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 1-4 of 4 results.

A093431 a(n) = Sum_{k=1..n} (lcm(n,n-1,...,n-k+2,n-k+1)/lcm(1,2,...,k)).

Original entry on oeis.org

1, 3, 7, 13, 31, 38, 113, 165, 265, 420, 1607, 1004, 3979, 6893, 4205, 8665, 40903, 49558, 315477, 162320, 79179, 269877, 1647123, 937552, 1810091, 8445653, 7791355, 3978237, 33071543, 19578860, 283536169, 327438713, 117635955, 742042966, 154748983, 88779588
Offset: 1

Views

Author

Amarnath Murthy, Mar 31 2004

Keywords

Comments

The following sequences all appear to have the same parity: A003071, A029886, A061297, A092524, A093431, A102393, A104258, A122248, A128975. - Jeremy Gardiner, Dec 28 2008

Links

Crossrefs

Cf. A093430, A093432, A093433.
Row sums of A093430.

Programs

  • Maple
    a:= n-> add(ilcm(seq(i,i=n-k+1..n))/ilcm(seq(j,j=1..k)),k=1..n): seq(a(n),n=1..40); # Emeric Deutsch, Jan 30 2006
# second Maple program:
b:= proc(n) option remember; `if`(n=1, 1, ilcm(b(n-1), n)) end:
a:= proc(n) option remember; local k, r, s; r, s:= 0, 1;
      for k to n do s:= ilcm(s,n-k+1); r:= r+s/b(k) od; r
    end:
seq(a(n), n=0..40);  # Alois P. Heinz, Mar 17 2018
  • Mathematica
    Table[Sum[(LCM@@(n-Range[0,k-1])/LCM@@Range[k]),{k,n}],{n,33}] (* Jayanta Basu, May 22 2013 *)

Extensions

Corrected and extended by Emeric Deutsch, Jan 30 2006

A093430 Triangle read by rows: T(n,k) = lcm(n, n-1, ..., n-k+2, n-k+1)/lcm(1, 2, ..., k) (1 <= k <= n).

Original entry on oeis.org

1, 2, 1, 3, 3, 1, 4, 6, 2, 1, 5, 10, 10, 5, 1, 6, 15, 10, 5, 1, 1, 7, 21, 35, 35, 7, 7, 1, 8, 28, 28, 70, 14, 14, 2, 1, 9, 36, 84, 42, 42, 42, 6, 3, 1, 10, 45, 60, 210, 42, 42, 6, 3, 1, 1, 11, 55, 165, 330, 462, 462, 66, 33, 11, 11, 1, 12, 66, 110, 165, 66, 462, 66, 33, 11, 11, 1, 1, 13
Offset: 1

Views

Author

Amarnath Murthy, Mar 31 2004

Keywords

Comments

An LCM-analog of the binomial coefficients. - N. J. A. Sloane, Aug 26 2015

Examples

			T(7,3) = lcm(7,6,5)/lcm(1,2,3) = 210/6 = 35.
Triangle starts:
  1;
  2,  1;
  3,  3,  1;
  4,  6,  2,  1;
  5, 10, 10,  5,  1;
  6, 15, 10,  5,  1,  1;
  ...

Links

Crossrefs

Cf. A067049 (same triangle with an additional leading column of ones).
Cf. A093431, A093432, A093433.
Row sums yield A093431.

Programs

Extensions

More terms from Emeric Deutsch, Jan 30 2006

A093432 a(n) = lcm_{k=1..n} (lcm(n,n-1,...,n-k+2,n-k+1)/lcm(1,2,...,k)).

Original entry on oeis.org

1, 2, 3, 12, 10, 30, 105, 280, 252, 1260, 2310, 4620, 4290, 6006, 15015, 240240, 680680, 6126120, 11639628, 2771340, 1763580, 19399380, 223092870, 178474296, 171609900, 743642900, 1434168450, 20078358300, 19409079690, 19409079690, 300840735195, 875173047840
Offset: 1

Views

Author

Amarnath Murthy, Mar 31 2004

Keywords

Examples

			a(4) = lcm(lcm(4)/lcm(1), lcm(4,3)/lcm(1,2), lcm(4,3,2)/lcm(1,2,3), lcm(4,3,2,1)/lcm(1,2,3,4)) = lcm(4,6,2,1) = 12.

Links

Crossrefs

Cf. A093430, A093431, A093433.
LCM of the terms in row n of the triangle in A093430.

Programs

  • Maple
    T:=(n,k)->lcm(seq(i,i=n-k+1..n))/lcm(seq(j,j=1..k)): seq(lcm(seq(T(n,k),k=1..n)),n=1..35); # Emeric Deutsch, Jan 30 2006
# second Maple program:
b:= proc(n) option remember; `if`(n=1, 1, ilcm(b(n-1), n)) end:
a:= proc(n) option remember; local k, r, s; r, s:= 1, 1;
      for k to n do s:= ilcm(s,n-k+1); r:= ilcm(r,s/b(k)) od; r
    end:
seq(a(n), n=1..40);  # Alois P. Heinz, Mar 17 2018
  • Mathematica
    b[n_] := b[n] = If[n == 1, 1, LCM[b[n - 1], n]];
a[n_] := a[n] = Module[{k, r = 1, s = 1}, For[k = 1, k <= n, k++, s = LCM[s, n - k + 1]; r = LCM[r, s/b[k]]]; r];
Array[a, 40] (* Jean-François Alcover, Jun 18 2018, after Alois P. Heinz *)

Extensions

Corrected and extended by Emeric Deutsch, Jan 30 2006

A093429 Number of distinct prime factors of (prime(1)*...*prime(n))+(prime(n+1)*...*prime(2n)), where prime(n) is the n-th prime.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 2, 3, 2, 3, 2, 2, 3, 2, 2, 4, 3, 2, 6, 3, 4, 4, 3, 1, 1, 3, 3, 3, 3, 2, 4, 3, 3, 3, 3, 5, 4, 2, 3, 3, 5, 3, 7, 4, 1, 4, 3, 4, 3, 6, 2, 4, 3, 3
Offset: 1

Views

Author

Jason Earls, May 12 2004

Keywords

Comments

Prime for n = 1, 2, 3, 4, 24, 25, 45, 59 and no more for n < 100 (A329532).

Examples

			a(31)=4 because 509102378439545188849067644696085192959414195658632710736111053092210207
= 3711597629 * 238694867020723 * 226814268663739929299 * 2533557617597929944840907379.

Links

Crossrefs

Cf. A001221, A002110, A093433, A329532.

Programs

  • Mathematica
    PrimeFactors[n_] := Flatten[ Table[ # [[1]], {1} ] & /@ FactorInteger[n]]; f[n_] := Length[ PrimeFactors[ Product[Prime[i], {i, n}] + Product[Prime[i + n], {i, n}]]]; Table[ f[n], {n, 20}]
  • PARI
    a(n) = omega(prod(k=1, n, prime(k)) + prod(k=n+1, 2*n, prime(k))); \\ Daniel Suteu, Nov 26 2019

Formula

a(n) = A001221(A002110(n) + A002110(2*n) / A002110(n)). - Daniel Suteu, Nov 26 2019

Extensions

a(40)-a(48) from Robert G. Wilson v, May 27 2004
a(49)-a(54) from Daniel Suteu, Nov 26 2019
Showing 1-4 of 4 results.

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