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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.

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A127644 a(1)=3. a(n) is the smallest positive integer not occurring earlier in the sequence such that (sum{k=1 to n} a(k)) divides product{j=1 to n} a(j).

Original entry on oeis.org

3, 6, 9, 18, 12, 16, 8, 24, 32, 34, 30, 48, 15, 1, 14, 2, 17, 11, 20, 4, 26, 7, 21, 22, 5, 27, 10, 13, 25, 40, 19, 28, 33, 37, 23, 42, 38, 44, 35, 31, 41, 29, 46, 49, 39, 50, 36, 43, 47, 45, 51, 54, 55, 53, 52, 56, 57, 62, 61, 60, 64, 68, 67, 58, 63, 70, 69, 71, 65, 77, 66, 72
Offset: 1

Views

Author

Leroy Quet, Jan 22 2007

Keywords

Comments

Is this a permutation of the positive integers? According to Hans Havermann, this sequence matches sequence A127562 beginning with the 104th term of each. Unlike sequence A127562, the sum of the first n terms of this sequence divides the product of the first n terms for every positive integer n.

Crossrefs

Cf. A127562, A127645, A127646.

Programs

  • Maple
    N:= 1000: # to get a(1) to a(m-1) where a(m) is the first term > N
a[1]:= 3:
R:= {$1..N} minus {3}:
P:= 3:
S:= 3:
success:= true:
for n from 2 while success and R <> {} do
   success := false;
   for r in R do
     if type((P*r)/(S+r),integer) then
        a[n]:= r;
        nmax:= n;
        R:= R minus {r};
        success:= true;
        P:= P * r;
        S:= S + r;
        break
     fi
   od:
od:
seq(a[i],i=1..nmax); # Robert Israel, Dec 13 2014
  • Mathematica
    f[l_List] := Block[{k = 1, s = Plus @@ l, p = Times @@ l},While[MemberQ[l, k] || Mod[k*p, k + s] > 0, k++ ];Append[l, k]];Nest[f, {3}, 75] (* Ray Chandler, Jan 22 2007 *)
  • PARI
    v=[3];print1(3,", ");n=1;while(n<100,p=prod(i=1,#v,v[i]);if(p*n\(vecsum(v)+n)==p*n/(vecsum(v)+n)&&!vecsearch(vecsort(v),n),v=concat(v,n);print1(n,", ");n=0);n++) \\ Derek Orr, Dec 13 2014

Extensions

Extended by Ray Chandler, Jan 22 2007

A128540 Triangle A127647 * A097806, read by rows.

Original entry on oeis.org

1, 1, 1, 0, 2, 2, 0, 0, 3, 3, 0, 0, 0, 5, 5, 0, 0, 0, 0, 8, 8, 0, 0, 0, 0, 0, 13, 13, 0, 0, 0, 0, 0, 0, 21, 21, 0, 0, 0, 0, 0, 0, 0, 34, 34, 0, 0, 0, 0, 0, 0, 0, 0, 55, 55, 0, 0, 0, 0, 0, 0, 0, 0, 0, 89, 89, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 144, 144
Offset: 1

Views

Author

Gary W. Adamson, Mar 10 2007

Keywords

Comments

Row sums = A094895 starting (1, 2, 4, 6, 10, 16, 26, ...). A128541 = A097806 * A127647.

Examples

			First few rows of the triangle:
  1;
  1, 1;
  0, 2, 2;
  0, 0, 3, 3;
  0, 0, 0, 5, 5;
  0, 0, 0, 0, 8, 8;
  ...

Links

Crossrefs

Cf. A127647, A097806, A128541.

