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 1-7 of 7 results.

A080383 Number of j (0 <= j <= n) such that the central binomial coefficient C(n,floor(n/2)) = A001405(n) is divisible by C(n,j).

Original entry on oeis.org

1, 2, 3, 4, 3, 6, 3, 6, 3, 6, 3, 6, 7, 10, 3, 6, 3, 6, 3, 6, 3, 6, 3, 6, 3, 8, 3, 6, 3, 6, 7, 10, 3, 6, 3, 6, 3, 8, 3, 6, 5, 10, 3, 6, 3, 6, 3, 6, 3, 6, 3, 6, 3, 6, 3, 6, 7, 10, 3, 6, 3, 6, 3, 6, 3, 6, 3, 6, 3, 6, 3, 6, 3, 6, 3, 6, 3, 6, 3, 6, 3, 6, 3, 6, 7, 10, 3, 6, 3, 6, 7, 10, 3, 6, 3, 6, 3, 6, 3, 6, 3, 6
Offset: 0

Views

Author

Labos Elemer, Mar 12 2003

Keywords

Examples

			For n <= 500 only a few values of a(n) arise: {1,2,3,4,5,6,7,8,10,11,14}.
From _Jon E. Schoenfield_, Sep 15 2019: (Start)
a(n)=1 occurs only at n=0.
a(n)=2 occurs only at n=1.
a(n)=3 occurs for all even n > 0 such that C(n,j) divides C(n,n/2) only at j = 0, n/2, and n. (This is the case for about 4/9 of the first 100000 terms, and there appear to be nearly as many terms for which a(n)=6.)
a(n)=4 occurs only at n=3.
For n <= 100000, the only values of a(n) that occur are 1..16, 18, 19, 22, 23, and 26.
   k | Indices n (up to 100000) at which a(n)=k
  ---+-------------------------------------------------------
   1 | 0
   2 | 1
   3 | 2, 4, 6, 8, 10, 14, 16, 18, 20, 22, 24, ...
   4 | 3
   5 | 40, 176, 208, 480, 736, 928, 1248, 1440, ... (A327430)
   6 | 5, 7, 9, 11, 15, 17, 19, 21, 23, 27, 29, ... (A080384)
   7 | 12, 30, 56, 84, 90, 132, 154, 182, 220, ...  (A080385)
   8 | 25, 37, 169, 199, 201, 241, 397, 433, ...    (A080386)
   9 | 1122, 1218, 5762, 11330, 12322, 15132, ...   (A327431)
  10 | 13, 31, 41, 57, 85, 91, 133, 155, 177, ...   (A080387)
  11 | 420, 920, 1892, 1978, 2444, 2914, 3198, ...
  12 | 1103, 1703, 2863, 7773, 10603, 15133, ...
  13 | 12324, 37444
  14 | 421, 921, 1123, 1893, 1979, 1981, 2445, ...
  15 | 4960, 6956, 13160, 16354, 18542, 24388, ...
  16 | 11289, 16483, 36657, 62653, 89183
  17 |
  18 | 4961, 6957, 12325, 13161, 16355, 18543, ...
  19 | 16356, 88510, 92004
  20 |
  21 |
  22 | 16357, 88511, 90305, 92005
  23 | 90306
  24 |
  25 |
  26 | 90307
(End)

Links

Crossrefs

Cf. A001405, A000225, A057977, A022292, A020475, A067348, A042996.
Cf. A327430, A080384, A080385, A080386, A327431, A080387.
Cf. A080393.

Programs

  • Magma
    [#[j:j in [0..n]| Binomial(n,Floor(n/2)) mod Binomial(n,j) eq 0]:n in [0..100]]; // Marius A. Burtea, Sep 15 2019
  • Mathematica
    Table[Count[Table[IntegerQ[Binomial[n, Floor[n/2]]/Binomial[n, j]], {j, 0, n}], True], {n, 0, 500}] (* adapted by Vincenzo Librandi, Jul 29 2017 *)
  • PARI
    a(n) = my(b=binomial(n, n\2)); sum(i=0, n, (b % binomial(n, i)) == 0); \\ Michel Marcus, Jul 29 2017
  • PARI
    a(n) = {if(n==0, return(1)); my(bb = binomial(n, n\2), b = n); res = 2 + !(n%2) + 2 * (n>2 && n%2 == 1); for(i = 2, (n-1)\2, res += 2*(bb%b==0); b *= (n + 1 - i) / i); res} \\ David A. Corneth, Jul 29 2017

Extensions

Edited by Dean Hickerson, Mar 14 2003
Offset corrected by David A. Corneth, Jul 29 2017

A080385 Numbers k such that there are exactly 7 numbers j for which binomial(k, floor(k/2)) / binomial(k,j) is an integer, i.e., A080383(k) = 7.

