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.

A032780 a(n) = n(n+1)(n+2)...(n+8) / (n+(n+1)+(n+2)+...+(n+8)).

Original entry on oeis.org

0, 8064, 67200, 316800, 1108800, 3203200, 8072064, 18345600, 38438400, 75398400, 140025600, 248312064, 423259200, 697132800, 1114220800, 1734163200, 2635928064, 3922512000, 5726448000, 8216208000, 11603592000, 16152200064, 22187088000, 30105712000
Offset: 0

Views

Author

Patrick De Geest, May 15 1998

Keywords

Comments

a(5n+1) == 4 modulo 10.
The product of any k consecutive integers is divisible by the sum of the same k integers for odd nonprime k's: 1 (trivial case), 9 (this sequence), 15, etc. - Zak Seidov, Mar 18 2014

Links

Crossrefs

Cf. A032765-A032798.
Cf. A104678.

Programs

  • Mathematica
    nn = 9; Table[c = Range[n, n + nn - 1]; Times @@ c/Total[c], {n, 0, 25}] (* T. D. Noe, Mar 18 2014 *)
  • PARI
    a(n) = prod(i=0, 8, n+i)/sum(i=0, 8, n+i); \\ Michel Marcus, Mar 18 2014

Formula

a(-n) = a(n-8) for all n in Z. - Michael Somos, Mar 18 2014
a(n) = 64 * A104678(n-1) = 64 * binomial(n+3, 4) * binomial(n+8, 4). - Michael Somos, Mar 18 2014
From Chai Wah Wu, Dec 17 2016: (Start)
a(n) = 9*a(n-1) - 36*a(n-2) + 84*a(n-3) - 126*a(n-4) + 126*a(n-5) - 84*a(n-6) + 36*a(n-7) - 9*a(n-8) + a(n-9) for n > 8.
G.f.: 64*x*(-x^4 + 9*x^3 - 36*x^2 + 84*x - 126)/(x - 1)^9. (End)

Extensions

Typo in name fixed by Zak Seidov, Mar 18 2014
More terms from Michel Marcus, Mar 18 2014

A032796 Numbers that are congruent to {1, 2, 3, 5, 6} mod 7.

Original entry on oeis.org

1, 2, 3, 5, 6, 8, 9, 10, 12, 13, 15, 16, 17, 19, 20, 22, 23, 24, 26, 27, 29, 30, 31, 33, 34, 36, 37, 38, 40, 41, 43, 44, 45, 47, 48, 50, 51, 52, 54, 55, 57, 58, 59, 61, 62, 64, 65, 66, 68, 69, 71, 72, 73, 75, 76, 78, 79, 80, 82, 83, 85, 86, 87, 89, 90, 92, 93, 94, 96, 97, 99
Offset: 1

Views

Author

Patrick De Geest, May 15 1998

Keywords

Comments

If k is a term, then k*(k+1)*(k+2)*...*(k+6)/(k+(k+1)+(k+2)+...+(k+6)) is a multiple of k.

Links

Crossrefs

Cf. A032765..A032798.
Cf. A010874.

Programs

  • Magma
    [n: n in [0..120] | n mod 7 in {1, 2, 3, 5, 6}]; // Vincenzo Librandi, Dec 29 2010
  • Mathematica
    #+{1,2,3,5,6}&/@(7*Range[0,15])//Flatten (* or *) LinearRecurrence[ {1,0,0,0,1,-1},{1,2,3,5,6,8},100] (* Harvey P. Dale, Oct 07 2018 *)

Formula

Equals natural numbers minus '4, 7, 11, 14, 18, ...' (= previous term +3, +4, +3, +4, ...).
G.f.: x*(x^5 + x^4 + 2*x^3 + x^2 + x + 1)/((1-x)*(1-x^5)).
a(n) = (m^3 - 6*m^2 + 17*m + 6*(7*floor(n/5)-1))/6, where m = n mod 5. - Luce ETIENNE,Oct 17 2018

A032767 a(n) = floor ( n(n+1)(n+2)(n+3) / (n+(n+1)+(n+2)+(n+3)) ).

Original entry on oeis.org

0, 2, 8, 20, 38, 64, 100, 148, 208, 282, 373, 480, 606, 753, 921, 1112, 1328, 1571, 1841, 2140, 2471, 2833, 3229, 3661, 4129, 4635, 5182, 5769, 6399, 7074, 7794, 8561, 9377, 10244, 11162, 12133, 13160, 14242, 15382, 16582, 17842, 19164
Offset: 0

Views

Author

Patrick De Geest, May 15 1998

Keywords

Links

Crossrefs

Cf. A032765-A032798.