Programs

  • Magma
    [k eq n select Fibonacci(n) else k eq n-1 select Fibonacci(n) else 0: k in [1..n], n in [1..15]]; // G. C. Greubel, Jul 11 2019
  • Mathematica
    Table[If[k==n || k==n-1, Fibonacci[n], 0], {n, 15}, {k, n}]//Flatten (* G. C. Greubel, Jul 11 2019 *)
  • PARI
    T(n,k) = if(k==n || k==n-1, fibonacci(n), 0); \\ G. C. Greubel, Jul 11 2019
  • Sage
    def T(n, k):
    if (k==n): return fibonacci(n)
    elif (k==n-1): return fibonacci(n)
    else: return 0
[[T(n, k) for k in (1..n)] for n in (1..15)] # G. C. Greubel, Jul 11 2019

Formula

A127646 * A097806 as infinite lower triangular matrices.

A128619 Triangle T(n, k) = A127647(n,k) * A128174(n,k), read by rows.

Original entry on oeis.org

1, 0, 1, 2, 0, 2, 0, 3, 0, 3, 5, 0, 5, 0, 5, 0, 8, 0, 8, 0, 8, 13, 0, 13, 0, 13, 0, 13, 0, 21, 0, 21, 0, 21, 0, 21, 34, 0, 34, 0, 34, 0, 34, 0, 34, 0, 55, 0, 55, 0, 55, 0, 55, 0, 55
Offset: 1

Views

Author

Gary W. Adamson, Mar 14 2007

Keywords

Comments

This triangle is different from A128618, which is equal to A128174 * A127647.

Examples

			First few rows of the triangle are:
   1;
   0,  1;
   2,  0,  2;
   0,  3,  0,  3;
   5,  0,  5,  0,  5;
   0,  8,  0,  8,  0,  8;
  13,  0, 13,  0, 13,  0, 13;
   0, 21,  0, 21,  0, 21,  0, 21,
  ...

Links

Crossrefs

Cf. A000045, A096140, A127646, A128174, A128610, A128618, A128620.
Cf. A128620 (row sums).

Programs

  • Magma
    [((n+k+1) mod 2)*Fibonacci(n): k in [1..n], n in [1..15]]; // G. C. Greubel, Mar 17 2024
  • Mathematica
    Table[Fibonacci[n]*Mod[n+k+1,2], {n,15}, {k,n}]//Flatten (* G. C. Greubel, Mar 16 2024 *)
  • SageMath
    flatten([[((n+k+1)%2)*fibonacci(n) for k in range(1,n+1)] for n in range(1,16)]) # G. C. Greubel, Mar 17 2024

Formula

T(n, k) = A127647 * A128174, an infinite lower triangular matrix. In odd rows, n terms of F(n), 0, F(n),...; in the n-th row. In even rows, n terms of 0, F(n), 0,...; in the n-th row.
Sum_{k=1..n} T(n, k) = A128620(n-1).
From G. C. Greubel, Mar 16 2024: (Start)
T(n, k) = Fibonacci(n)*(1 + (-1)^(n+k))/2.
Sum_{k=1..n} (-1)^(k-1)*T(n, k) = (-1)^n*A128620(n-1).
Sum_{k=1..floor((n+1)/2)} T(n-k+1, k) = (1/2)*(1-(-1)^n)*A096140(floor((n + 1)/2)).
Sum_{k=1..floor((n+1)/2)} (-1)^(k-1)*T(n-k+1, k) = (1/2)*(1 - (-1)^n)*( Fibonacci(n-1) + (-1)^floor((n-1)/2) * Fibonacci(floor((n-3)/2)) ). (End)

A127645 a(n) = sum of first n terms of sequence A127644.

Original entry on oeis.org

3, 9, 18, 36, 48, 64, 72, 96, 128, 162, 192, 240, 255, 256, 270, 272, 289, 300, 320, 324, 350, 357, 378, 400, 405, 432, 442, 455, 480, 520, 539, 567, 600, 637, 660, 702, 740, 784, 819, 850, 891, 920, 966, 1015, 1054, 1104, 1140, 1183, 1230, 1275, 1326, 1380
Offset: 1

Views

Author

Leroy Quet, Jan 22 2007

Keywords

Comments

a(n) divides A127646(n).

Crossrefs

Cf. A127644, A127646.

Extensions

Extended by Ray Chandler, Jan 22 2007
Showing 1-4 of 4 results.

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