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 1-5 of 5 results.

A280377 Partial sums of A119619.

Original entry on oeis.org

1, 2, 4, 7, 31, 41, 761, 1076, 5556, 10092, 3638892, 3650442, 482652042, 496551642, 539602650, 1178115525, 20923968003525, 20925873907525, 6423299579635525, 6423418373678893, 6491526825518893, 11027299390478893, 1124011755076998158893, 1124011800169644507643
Offset: 1

Views

Author

Jaroslav Krizek, Jan 07 2017

Keywords

Comments

A119619(n) = Product_{d|n} pxi(d), where pxi(m) is the product of totatives of m (A001783).

Crossrefs

Cf. A119619, A280378.

Programs

  • Magma
    [&+[&*[&*[h: h in [1..d] | GCD(h,d) eq 1]: d in Divisors(k)]: k in [1..n]]: n in [1..100]]

Formula

a(n) = Sum_{i=1..n} A119619(i).

A280378 Partial products of A119619.

Original entry on oeis.org

1, 1, 2, 6, 144, 1440, 1036800, 326592000, 1463132160000, 6636767477760000, 24083501823295488000000, 278164446059062886400000000, 133241214725404817086218240000000000, 1851999588197236795571599048704000000000000, 79730449087475946864047275218548293632000000000000
Offset: 1

Views

Author

Jaroslav Krizek, Jan 08 2017

Keywords

Comments

A119619(n) = Product_{d|n} pxi(d), where pxi(m) is the product of totatives of m (A001783).

Crossrefs

Cf. A119619, A280377.

Programs

  • Magma
    [&*[&*[&*[h: h in [1..d] | GCD(h,d) eq 1]: d in Divisors(k)]: k in [1..n]]: n in [1..100]]
  • Mathematica
    FoldList[#1 #2 &, Table[Product[Times @@ Select[Range@ d, CoprimeQ[d, #] &], {d, Divisors@ n}], {n, 15}]] (* Michael De Vlieger, Jan 09 2017 *)

Formula

a(n) = Product_{i=1..n} A119619(i).

A203904 Triangular array T; for n>0, row n shows the coefficients of a reduced polynomial having zeros -k/(n+1) for k=1,2,...,n.

Original entry on oeis.org

1, 1, 2, 2, 9, 9, 3, 22, 48, 32, 24, 250, 875, 1250, 625, 10, 137, 675, 1530, 1620, 648, 720, 12348, 79576, 252105, 420175, 352947, 117649, 315, 6534, 52528, 216608, 501760, 659456, 458752, 131072, 4480, 109584, 1063116, 5450004, 16365321
Offset: 1

Views

Author

Clark Kimberling, Jan 08 2012

Keywords

Comments

For n>0, the zeros of the polynomial represented by row n+1 interlace the zeros of the polynomial for row n; see the Example section.
...
T(n,1): A119619
T(n,n): A056916.

Examples

			First five rows(counting the top row as row 0):
1
1...2.................representing 1+2x
1...9...9.............representing 2+9x+9x^2
3...22..48...32
24...250...875...1250...625
Zeros corresponding to rows 1 to 4:
.................-1/2
............-2/3......-1/3
......-3/4.......-1/2.......-1/4
-4/5........-3/5......-2/5.......-1/5
Interlace property for successive rows illustrated by
  1/5 < 1/4 < 2/5 < 1/2 < 3/5 < 3/4 < 4/5.

Crossrefs

Cf. A056856, A119619, A056916, A007305/A007306 (Farey fractions).

Programs

  • Mathematica
    p[n_, x_] := Product[(n*x + k)/GCD[n, k], {k, 1, n - 1}]
Table[CoefficientList[p[n, x], x], {n, 1, 10}]
TableForm[%]  (* A203904 triangle *)
Flatten[%%]   (* A203904 sequence *)

A322550 Table read by ascending antidiagonals: T(n, k) is the minimum number of cubes necessary to fill a right square prism with base area n^2 and height k.

