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.

A225636 a(n) = A225627(n)/A000793(n).

Original entry on oeis.org

1, 1, 1, 2, 3, 5, 5, 7, 8, 9, 7, 14, 11, 13, 15, 44, 39, 26, 26, 22, 33, 33, 39, 39, 143, 143, 143, 153, 78, 187, 221, 221, 209, 209, 247, 247, 323, 323, 418, 418, 391, 646, 437, 437, 646, 969, 969, 969, 969, 782, 874, 874, 1292, 667, 713, 713, 782, 5681, 3496
Offset: 0

Views

Author

Antti Karttunen, May 13 2013

Keywords

Comments

a(n) divides A225558(n) for all n.

Crossrefs

Programs

A225655 a(n) = largest LCM of partitions of n divisible by n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 30, 11, 60, 13, 84, 105, 16, 17, 180, 19, 420, 420, 330, 23, 840, 25, 780, 27, 1540, 29, 4620, 31, 32, 4620, 3570, 9240, 13860, 37, 7980, 16380, 27720, 41, 32760, 43, 60060, 45045, 19320, 47, 55440, 49, 23100, 157080, 180180, 53
Offset: 1

Views

Author

Antti Karttunen, May 19 2013

Keywords

Comments

a(n) = lcm(p1,p2,...,pk) for that partition of n for which the LCM is a multiple of n, and which maximizes this value among all such partitions [p1,p2,...,pk] of n.

Crossrefs

For all n, a(A225651(n)) = A000793(A225651(n)).
A225657 lists the values of n for which a(n) = n.

Programs

  • Maple
    b:= proc(n, i) option remember; `if`(n=0, {1},
          `if`(i<1, {}, {seq(map(x->ilcm(x, `if`(j=0, 1, i)),
           b(n-i*j, i-1))[], j=0..n/i)}))
        end:
    a:= n-> max(select(x-> irem(x, n)=0, b(n$2))[]):
    seq(a(n), n=1..50);  # Alois P. Heinz, May 26 2013
  • Mathematica
    b[n_, i_] := b[n, i] = If[n==0, {1}, If[i<1, {}, Union @ Flatten @ Table[ Map[ Function[{x}, LCM[x, If[j==0, 1, i]]], b[n-i*j, i-1]], {j, 0, n/i}]]]; a[n_] := Max[Select[b[n, n], Mod[#, n]==0&]]; Table[a[n], {n, 1, 50}] (* Jean-François Alcover, Jul 29 2015, after Alois P. Heinz *)

A225646 a(n) = lcm(n,p1,p2,...,pk) for such a partition of n which maximizes this value among all partitions {p1+p2+...pk} of n.

Original entry on oeis.org

1, 1, 2, 6, 12, 30, 30, 84, 120, 180, 210, 330, 420, 780, 630, 840, 1680, 3570, 1386, 7980, 1980, 4620, 6930, 19320, 9240, 23100, 30030, 41580, 16380, 73080, 10920, 143220, 110880, 120120, 157080, 120120, 180180, 512820, 311220, 240240, 360360, 1231230, 180180
Offset: 0

Views

Author

Antti Karttunen, May 15 2013

Keywords

Comments

Second row of table A225640.
a(0)=1 by convention.

Crossrefs

Programs

  • Scheme
    (define (A225646 n) (let ((maxlcm (list 1))) (fold_over_partitions_of n n lcm (lambda (p) (set-car! maxlcm (max (car maxlcm) p)))) (car maxlcm)))
    (define (fold_over_partitions_of m initval addpartfun colfun) (let recurse ((m m) (b m) (n 0) (partition initval)) (cond ((zero? m) (colfun partition)) (else (let loop ((i 1)) (recurse (- m i) i (+ 1 n) (addpartfun i partition)) (if (< i (min b m)) (loop (+ 1 i))))))))

A225656 a(n) = (1/n) * lcm(p1,p2,...,pk) for such a partition of n that this formula produces an integer and maximizes this value among all such partitions [p1,p2,...,pk] of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 5, 1, 6, 7, 1, 1, 10, 1, 21, 20, 15, 1, 35, 1, 30, 1, 55, 1, 154, 1, 1, 140, 105, 264, 385, 1, 210, 420, 693, 1, 780, 1, 1365, 1001, 420, 1, 1155, 1, 462, 3080, 3465, 1, 770, 6552, 6435, 5460, 1260, 1, 17017, 1, 2310, 12155
Offset: 1

Views

Author

Antti Karttunen, May 19 2013

Keywords

Crossrefs

Cf. A225652, A225636, A225657 (position of ones).

Programs

  • Maple
    b:= proc(n, i) option remember; `if`(n=0, {1},
          `if`(i<1, {}, {seq(map(x->ilcm(x, `if`(j=0, 1, i)),
           b(n-i*j, i-1))[], j=0..n/i)}))
        end:
    a:= n-> max(select(x-> irem(x, n)=0, b(n$2))[])/n:
    seq(a(n), n=1..50);  # Alois P. Heinz, May 25 2013
  • Mathematica
    b[n_, i_] := b[n, i] = If[n==0, {1}, If[i<1, {}, Union @ Flatten @ Table[ Map[ Function[{x}, LCM[x, If[j==0, 1, i]]], b[n-i*j, i-1]], {j, 0, n/i}]]]; a[n_] := Max[Select[b[n, n], Mod[#, n]==0&]]/n; Table[a[n], {n, 1, 50}] (* Jean-François Alcover, Jul 29 2015, after Alois P. Heinz *)
  • Scheme
    (define (A225656 n) (/ (A225655 n) (max 1 n)))

Formula

a(n) = A225655(n)/n.

A225637 a(n) = A003418(n)/A225629(n).

Original entry on oeis.org

1, 1, 2, 2, 3, 2, 2, 5, 7, 2, 3, 2, 2, 2, 2, 2, 3, 2, 2, 2, 4, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 2, 3, 2, 2, 2, 3, 3, 2, 2, 3, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 6, 6, 2, 2, 2, 2, 4, 4, 4, 4, 2, 2, 2, 2, 2, 2, 4, 4, 4, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2
Offset: 0

Views

Author

Antti Karttunen, May 13 2013

Keywords

Comments

For n >= 2, a(n) is the final factor by which the A225629(n) needs to be multiplied that it finally reaches the fixed point A003418(n) of the column n of A225630.
The first composite, 4, occurs at n=20. The first composite which is not power of prime, namely 6, occurs at n=61.
For all n >= 3, a(n) divides A225558(n).

Crossrefs

Programs

Showing 1-5 of 5 results.