Programs

  • Maple
    A032767 := proc(n)
        n*(n+1)*(n+2)*(n+3) /(4*n+6) ;
        floor(%) ;
end proc: # R. J. Mathar, May 20 2013
  • Mathematica
    Table[Floor[Times@@(n+Range[0,3])/Total[n+Range[0,3]]],{n,0,50}] (* Harvey P. Dale, Oct 18 2024 *)

Formula

The g.f. has denominator (1-x)^4(1-x^16). - Ralf Stephan, May 16 2005

A032784 Numbers k such that k(k+1)(k+2)...(k+11) / (k+(k+1)+(k+2)+...+(k+11)) is an integer.

Original entry on oeis.org

0, 2, 5, 7, 8, 11, 12, 17, 19, 22, 26, 32, 33, 35, 44, 47, 55, 62, 68, 77, 82, 89, 107, 110, 116, 117, 132, 143, 152, 176, 187, 197, 215, 242, 257, 264, 278, 297, 332, 341, 362, 407, 418, 440, 467, 539, 572, 602, 607, 656, 737, 782, 803, 845, 902, 957, 1007, 1034
Offset: 1

Views

Author

Patrick De Geest, May 15 1998

Keywords

Comments

(d-11)/2 where d>=7 is a divisor of 36018675. In particular, the sequence is finite. - Robert Israel, Jul 12 2018

Links

Crossrefs

Cf. A032765-A032798.

Programs

  • Maple
    sort(convert(select(type,map(t -> (t-11)/2,numtheory:-divisors(36018675)),nonnegint),list));
  • Mathematica
    Select[Range[0,1100],IntegerQ[Times@@Range[#,#+11]/Total[Range[#,#+11]]]&] (* Harvey P. Dale, Sep 02 2016 *)

Extensions

Definition corrected by Harvey P. Dale, Sep 02 2016
Offset changed by Robert Israel, Jul 12 2018

A032785 Numbers k such that k*(k+1)*(k+2) ... (k+13) / (k+(k+1)+(k+2)+ ... +(k+13)) is an integer.

Original entry on oeis.org

0, 1, 4, 6, 7, 10, 11, 13, 16, 21, 25, 26, 31, 32, 34, 39, 43, 46, 52, 54, 61, 65, 76, 78, 81, 88, 91, 106, 109, 115, 130, 131, 142, 151, 156, 169, 175, 186, 196, 208, 221, 241, 247, 256, 277, 286, 296, 331, 340, 351, 358, 403, 406, 416, 417, 439, 466, 481, 494
Offset: 1

Views

Author

Patrick De Geest, May 15 1998

Keywords

Comments

(d-13)/2 for divisors d>=13 of 2608781175. In particular, the sequence is finite. - Robert Israel, Jul 13 2018

Links

Crossrefs

Cf. A032765-A032798.

Programs

  • Maple
    seq((t-13)/2, t=select(`>=`,numtheory:-divisors(2608781175),13)); # Robert Israel, Jul 13 2018
  • Mathematica
    Select[Range[0,500],IntegerQ[Times@@Range[#,#+13]/Total[Range[ #, #+13]]]&] (* Harvey P. Dale, Sep 02 2016 *)

Extensions

Definition corrected by Harvey P. Dale, Sep 02 2016

A032786 Numbers k such that k(k+1)(k+2)...(k+15) / (k+(k+1)+(k+2)+...+(k+15)) is an integer.

Original entry on oeis.org

0, 3, 5, 6, 9, 10, 12, 15, 17, 20, 24, 25, 30, 31, 33, 38, 42, 45, 51, 53, 55, 60, 64, 66, 75, 77, 80, 87, 90, 105, 108, 114, 115, 129, 130, 141, 150, 155, 168, 174, 180, 185, 195, 207, 213, 220, 240, 246, 255, 262, 276, 285, 295, 305, 311, 330, 339, 350, 357
Offset: 1

Views

Author

Patrick De Geest, May 15 1998

Keywords

Comments

(d-15)/2 where d >= 15 divides 4108830350625. In particular, the sequence is finite. - Robert Israel, Jul 13 2018

Links

Crossrefs

Cf. A032765-A032798.

Programs

  • Maple
    sort([seq((t-15)/2, t=select(`>=`,numtheory:-divisors(4108830350625),15))]); # Robert Israel, Jul 13 2018
  • Mathematica
    Select[Range[0,500],IntegerQ[Times@@Range[#,#+15]/Total[Range[ #,#+15]]]&] (* Harvey P. Dale, Sep 02 2016 *)

Extensions

Definition corrected by Harvey P. Dale, Sep 02 2016
Offset changed by Robert Israel, Jul 13 2018

A032787 Numbers k such that k(k+1)(k+2)...(k+17) / (k+(k+1)+(k+2)+ ... +(k+17)) is an integer.

