691/3628800, ... = A227831/A095996 (LambertW)

First nontrivial term in range (3/2, 2]

  • 1, 5/3, 23/15, 45/29, 925/597, 7285/4701, ... = A053518/A053519
  • 1, 7/4, 137/72, 2341/1200, 38629/19600, 1257937/635040, 50881679/25613280, 164078209/82450368, 18480100619/9275666400, 1187779852639/595703908800, ... = A100520/A100521
  • 0, -1, 0, 1, -2, -1/2, 0, 1/2, 2, -3, -1, -1/3, 0, 1/3, 1, 3, -1, -3/2, ... = A196199/A004737 (another listing of rationals, with repetition)
  • 0/1, -1/1, 1/1,  -2/1, -1/2, 1/2, 2/1,  -3/1, -1/3, 1/3, 3/1,  -4/1, -3/2, -2/3, -1/4, 1/4, 2/3, 3/2, 4/1,  -5/1, -1/5, 1/5, 5/1, ... = A037161/A037162 ("well ordered" enumeration of all fractions: see there)
  • -1, 0, 1, -2, -1/2, 1/2, 2, -3, -3/2, -2/3, -1/3, 1/3, 2/3, 3/2, 3, -4, -4/3, -3/4, -1/4, 1/4, 3/4, 4/3, 4, ... = A113136/A113137 (rationals ordered by height)
  • 1, 2, 1/2, 3, 1/3, 3/2, 2/3, 4, 1/4, 4/3, ... = A020650/A020651 (another enumeration of rationals)
  • 0, 1, 2, 1/2, 3, 3/2, 1/3, 2/3, 4, 5/2, 4/3, 5/3, 1/4, 2/5, 3/4, 3/5, 5, 7/2, 7/3, 8/3, 5/4, 7/5, 7/4, 8/5, ... = A229742/A071766
  • 1, 1, 2, 1, 4/3, 2, 6/5, 1, 2, 4/3, 10/9, 2, 12/11, 6/5, 8/3, 1, 16/15, 2, 18/17, 4/3, 12/5, 10/9, 22/21, 2, ... = A173557/A305444
  • 1, 1, 2, 1, 3/2, 1, 3/2, 2, 6/5, 1, 5/4, 1, 3/2, 4/3, 1, 4/3, 3/2, 5/4, 8/7, 1, ... = A224762/A224763 (fractional Gijswijt)
  • 1, 2, -1, 3, -1/2, 0, 4, -1/3, 1/2, 5, -1/4, 2/3, 3/2, -2, 6, -1/5, 3/4, 5/3, -3/2, 5/2, -2/3, 7, ... = A226131/A226130 (Fibonacci ordering of rationals)
  • 1, 2, 9/8, 176/2835, 23590375/167382319104,... = A078524/A078525 (Birkhoff polytopes)
  • 2, 4/3, 8/7, 16/15, 32/31, 64/63, 128/127, 256/255, 512/511, 1024/1023, 2048/2047, 4096/4095, 8192/8191, 16384/16383, ... = A000079/A000225
  • 2, 4/3, 16/9, 64/45, 384/225, 2304/1575, 18432/11025, 147456/99225, 1474560/893025, 14745600/9823275, 176947200/108056025, ... = A001900/A000246
  • 2, 3/2, 5/4, 9/8, 17/16, 33/32, 65/64, 129/128, 257/256, 513/512, 1025/1024, 2049/2048, 4097/4096, 8193/8192, 16385/16384, ... = A000051/A000079
  • 1, 2, 5/3, 7/4, 19/11, 26/15, 71/41, 97/56, 265/153, 362/209, 989/571, 1351/780, 3691/2131, ... = A002531/A002530 (sqrt(3))
  • 2, 2, 4/3, 16/13, 256/217, 65536/57073, 4294967296/3811958497, 18446744073709551616/16605534578235736513, ... = A001146/A100441
  • 1, 1, 1, 2, 2, 3/2, 1, 3, 3, 4/3, 2, 2, 2, 3/2, 1, 4, 4, 5/4, 5/3, ... = A224764/A224765 (fractional exponent of n)
  • 1, 2, 41/20, 85/42, 9287/4620, 10034/5005, 4089347/2042040, 3529889/1763580, 119042647/59491432, 191288533/95611230, 1553111566613/776363187600, ... = A100514/A100515
  • 1, 2, 13/6, 32/15, 73/35, 647/315, 28211/13860, 6080/3003, 18181/9009, 1542158/765765, 2786599/1385670, 29229544/14549535, 134354573/66927861, ... = A100512/A100513.
  • 1, 2, 17/8, 56/27, 1759/864, 1009/500, 86831/43200, 2322304/1157625, 85922/42875, 1144667/571536, 16019198113/8001504000, 123357293/61631955, ... = A100518/A100519
  • 1, 2, 9/4, 20/9, 155/72, 21/10, 7441/3600, 3224/1575, 5697/2800, 3575/1764, 28523/14112, 27183/13475, 70357417/34927200, 4661447/2316600, ... = A100516/A100517
  • 2, 2, 4/3, 16/13, 256/217, 65536/57073, 4294967296/3811958497, 18446744073709551616/16605534578235736513, ... = A001146/A100441
  • 2, 5/2, 12/5, 113/47, 351/146, 3623/1507, 3974/1653, 19519/8119, 355316/147795, 374835/155914, ... = A060807/A060808
  • 1, 2, 5/2, 8/3, 8/3, 13/5, 151/60, 256/105, 83/35, 146/63, 1433/630, 2588/1155, 15341/6930, 28211/12870, 52235/24024, 19456/9009, 19345/9009, ... = A046825/A046826
  • 1, 2, 5/2, 17/6, 91/30, 379/120, 5047/1560, 35849/10920, 614893/185640, 6800951/2042040, 607326679/181741560, ... = A059248/A035105
  • 2, 5/2, 26/9, 103/32, 2194/625, 1223/324, 472730/117649, 556403/131072, 21323986/4782969, ... = A090878/A036505
  • 1, 2, 5/2, 29/10, 941/290, 969581/272890, 1014556267661/264588959090, 1099331737522548368039021/268440386798659418988490, ... = A073833/A073834
  • 1, 2, 3, 1/2, 4, 1/3, 3/2, 5, 1/4, 4/3, 5/2, 2/3, 6, ... = A226081/A226080 (another Fibonacci ordering of rationals)
  • 1, 2, 3, 3/2, 4, 5/2, 4/3, 5/3, 5, 7/2, 7/3, 8/3, 5/4, 7/5, 7/4, 8/5, 6, ... = A071585/A071766 (rationals > 1)
  • 2, 3, 8/3, 11/4, 19/7, 87/32, 106/39, 193/71, 1264/465, 1457/536, 2721/1001, 23225/8544, 25946/9545, 49171/18089, ... = A007676/A007677 (e)
  • 1, 2, 3, 15/4, 35/8, 77/16, 1001/192, 17017/3072, 323323/55296, 676039/110592, 2800733/442368, 86822723/13271040, ... = A060753/A038110
  • 2, 4, 20/3, 32/3, 256/15, 416/15, 4832/105, 8192/105, 42496/315, 74752/315, 1467392/3465, 2650112/3465, 62836736/45045, 115552256/45045, 42790912/9009, ... = A108866/A229726(repeated) (sum 2^k/k)
  • 2, 20/3, 256/15, 4832/105, 42496/315, 1467392/3465, 62836736/45045, 42790912/9009, 2535587840/153153, 170851041280/2909907, ... = A229727/A229726 (sum 2^k/k)

