A225629 -id:A225629 - cp's OEIS frontend

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

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

Antti Karttunen, May 13 2013

nonn

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).

Cf. A003418, A225629, A225636, A225558, A225652, A225656.

(define (A225637 n) (/ (A003418 n) (A225629 n)))

A225630 Array of iterated Landau-like functions, starting maximization of LCM from the partition {1+1+...+1} of n, read downwards antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 1, 4, 6, 2, 1, 1, 1, 6, 12, 6, 2, 1, 1, 1, 6, 30, 12, 6, 2, 1, 1, 1, 12, 30, 60, 12, 6, 2, 1, 1, 1, 15, 84, 60, 60, 12, 6, 2, 1, 1, 1, 20, 120, 420, 60, 60, 12, 6, 2, 1, 1, 1, 30, 180, 840, 420, 60, 60, 12, 6, 2, 1, 1

Offset: 0

Antti Karttunen, May 13 2013

Row 0 consists of all 1's (corresponding to lcm(1,1,...,1) computed from the {1+1+...+1} partition), after which, on each succeeding row, the entry A(row,n) is computed by finding such a partition {p1+p2+...+pk} of n that value of lcm(A(row-1,n),p1,p2,...,pk) is maximized.
This will produce the ordinary Landau's function (A000793) for row 1, the "second order Landau's function" (A225627) for row 2, etc.
For each column n, only finite number of distinct values (A225634(n)) occur, after which the fixed point A003418(n) that has been reached repeats forever.

			Table begins:
  1, 1, 1, 1,  1,  1,  1,   1,   1,  1, ...
  1, 1, 2, 3,  4,  6,  6,  12,  15, 20, ...
  1, 1, 2, 6, 12, 30, 30,  84, 120, ...
  1, 1, 2, 6, 12, 60, 60, 420, 840, ...
  ...

Index entries for sequences related to lcm's

Transpose: A225631. Cf. also A225632, A225634.
Row 0: A000012, row 1: A000793, row 2: A225627, row 3: A225628. Cf. also A225629.
Rows converge towards A003418, which is also the main diagonal of this array.
See A225640 for a variant which uses a similar process, but where the "initial seed" in column n is n instead of 1.

(define (A225630 n) (A225630bi (A025581 n) (A002262 n)))
(define (A225630bi col row) (let ((maxlcm (list 0))) (let loop ((prevmaxlcm 1) (stepsleft row)) (if (zero? stepsleft) prevmaxlcm (begin (gen_partitions col (lambda (p) (set-car! maxlcm (max (car maxlcm) (apply lcm (cons prevmaxlcm p)))))) (loop (car maxlcm) (- stepsleft 1)))))))
(define (gen_partitions m colfun) (let recurse ((m m) (b m) (n 0) (partition (list))) (cond ((zero? m) (colfun partition)) (else (let loop ((i 1)) (recurse (- m i) i (+ 1 n) (cons i partition)) (if (< i (min b m)) (loop (+ 1 i))))))))

A225632 Irregular table read by rows: n-th row gives distinct values of successively iterated Landau-like functions for n, starting with the initial value 1.

Original entry on oeis.org

1, 1, 2, 1, 3, 6, 1, 4, 12, 1, 6, 30, 60, 1, 6, 30, 60, 1, 12, 84, 420, 1, 15, 120, 840, 1, 20, 180, 1260, 2520, 1, 30, 210, 840, 2520, 1, 30, 420, 4620, 13860, 27720, 1, 60, 660, 4620, 13860, 27720, 1, 60, 780, 8580, 60060, 180180, 360360

Offset: 1

Antti Karttunen, May 13 2013

The leftmost column of table (the initial term of each row, T(n,1)) is 1, corresponding to lcm(1,1,...,1) computed from the {1+1+...+1} partition of n, after which, on the same row, each further term T(n,i) is computed by finding such a partition [p1,p2,...,pk] of n so that value of lcm(T(n,i-1), p1,p2,...,pk) is maximized, until finally A003418(n) is reached, which will be listed as the last term of row n (as the result would not change after that, if we continued the same process).

			The first fifteen rows of table are:
  1;
  1,   2;
  1,   3,    6;
  1,   4,   12;
  1,   6,   30,    60;
  1,   6,   30,    60;
  1,  12,   84,   420;
  1,  15,  120,   840;
  1,  20,  180,  1260,   2520;
  1,  30,  210,   840,   2520;
  1,  30,  420,  4620,  13860,  27720;
  1,  60,  660,  4620,  13860,  27720;
  1,  60,  780,  8580,  60060, 180180, 360360;
  1,  84, 1260, 16380, 180180, 360360;
  1, 105, 4620, 60060, 180180, 360360;

Alois P. Heinz, Rows n = 1..150, flattened
Index entries for sequences related to lcm's

Cf. A225634 (length of n-th row), A000793 (n>=2 gives the second column).
Cf. A225629 (second largest/rightmost term of n-th row).
Cf. A003418 (largest/rightmost term of n-th row).
Cf. A225630, A225631, A225635, A212721.
Cf. A225642 (row n starts from n instead of 1).
Cf. A226055 (the first term common with A225642 on the n-th row).
Cf. A225638 (distance to that first common term from the beginning of the row n).
Cf. A226056 (number of trailing terms common with A225642 on the n-th row).