Original entry on oeis.org

0, 2, 4, 5, 8, 9, 11, 14, 16, 17, 19, 23, 24, 29, 30, 32, 34, 37, 41, 44, 50, 51, 52, 54, 59, 63, 65, 68, 74, 76, 79, 85, 86, 89, 102, 104, 107, 113, 114, 119, 128, 129, 136, 140, 149, 154, 167, 170, 173, 179, 184, 194, 204, 206, 212, 219, 221, 239, 245, 254
Offset: 1

Views

Author

Patrick De Geest, May 15 1998

Keywords

Comments

(d-17)/2 where d >= 17 divides 131939107925625. In particular, the sequence is finite. - Robert Israel, Jul 13 2018

Links

Crossrefs

Cf. A032765-A032798.

Programs

  • Maple
    sort([seq((t-17)/2, t=select(`>=`,numtheory:-divisors(131939107925625),17))]); # Robert Israel, Jul 13 2018
  • Mathematica
    Select[Range[0,300],IntegerQ[Times@@Range[#,#+17]/Total[Range[#,#+17]]]&] (* Harvey P. Dale, Sep 02 2016 *)

Extensions

Definition corrected by Harvey P. Dale, Sep 02 2016

A032788 Numbers k such that k(k+1)(k+2)...(k+19) / (k+(k+1)+(k+2)+...+(k+19)) is an integer.

Original entry on oeis.org

0, 1, 3, 4, 7, 8, 10, 13, 15, 16, 18, 19, 22, 23, 28, 29, 31, 33, 36, 38, 40, 43, 49, 50, 51, 53, 57, 58, 62, 64, 67, 73, 75, 76, 78, 84, 85, 88, 95, 101, 103, 106, 112, 113, 114, 118, 127, 128, 133, 135, 139, 148, 152, 153, 166, 169, 171, 172, 178, 183, 190
Offset: 1

Views

Author

Patrick De Geest, May 15 1998

Keywords

Comments

(d-19)/2 where d >= 19 divides 85734032330071125. In particular, the sequence is finite. - Robert Israel, Jul 13 2018

Links

Crossrefs

Cf. A032765-A032798.

Programs

  • Maple
    sort([seq((t-19)/2, t=select(`>=`,numtheory:-divisors(85734032330071125),19))]); # Robert Israel, Jul 13 2018
  • Mathematica
    Select[Range[0,190],IntegerQ[Times@@Range[#,#+19]/Total[Range[#,#+19]]]&] (* Harvey P. Dale, Sep 02 2016 *)
Select[Range[0,190],With[{c=Range[#,#+19]},Mod[Times@@c,Total[c]]==0&]] (* Harvey P. Dale, Nov 20 2024 *)

Extensions

Definition corrected by Harvey P. Dale, Sep 02 2016
Offset changed by Robert Israel, Jul 13 2018

A032797 Numbers n such that n(n+1)(n+2)...(n+10) /(n+(n+1)+(n+2)+...+(n+10)) is a multiple of n.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 10, 12, 13, 14, 15, 16, 18, 19, 20, 21, 23, 24, 25, 26, 27, 29, 30, 31, 32, 34, 35, 36, 37, 38, 40, 41, 42, 43, 45, 46, 47, 48, 49, 51, 52, 53, 54, 56, 57, 58, 59, 60, 62, 63, 64, 65, 67, 68, 69, 70, 71, 73, 74, 75, 76, 78, 79, 80, 81, 82, 84, 85, 86
Offset: 1

Views

Author

Patrick De Geest, May 15 1998

Keywords

Comments

Equals natural numbers minus '6,11,17,22,28,...' (= previous term +5,+6,+5,+6,...).

Links

Crossrefs

Cf. A032765-A032798.

Programs

  • Mathematica
    nmnQ[n_]:=With[{c=n+Range[0,10]},Divisible[Times@@c/Total[c],n]]; Select[ Range[ 100],nmnQ] (* or *) LinearRecurrence[{1,0,0,0,0,0,0,0,1,-1},{1,2,3,4,5,7,8,9,10,12},80] (* Harvey P. Dale, May 07 2017 *)
  • PARI
    is(n)=factorback(vector(10,i,n+i))%(11*n+55)==0 \\ Charles R Greathouse IV, Aug 07 2016

Formula

From Chai Wah Wu, Dec 17 2016: (Start)
a(n) = a(n-1) + a(n-9) - a(n-10) for n > 10.
G.f.: x*(x^9 + x^8 + x^7 + x^6 + 2*x^5 + x^4 + x^3 + x^2 + x + 1)/(x^10 - x^9 - x + 1). (End)

Extensions

Typo in definition corrected by Zak Seidov, Aug 06 2016
Showing 1-9 of 9 results.

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