A355403 -id:A355403 - cp's OEIS frontend

A355402 Maximal GCD of seven positive integers with sum n.

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

Offset: 7

Wesley Ivan Hurt, Jun 30 2022

Also largest divisor <= n/7 of n. - David A. Corneth, Jul 24 2022

Alois P. Heinz, Table of n, a(n) for n = 7..5000

Cf. A009641, A355403.

Maximal GCD of k positive integers with sum n for k = 2..10: A032742 (k=2,n>=2), A355249 (k=3), A355319 (k=4), A355366 (k=5), A355368 (k=6), this sequence (k=7), A354598 (k=8), A354599 (k=9), A354601 (k=10).

b:= proc(n, i, t) option remember; `if`(n=0, signum(t),
     `if`(min(i, t)<1, 1, max(b(n, i-1, t),
      igcd(b(n-i, min(n-i, i), t-1), i))))
    end:
a:= n-> `if`(n<7, 0, b(n$2, 7)):
seq(a(n), n=7..200);  # Alois P. Heinz, Jul 13 2022

b[n_, i_, t_] := b[n, i, t] = If[n == 0, Sign[t],
   If[Min[i, t] < 1, 1, Max[b[n, i - 1, t],
   GCD[b[n - i, Min[n - i, i], t - 1], i]]]];
a[n_] := If[n < 7, 0, b[n, n, 7]];
Table[a[n], {n, 7, 100}] (* Jean-François Alcover, Jul 24 2022, after Alois P. Heinz *)

a(n) = my(d = divisors(n)); d = select(x->x <= n\7,d); d[#d] \\ David A. Corneth, Jul 24 2022

A129647 Largest order of a permutation of n elements with exactly 2 cycles. Also the largest LCM of a 2-partition of n.

Original entry on oeis.org

0, 1, 2, 3, 6, 5, 12, 15, 20, 21, 30, 35, 42, 45, 56, 63, 72, 77, 90, 99, 110, 117, 132, 143, 156, 165, 182, 195, 210, 221, 240, 255, 272, 285, 306, 323, 342, 357, 380, 399, 420, 437, 462, 483, 506, 525, 552, 575, 600, 621, 650, 675, 702, 725, 756, 783, 812, 837

Offset: 1

Nickolas Reynolds (nickels(AT)gmail.com), Apr 25 2007

a(n) is asymptotic to (n^2)/4.

a(n) = A116921(n)*A116922(n). - Mamuka Jibladze, Aug 22 2019

			a(26) = 165 because 26 = 11+15 and lcm(11,15) = 165 is maximal.

Alois P. Heinz, Table of n, a(n) for n = 1..1000
Index entries for sequences related to lcm's
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,1,-2,1).

Cf. A000793, A129651.
Cf. A116921, A116922.
Maximal LCM of k positive integers with sum n for k = 2..7: this sequence (k=2), A129648 (k=3), A129649 (k=4), A129650 (k=5), A355367 (k=6), A355403 (k=7).

a:= n-> `if`(n<2, 0, max(seq(ilcm(i, n-i), i=1..n/2))):
seq(a(n), n=1..60);  # Alois P. Heinz, Feb 16 2013

Join[{0}, Rest[With[{n = 60}, Max[LCM @@@ IntegerPartitions[#, {2}]] & /@ Range[1, n]]]] (* Modified by Philip Turecek, Mar 25 2023 *)
a[n_] := If[n<2, 0, Max[Table[LCM[i, n-i], {i, 1, n/2}]]]; Table[a[n], {n, 1, 60}] (* Jean-François Alcover, Jul 15 2015, after Alois P. Heinz *)

G.f.: t^2*(1 + 2*t^3 - 5*t^4 + 8*t^5 - 4*t^6)/((1-t)^2*(1-t^4)). - Mamuka Jibladze, Aug 22 2019

A129648 Largest order of a permutation of n elements with exactly 3 cycles. Also the largest LCM of a 3-partition of n.

Original entry on oeis.org

0, 0, 1, 2, 3, 6, 6, 12, 15, 30, 21, 60, 35, 84, 105, 140, 84, 210, 165, 280, 315, 360, 385, 504, 495, 630, 693, 792, 819, 990, 1001, 1170, 1287, 1430, 1365, 1716, 1683, 2002, 2145, 2310, 2431, 2730, 2805, 3120, 3315, 3570, 3705, 4080, 4199, 4560, 4845, 5168

Offset: 1

Nickolas Reynolds (nickels(AT)gmail.com), Apr 25 2007

nonn

a(n) is asymptotic to (n^3)/27.

			a(9) = 15 because 9 = 5+3+1 and lcm(1,3,5) = 15 is maximal.

Alois P. Heinz, Table of n, a(n) for n = 1..1000
Index entries for sequences related to lcm's

Cf. A000793, A129651.

