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 1-9 of 9 results.

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

A080384 Numbers k such that there are exactly 6 numbers j for which binomial(k, floor(k/2)) / binomial(k,j) is an integer, i.e., A080383(k) = 6.

Original entry on oeis.org

5, 7, 9, 11, 15, 17, 19, 21, 23, 27, 29, 33, 35, 39, 43, 45, 47, 49, 51, 53, 55, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 87, 89, 93, 95, 97, 99, 101, 103, 105, 107, 109, 111, 113, 115, 117, 119, 121, 123, 125, 127, 129, 131, 135, 137, 139, 141, 143, 145
Offset: 1

Views

Author

Labos Elemer, Mar 12 2003

Keywords

Examples

			For n=9, the central binomial coefficient (C(9,4) = 126) is divisible by C(9,0), C(9,1), C(9,4), C(9,5), C(9,8), and C(9,9); certain primes are missing, certain composites are here.

Links

Crossrefs

Cf. A327430, A080385, A080386, A327431, A080387.
Cf. A001405, A057977.

Programs

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

A080392 Numbers k such that A000984(k) mod k = 0 and A080383(k) != 7.

Original entry on oeis.org

2, 420, 920, 1122, 1218, 1892, 1978, 2444, 2914, 3198, 3782, 4028, 4136, 4292, 4664, 4958, 4960, 5330, 5762, 5986, 6020, 6032, 6710, 6834, 6864, 6882, 6954, 6956, 6968, 7106, 7130, 7140, 7238, 7254, 7448, 7616, 8178, 8190, 8400, 8692, 9462, 9506, 10712, 11060, 11288
Offset: 1

Views

Author

Labos Elemer, Mar 17 2003

Keywords

Comments

Numbers arising in A067348 and not present in A080385.
Even numbers n such that n divides binomial(n, [n/2]) and A010551(n) does not divide j!*(n-j)! exactly 7 times for j = 0..n. - Peter Luschny, Aug 04 2017

Examples

			A080383(2) = 3;
A080383(420) = 11;
A080383(920) = 11;
A080383(1122) = 9;
A080383(1218) = 9.

Links

Crossrefs

Cf. A000984, A001405, A067348, A080383, A080385.

Programs

  • Maple
    isa := proc(n)  local bn, bm;
if n mod 2 = 0 then bn := binomial(n, iquo(n,2)):
if modp(bn, n) = 0 then
   bm := (n, j) -> `if`(modp(bn, binomial(n, j)) = 0, 1, 0):
   return 1 <> add(bm(n, j), j=2..iquo(n,2)-1)
fi fi; false end:
select(isa, [$1..5000]); # Peter Luschny, Aug 04 2017
  • Mathematica
    Do[s=Count[Table[IntegerQ[Binomial[n, Floor[n/2]]/ Binomial[n, j]], {j, 0, n}], True]; s1=IntegerQ[Binomial[n, n/2]/n]; If[ !Equal[s, 7] && Equal[s1, True], Print[n]], {n, 1, 10000}]
(* Second program: *)
Select[Range@ 5000, Function[n, And[Divisible[Binomial[n, n/2], n], Count[Table[Divisible[Binomial[n, Floor[n/2]], Binomial[n, j]], {j, 0, n}], True] != 7]]] (* Michael De Vlieger, Jul 30 2017 *)

Extensions

More terms from Michael De Vlieger, Jul 30 2017

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

A067348 Even numbers n such that binomial(n, [n/2]) is divisible by n.

Original entry on oeis.org

2, 12, 30, 56, 84, 90, 132, 154, 182, 220, 252, 280, 306, 312, 340, 374, 380, 408, 418, 420, 440, 456, 462, 476, 532, 552, 598, 616, 624, 630, 644, 650, 660, 690, 756, 828, 840, 858, 870, 880, 884, 900, 918, 920, 936, 952, 966, 986, 992, 1020, 1054, 1102
Offset: 1

Views

Author

Dean Hickerson, Jan 16 2002

Keywords

Comments

This sequence has a surprisingly large overlap with A080385(n); a few values, 2, 420, 920 are exceptional. This means that usually A080383(A067348(n))=7. - Labos Elemer, Mar 17 2003
Conjecture: sequence contains most of 2*A000384(k). Exceptions are k = 8, 18, 20, 23, 35, ... - Ralf Stephan, Mar 15 2004

Links

Crossrefs

Subsequence of A042996.
Cf. A000984, A067315, A080385.

Programs

  • Mathematica
    Select[Range[2, 1200, 2], Mod[Binomial[ #, #/2], # ]==0&]
  • PARI
    val(n, p) = my(r=0); while(n, r+=n\=p);r
is(n) = {if(valuation(n, 2) == 0, return(0)); my(f = factor(n)); for(i=1, #f~, if(val(n, f[i, 1]) - 2 * val(n/2, f[i, 1]) - f[i, 2] < 0, return(0))); return(1)} \\ David A. Corneth, Jul 29 2017

Formula

a(n) = 2*A002503(n-2) + 2.
Appears to be 2*A058008(n). - Benoit Cloitre, Mar 21 2003

Extensions

Name clarified by Peter Luschny, Aug 04 2017
Showing 1-9 of 9 results.

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