A032765 -id:A032765 - cp's OEIS frontend

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

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

Patrick De Geest, May 15 1998

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

Robert Israel, Table of n, a(n) for n = 1..1208

Cf. A032765-A032798.

sort([seq((t-15)/2, t=select(`>=`,numtheory:-divisors(4108830350625),15))]); # Robert Israel, Jul 13 2018

Select[Range[0,500],IntegerQ[Times@@Range[#,#+15]/Total[Range[ #,#+15]]]&] (* Harvey P. Dale, Sep 02 2016 *)

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

Patrick De Geest, May 15 1998

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

Robert Israel, Table of n, a(n) for n = 1..2827

Cf. A032765-A032798.

sort([seq((t-17)/2, t=select(`>=`,numtheory:-divisors(131939107925625),17))]); # Robert Israel, Jul 13 2018

Select[Range[0,300],IntegerQ[Times@@Range[#,#+17]/Total[Range[#,#+17]]]&] (* Harvey P. Dale, Sep 02 2016 *)

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

Patrick De Geest, May 15 1998

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

Robert Israel, Table of n, a(n) for n = 1..8739

Cf. A032765-A032798.

sort([seq((t-19)/2, t=select(`>=`,numtheory:-divisors(85734032330071125),19))]); # Robert Israel, Jul 13 2018

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 *)

Definition corrected by Harvey P. Dale, Sep 02 2016

Offset changed by Robert Israel, Jul 13 2018

A071408 a(n+1) - 2a(n) + a(n-1) = (2/3)(1 + w^(n+1) + w^(2*n+2)) with a(1)=0, a(2)=1, and where w is the imaginary cubic root of unity.

Original entry on oeis.org

0, 1, 4, 7, 10, 15, 20, 25, 32, 39, 46, 55, 64, 73, 84, 95, 106, 119, 132, 145, 160, 175, 190, 207, 224, 241, 260, 279, 298, 319, 340, 361, 384, 407, 430, 455, 480, 505, 532, 559, 586, 615, 644, 673, 704, 735, 766, 799, 832, 865, 900, 935, 970, 1007, 1044

Offset: 1

Robert G. Wilson v, Jun 24 2002

w = exp(2*Pi*i/3)= (-1 - sqrt(-3))/2. Beginning with a(2) the first differences are 3,3,3,5,5,5,7,7,7,9,9,9,11, etc.

Index entries for linear recurrences with constant coefficients, signature (2,-1,1,-2,1).

Cf. A071618.

a[1] = 0; a[2] = 1; w = Exp[2Pi*I/3]; a[n_] := a[n] = Simplify[(2/3)(1 + w^n + w^(2n)) + 2a[n - 1] - a[n - 2]]; Table[ a[n], {n, 1, 60}]
Table[If[n<3,n-1,Floor[((n+1)^2-4)/3]],{n,1,100}] (*  Vladimir Joseph Stephan Orlovsky, Jan 30 2012 *)
LinearRecurrence[{2,-1,1,-2,1},{0,1,4,7,10},60] (* Harvey P. Dale, Jun 10 2016 *)

a(n)=n*(n+2)\3 - 1 \\ Charles R Greathouse IV, Mar 02 2017

a(n) = A032765(n)-1.
a(n) = floor((n-1)*(n+1)*(n+3)/(3*n+3)). - Gary Detlefs, Jul 13 2010
a(n) = (n-1)^2 - A030511(n-1). - Wesley Ivan Hurt, Jun 19 2013
G.f.: x^2*(1+x)*(x^2-x-1) / ( (1+x+x^2)*(x-1)^3 ). - R. J. Mathar, Jun 23 2013
a(n) = n + floor(n*(n-1)/3) - 1. - Bruno Berselli, Mar 02 2017

A278814 a(n) = ceiling(sqrt(3n+1)).

Original entry on oeis.org

1, 2, 3, 4, 4, 4, 5, 5, 5, 6, 6, 6, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 13, 13, 13, 13, 14, 14, 14, 14, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15, 15, 15, 15, 15, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 18, 18, 18, 18

Offset: 0

Mohammad K. Azarian, Nov 28 2016

Cf. A002162, A016777, A016789, A016933, A017569, A032765, A058183, A131033, A007494, A051536, A007559.

PROG(y := [], n := 100, LOOP(IF(n = -1, RETURN y), y := ADJOIN(CEILING(SQRT(1 + 3·n)), y), n := n - 1))

seq(ceil(sqrt(3*k+1)), k=0..100); # Robert Israel, Nov 28 2016

Mathematica
```
Table[Ceiling[Sqrt[3n+1]],{n,0,100}]
```

a(n)=sqrtint(3*n)+1 \\ Charles R Greathouse IV, Nov 29 2016

from math import isqrt
def A278814(n): return 1+isqrt(3*n) # Chai Wah Wu, Jul 28 2022

a(n) = ceiling(sqrt(3n+1)).
From Robert Israel, Nov 28 2016: (Start)
G.f.: (1-x)^(-1)*Sum_{k>=0} (x^(3*k^2)+x^(3*k^2+2*k+1)+x^(3*k^2+4*k+2)).
a(n+1) = a(n)+1 if n is in A032765, otherwise a(n+1) = a(n). (End)
Sum_{n>=0} (-1)^n/a(n) = log(2) (A002162). - Amiram Eldar, Jun 18 2025

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

Patrick De Geest, May 15 1998

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

Index entries for linear recurrences with constant coefficients, signature (1, 0, 0, 0, 0, 0, 0, 0, 1, -1).

Cf. A032765-A032798.

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 *)

is(n)=factorback(vector(10,i,n+i))%(11*n+55)==0 \\ Charles R Greathouse IV, Aug 07 2016

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)

Typo in definition corrected by Zak Seidov, Aug 06 2016

cp's OEIS Frontend

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

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

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

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A071408 a(n+1) - 2*a(n) + a(n-1) = (2/3)*(1 + w^(n+1) + w^(2*n+2)) with a(1)=0, a(2)=1, and where w is the imaginary cubic root of unity.

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A278814 a(n) = ceiling(sqrt(3n+1)).

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

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A071408 a(n+1) - 2a(n) + a(n-1) = (2/3)(1 + w^(n+1) + w^(2*n+2)) with a(1)=0, a(2)=1, and where w is the imaginary cubic root of unity.