Original entry on oeis.org

12, 30, 56, 84, 90, 132, 154, 182, 220, 252, 280, 306, 312, 340, 374, 380, 408, 418, 440, 456, 462, 476, 532, 552, 598, 616, 624, 630, 644, 650, 660, 690, 756, 828, 840, 858, 870, 880, 884, 900, 918, 936, 952, 966, 986, 992, 1020, 1054, 1102, 1116, 1140, 1160
Offset: 1

Views

Author

Labos Elemer, Mar 12 2003

Keywords

Examples

			For n=12, the central binomial coefficient (C(12,6) = 924) is divisible by C(12,0), C(12,1), C(12,2), C(12,6), C(12,10), C(12,11), and C(12,12).

Links

Crossrefs

Cf. A327430, A080384, A080386, A327431, A080387.
Cf. A001405, A057977.

Extensions

More terms from Vaclav Kotesovec, Sep 06 2019

A080386 Numbers k such that there are exactly 8 numbers j for which binomial(k, floor(k/2)) / binomial(k,j) is an integer, i.e., A080383(k) = 8.

Original entry on oeis.org

25, 37, 169, 199, 201, 241, 397, 433, 547, 685, 865, 1045, 1081, 1585, 1657, 1891, 1951, 1969, 2071, 2143, 2647, 2901, 3011, 3025, 3097, 3151, 3251, 3421, 3511, 3727, 4105, 4213, 4453, 4771, 4885, 5581, 5857, 6019, 6031, 6265, 6397, 6967, 7345, 7615, 7831, 8425, 8857, 8929
Offset: 1

Views

Author

Labos Elemer, Mar 12 2003

Keywords

Examples

			For n=25, the central binomial coefficient (C(25,12) = 5200300) is divisible by C(25,0), C(25,1), C(25,3), C(25,12), C(25,13), C(25,22), C(25,24), and C(25,25).

Links

Crossrefs

Cf. A327430, A080384, A080385, A327431, A080387.
Cf. A001405, A057977.

Extensions

More terms from Michel Marcus, Aug 23 2019

A080387 Numbers k such that there are exactly 10 numbers j for which binomial(k, floor(k/2)) / binomial(k,j) is an integer, i.e., A080383(k) = 10.

Original entry on oeis.org

13, 31, 41, 57, 85, 91, 133, 155, 177, 183, 209, 221, 253, 281, 307, 313, 341, 375, 381, 409, 419, 441, 457, 463, 477, 481, 533, 553, 599, 617, 625, 631, 645, 651, 661, 691, 737, 757, 829, 841, 859, 871, 881, 885, 901, 919, 929, 937, 953, 967, 987, 993
Offset: 1

Views

Author

Labos Elemer, Mar 12 2003

Keywords

Examples

			For n=13, the central binomial coefficient (C(13,6) = 1716) is divisible by 10 binomial coefficients C(13,j); the 4 nondivisible cases are C(13,4), C(13,5), C(13,8), and C(13,9).

Links

Crossrefs

Cf. A327430, A080384, A080385, A080386, A327431.
Cf. A001405, A057977.

A327430 Numbers k such that there are exactly 5 numbers j for which binomial(k, floor(k/2)) / binomial(k,j) is an integer, i.e., A080383(k) = 5.

Original entry on oeis.org

40, 176, 208, 480, 736, 928, 1248, 1440, 1632, 1824, 2128, 2400, 2464, 2720, 3008, 3360, 3520, 3776, 3904, 4144, 4240, 4320, 4704, 5280, 5664, 6432, 7040, 7200, 7360, 7488, 7992, 8064, 8544, 9504, 9792, 10336, 10400, 10944, 12160, 12992, 13158, 13392, 15744
Offset: 1

Views

Author

Vaclav Kotesovec, Sep 10 2019

Keywords

Examples

			C(40,20) is divisible by 5 binomial coefficients: C(40,0), C(40,2), C(40,20), C(40,38) and C(40,40).