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:
T:= proc(n) option remember; local d, h, l, ll;
      l:= b(n$2); ll:= NULL; d:=1; h:=0;
      while d<>h do ll:= ll, d; h:= d;
        d:= max(seq(ilcm(h, i), i=l))
      od; ll
    end:
seq(T(n), n=1..20);  # Alois P. Heinz, May 29 2013

b[n_, i_] := b[n, i] = If[n==0, {1}, If[i<1, {}, Table[Map[Function[{x}, LCM[x, If[j==0, 1, i]]], b[n-i*j, i-1]], {j, 0, n/i}]]]; T[n_] := T[n] = Module[{d, h, l, ll}, l=b[n, n]; ll={}; d=1; h=0; While[d != h, AppendTo[ll, d]; h=d; d = Max[ Table[LCM[h, i], {i, l}]]]; ll]; Table[T[n], {n, 1, 20}] // Flatten (* Jean-François Alcover, Jul 29 2015, after Alois P. Heinz *)

A225627 a(n) = lcm(A000793(n),p1,p2,...,pk) for such a partition {p1+p2+...+pk} of n that maximizes this value among all partitions of n.

Original entry on oeis.org

1, 1, 2, 6, 12, 30, 30, 84, 120, 180, 210, 420, 660, 780, 1260, 4620, 5460, 5460, 5460, 9240, 13860, 13860, 16380, 32760, 120120, 180180, 180180, 235620, 180180, 471240, 1021020, 1021020, 1141140, 1141140, 2282280, 2282280, 4476780, 4476780, 6846840, 6846840

Offset: 0

Antti Karttunen, May 13 2013

nonn

Row 2 of A225630.

This could be called a "twice-iterated Landau's function."

Index entries for sequences related to lcm's

Cf. A000793, A225630, A225628, A225629, A225636.

(define (A225627 n) (let ((maxlcm (list 0))) (fold_over_partitions_of n (A000793 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))))))))

a(n) = A225636(n)*A000793(n).

A225628 a(n) = lcm(A225627(n),p1,p2,...,pk) for such a partition {p1+p2+...+pk} of n which maximizes this value among all partitions of n.

Original entry on oeis.org

1, 1, 2, 6, 12, 60, 60, 420, 840, 1260, 840, 4620, 4620, 8580, 16380, 60060, 60060, 92820, 92820, 175560, 263340, 360360, 360360, 753480, 2762760, 6126120, 6126120, 8953560, 6846840, 13665960, 58198140, 58198140, 78738660, 78738660, 157477320, 157477320

Offset: 0

Antti Karttunen, May 13 2013

nonn

Row 3 of A225630.

This could be called a "thrice-iterated Landau's function."

Index entries for sequences related to lcm's

Cf. A225630, A225627, A225629.

(define (A225628 n) (let ((maxlcm (list 0))) (fold_over_partitions_of n (A225627 n) lcm (lambda (p) (set-car! maxlcm (max (car maxlcm) p)))) (car maxlcm)))
;; Adapted by AK from Kreher & Stinson, CAGES-book, p. 68, Algorithm 3.1:
(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))))))))

A225633 Number of steps to reach a fixed point (A003418(n)), when starting from partition {1+1+1+...+1} of n and continuing with the process described in A225632.

Original entry on oeis.org

0, 0, 1, 2, 2, 3, 3, 3, 3, 4, 4, 5, 5, 6, 5, 5, 5, 6, 6, 7, 6, 6, 6, 7, 7, 7, 7, 8, 8, 8, 8, 9, 9, 9, 9, 9, 8, 9, 10, 10, 9, 10, 9, 11, 11, 11, 11, 12, 11, 12, 12, 12, 11, 12, 12, 12, 11, 11, 11, 12, 12, 13, 12, 12, 13, 13, 12, 13, 12, 12, 12, 13, 13, 14, 14

Offset: 0

Antti Karttunen, May 13 2013

nonn

a(0)=0, as its only partition is an empty partition {}, and by convention lcm()=1, thus it takes no steps to reach from 1 to A003418(0)=1.

The records occur at positions 0, 2, 3, 5, 9, 11, 13, 19, 27, 31, 38, 43, 47, 61, 73, 81, ... and they seem to occur in order, i.e., as A001477. Thus the record-positions probably also give the left inverse function for this sequence. It also seems that each integer occurs only finite times in this sequence, so there should be a right inverse function as well.

Cf. A225634, A225643, A225629, A225654, A226056.

a(n) = A225634(n) - 1.

cp's OEIS Frontend

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

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A225630 Array of iterated Landau-like functions, starting maximization of LCM from the partition {1+1+...+1} of n, read downwards antidiagonals.

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A225632 Irregular table read by rows: n-th row gives distinct values of successively iterated Landau-like functions for n, starting with the initial value 1.

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Mathematica

A225627 a(n) = lcm(A000793(n),p1,p2,...,pk) for such a partition {p1+p2+...+pk} of n that maximizes this value among all partitions of n.

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Formula

A225628 a(n) = lcm(A225627(n),p1,p2,...,pk) for such a partition {p1+p2+...+pk} of n which maximizes this value among all partitions of n.

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A225633 Number of steps to reach a fixed point (A003418(n)), when starting from partition {1+1+1+...+1} of n and continuing with the process described in A225632.

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