Original entry on oeis.org

1, 4, 2, 9, 1, 3, 16, 18, 12, 4, 25, 4, 1, 2, 5, 36, 50, 48, 36, 20, 6, 49, 9, 75, 1, 45, 3, 7, 64, 98, 4, 100, 80, 2, 28, 8, 81, 16, 147, 18, 1, 12, 63, 4, 9, 100, 162, 192, 196, 180, 150, 112, 72, 36, 10, 121, 25, 9, 4, 245, 1, 175, 2, 3, 5, 11, 144, 242, 300, 324, 320, 294, 252, 200, 144, 90, 44, 12
Offset: 1

Views

Author

Stefano Spezia, Dec 15 2018

Keywords

Examples

			The table T starts in row n = 1 with columns k >= 1 as:
   1     2     3     4     5     6     7     8     9 ...
   4     1    12     2    20     3    28     4    36 ...
   9    18     1    36    45     2    63    72     3 ...
  16     4    48     1    80    12   112     2   144 ...
  25    50    75   100     1   150   175   200   225 ...
  36     9     4    18   180     1   252    36    12 ...
  49    98   147   196   245   294     1   392   441 ...
  64    16   192     4   320    48   448     1   576 ...
  81   162     9   324   405    18   567   648     1 ...
...
The triangle X(n, k) begins
  n\k|   1     2     3     4     5     6     7     8     9
  ---+----------------------------------------------------
   1 |   1
   2 |   4     2
   3 |   9     1     3
   4 |  16    18    12     4
   5 |  25     4     1     2     5
   6 |  36    50    48    36    20     6
   7 |  49     9    75     1    45     3     7
   8 |  64    98     4   100    80     2    28     8
   9 |  81    16   147    18     1    12    63     4     9
...

Links

Crossrefs

Cf. A000012 (main diagonal of the table), A000027 (1st row of the table or diagonal of the triangle), A000290 (k=1), A000578, A011379 (superdiagonal of the table), A045991 (subdiagonal of the table), A050873, A119619, A320043 (row sums of the triangle).

Programs

  • GAP
    Flat(List([1..12], n->List([1..n], k->(n+1-k)^2*k/GcdInt(n+1-k,k)^3)));
  • Magma
    [[(n+1-k)^2*k/Gcd(n+1-k,k)^3: k in [1..n]]: n in [1..12]]; // triangle output
  • Maple
    a := (n, k) -> (n+1-k)^2*k/gcd(n+1-k, k)^3: seq(seq(a(n, k), k = 1 .. n), n = 1 .. 12)
  • Mathematica
    T[n_,k_]:=n^2*k/GCD[n,k]^3; Flatten[Table[T[n-k+1,k], {n, 12}, {k, n}]]
  • Maxima
    sjoin(v, j) := apply(sconcat, rest(join(makelist(j, length(v)), v)))$ display_triangle(n) := for i from 1 thru n do disp(sjoin(makelist((i+1-j)^2*j/gcd(i+1-j,j)^3, j, 1, i), " ")); display_triangle(12);
  • PARI
    T(n, k) = (n+1-k)^2*k/gcd(n+1-k,k)^3;
tabl(nn) = for(i=1, nn, for(j=1, i, print1(T(i, j), ", ")); print);
tabl(12) \\ triangle output

Formula

T(n, k) = n^2*k/gcd(n, k)^3.
T(n, k) = A000290(n)*k/A000578(A050873(n,k)).
X(n, k) = T(n + 1 - k, k).
X(2*n - 1, n) = A000012(n).
Product_{k=1..n} X(n, k)^(1/3) = A119619(n+1). - Stefano Spezia, Jun 24 2024

A308944 a(n) = Product_{k=1..n} lcm(n,k) / (k * gcd(n,k)).

Original entry on oeis.org

1, 1, 3, 4, 125, 9, 16807, 1024, 59049, 15625, 2357947691, 5184, 1792160394037, 282475249, 474609375, 17179869184, 2862423051509815793, 3486784401, 5480386857784802185939, 250000000000, 10382917022245341, 5559917313492231481, 39471584120695485887249589623
Offset: 1

Views

Author

Ilya Gutkovskiy, Jul 01 2019

Keywords

Crossrefs

Cf. A000010, A051190, A056916, A067911, A071248, A119619, A127553.

Programs

  • Mathematica
    Table[Product[LCM[n, k]/(k GCD[n, k]), {k, 1, n}], {n, 1, 23}]
Table[Product[d^(EulerPhi[d] - EulerPhi[n/d]), {d, Divisors[n]}], {n, 1, 23}]
  • PARI
    a(n) = prod(k=1, n, lcm(n, k)/(k*gcd(n, k))); \\ Michel Marcus, Jul 02 2019

Formula

a(n) = Product_{d|n} d^(phi(d)-phi(n/d)).
a(n) = n^n / Product_{d|n} d^(2*phi(n/d)).
a(n) = n^(-n) * Product_{d|n} d^(2*phi(d)).
a(n) = n^n / Product_{k=1..n} gcd(n,k)^2.
a(n) = n^(-n) * Product_{k=1..n} lcm(n,k)^2/k^2.
a(n) = A127553(n)/n!.
a(n) = A056916(n)/A067911(n).
a(p) = p^(p-2), where p is a prime.
Showing 1-5 of 5 results.

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