Links

Crossrefs

Cf. A080383, A080384, A080385, A080386, A327431, A080387.

A327431 Numbers k such that there are exactly 9 numbers j for which binomial(k, floor(k/2)) / binomial(k,j) is an integer, i.e., A080383(k) = 9.

Original entry on oeis.org

1122, 1218, 5762, 11330, 12322, 15132, 16482, 26690, 37442, 40994, 57090, 61184, 77184, 94978, 103170, 107072, 108290, 114818, 121346, 124662, 136308, 138370, 142400, 148610, 149250, 149634, 177410, 198018, 221314, 221442, 233730, 246530, 259074, 264578
Offset: 1

Views

Author

Vaclav Kotesovec, Sep 10 2019

Keywords

Examples

			C(1122,561) is divisible by 9 binomial coefficients C(1122,0), C(1122,1), C(1122,2), C(1122,4), C(1122,561), C(1122,1118), C(1122,1120), C(1122,1121) and C(1122,1122).

Crossrefs

Cf. A080383, A327430, A080384, A080385, A080386, A080387.

Programs

  • Magma
    a:=[]; kMax:=265000; cbc:=2; for k in [4..kMax by 2] do cbc:=(cbc*(4*k-4)) div k; count:=3; p:=PreviousPrime((k div 2) + 1); b:=1; for j in [1..k-2*p] do b:=(b*(k+1-j)) div j; if cbc mod b eq 0 then count+:=2; end if; end for; r:=1/1; for j in [(k div 2)-1..p by -1] do r:=r*(j+1)/(k-j); end for; if r le 1/2 then b:=cbc; for j in [(k div 2)-1..p by -1] do b:=(b*(j+1)) div (k-j); if cbc mod b eq 0 then count+:=2; end if; end for; end if; if count eq 9 then a[#a+1]:=k; end if; end for; a // Jon E. Schoenfield, Sep 15 2019

Extensions

Terms > 100000 from Jon E. Schoenfield, Sep 15 2019

A080393 a(n) is the smallest integer such that A080383(a(n)) = n.

Original entry on oeis.org

0, 1, 2, 3, 40, 5, 12, 25, 1122, 13, 420, 1103, 12324, 421, 4960, 11289, 232582, 4961, 16356, 107073
Offset: 1

Views

Author

Labos Elemer, Mar 17 2003

Keywords

Comments

Parity of n and a(n) is opposite.
It is unknown whether all positive integers arise in A080383 or not.
a(22)=16357, a(23)=90306, a(26)=90307. - Vaclav Kotesovec, Sep 10 2019
For each n > 20 except 22, 23, and 26, a(n) > 10^6 (if it exists). - Jon E. Schoenfield, Sep 15 2019

Examples

			a(10)=13 because in A080383 10 appears first as the 13th term.

Crossrefs

Cf. A080383, A080384(1)=a(6), A080385(1)=a(7), A080386(1)=a(8), A080387(1)=a(10).

Programs

  • Mathematica
    f[x_] := Count[Table[IntegerQ[Binomial[x, Floor[x/2]]/ Binomial[x, j]], {j, 0, n}], True]; t=Table[0, {20}]; Do[s=f[n]; If[s<21&&t[[s]]==0, t[[s]]=n], {n, 1, 1300}]; t
  • PARI
    f(n) = my(b=binomial(n, n\2)); sum(i=0, n, (b % binomial(n, i)) == 0); \\ A080383
a(n) = my(k=0); while(f(k) != n, k++); k; \\ Michel Marcus, Aug 23 2019

Extensions

a(13)-a(16) from Michel Marcus, Aug 23 2019
a(17) from Jon E. Schoenfield, Sep 15 2019
a(18) from Michel Marcus, Aug 23 2019
a(19) from Vaclav Kotesovec, Sep 10 2019
a(20) from Jon E. Schoenfield, Sep 15 2019
Showing 1-7 of 7 results.

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