First nontrivial term > 2

  • 2, 5/2, 10/3, 17/4, 28/5, 41/6, 58/7, 77/8, 100/9, 129/10, 160/11, 197/12, 238/13, 281/14, 328/15, 381/16, 440/17, 167/6, 568/19, ... = A301273/A301274 (mean of primes)
  • 0, 1, 8/3, 29/6, 37/5, 103/10, 472/35, 2369/140, 2593/126, ... = A115107/A096620 (quicksort)
  • 1, 1, 3, 3, 15, 1, 63, 5, 21, 1, 1023, 5/3, 4095, 1, 17/3, 17, 65535, 1, 262143, 17/3, 65/3, 1, 4194303, 17/5, ... = A226305/A226306
  • 3, 22/7, 333/106, 355/113, 103993/33102, ... = A002485/A002486 (pi)
  • 3, 9/2, 53/9, 231/32, 5319/625, 3167/324, 1296273/117649, 1604979/131072, ... = A226931/A036505
  • 0, 1, 3, 17/3, 53/6, 62/5, 163/10, 717/35, 3489/140, ... = A093418/A096620
  • 1, 1, 1, 1, -432000/691, 1, -3456000/3617, -9504000/43867, ... = A085092/A001067


Comments

  • Updates to this Index are welcomed, but please be very careful. Remember that this is a scientific database. Please preserve the ordering (which is described at the top of this page).
  • The format is as follows: One line for each sequence of fractions, beginning with * followed by a space, then the first few fractions as numerator/denominator, separated by commas and spaces, followed by " ... = A000001/A000002" (say), where A000001 and A000002 are the A-numbers for the numerator and denominator sequences respectively. Unless the definition is very complicated, include a brief description in parentheses at the end of the line.
  • Several of the lines are too short. If you would like to help, please add more terms.
  • Interpolated zeros, and sometimes signs, have been ignored when matching with the A-numbers.
  • When you submit a pair of numerator-and-denominator sequences as new sequences to the OEIS, you should include a line like the following in the Example section of both the numerator and denominator sequences:
1, -1/3, -1/45, -2/945, -1/4725, -2/93555, -1382/638512875, -4/18243225, -3617/162820783125, -87734/38979295480125, -349222/1531329465290625, ... = A002431/A036278
and make a corresponding entry here.
  • Fractions are written as 7/4 (say), rather than 1 3/4 (i.e. no mixed fractions).
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