Maximal LCM of k positive integers with sum n for k = 2..7: A129647 (k=2), this sequence (k=3), A129649 (k=4), A129650 (k=5), A355367 (k=6), A355403 (k=7).

Max[LCM @@@ Compositions[ #, 3]] & /@ Range[1, n]

A129649 Largest order of a permutation of n elements with exactly 4 cycles. Also the largest LCM of a 4-partition of n.

Original entry on oeis.org

0, 0, 0, 1, 2, 3, 6, 6, 12, 15, 30, 30, 60, 60, 84, 105, 210, 140, 420, 210, 330, 420, 840, 420, 1260, 1155, 1540, 1365, 2520, 1320, 3080, 3465, 3960, 4095, 5544, 5005, 6930, 6435, 8190, 9009, 10296, 8415, 12870, 11781, 13464, 15015, 18018, 17017, 20592, 21879

Offset: 1

Nickolas Reynolds (nickels(AT)gmail.com), Apr 25 2007

nonn

a(n) is asymptotic to (n^4)/256.

			a(18)=140 because 18 = 7+5+2+2 and lcm(2,2,5,7) = 140 is maximal.

Philip Turecek, Table of n, a(n) for n = 1..1000
Index entries for sequences related to lcm's

Cf. A000793, A129651.

Maximal LCM of k positive integers with sum n for k = 2..7: A129647 (k=2), A129648 (k=3), this sequence (k=4), A129650 (k=5), A355367 (k=6), A355403 (k=7).

Max[LCM @@@ Compositions[ #, 4]] & /@ Range[1, n] (* needs Combinatorica *)
Join[{0,0,0},Table[Max[LCM@@#&/@IntegerPartitions[n,{4}]],{n,4,50}]] (* Harvey P. Dale, Feb 25 2012 *)

A129650 Largest order of a permutation of n elements with exactly 5 cycles. Also the largest LCM of a 5-partition of n.

Original entry on oeis.org

0, 0, 0, 0, 1, 2, 3, 6, 6, 12, 15, 30, 30, 60, 60, 84, 105, 210, 210, 420, 420, 420, 420, 840, 840, 1260, 1260, 2310, 1540, 4620, 2520, 5460, 4620, 9240, 5460, 13860, 9240, 16380, 15015, 27720, 13860, 32760, 19635, 40040, 45045, 51480, 32760, 72072, 58905

Offset: 1

Nickolas Reynolds (nickels(AT)gmail.com), Apr 25 2007

nonn

a(n) is asymptotic to n^5/3125.

			a(29)=1540 because 29 = 11+7+5+4+2 and lcm(2,4,5,7,11) = 1540 is maximal.

Index entries for sequences related to lcm's

Cf. A000793, A129651.

Maximal LCM of k positive integers with sum n for k = 2..7: A129647 (k=2), A129648 (k=3), A129649 (k=4), this sequence (k=5), A355367 (k=6), A355403 (k=7).

Max[LCM @@@ Compositions[ #, 5]] & /@ Range[1, n]

A355367 Maximal LCM of six positive integers with sum n.

Original entry on oeis.org

1, 2, 3, 6, 6, 12, 15, 30, 30, 60, 60, 84, 105, 210, 210, 420, 420, 420, 420, 840, 840, 1260, 1260, 2310, 2310, 4620, 4620, 5460, 5460, 9240, 9240, 13860, 13860, 16380, 16380, 30030, 27720, 60060, 32760, 40040, 60060, 120120, 60060, 180180, 120120, 157080, 120120, 360360

Offset: 6

Wesley Ivan Hurt, Jun 29 2022

nonn

Index entries for sequences related to lcm's

Cf. A008881.

Maximal LCM of k positive integers with sum n for k = 2..7: A129647 (k=2), A129648 (k=3), A129649 (k=4), A129650 (k=5), this sequence (k=6), A355403 (k=7).

Table[Max[LCM@@@IntegerPartitions[n,{6}]],{n,6,60}] (* Harvey P. Dale, Jun 23 2023 *)

a(n) = { my (v=0); forpart(p=n, v=max(v, lcm(Vec(p))),, [6,6]); v } \\ Rémy Sigrist, Jul 01 2022

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A355402 Maximal GCD of seven positive integers with sum n.

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A129647 Largest order of a permutation of n elements with exactly 2 cycles. Also the largest LCM of a 2-partition of n.

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A129648 Largest order of a permutation of n elements with exactly 3 cycles. Also the largest LCM of a 3-partition of n.

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A129649 Largest order of a permutation of n elements with exactly 4 cycles. Also the largest LCM of a 4-partition of n.

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A129650 Largest order of a permutation of n elements with exactly 5 cycles. Also the largest LCM of a 5-partition of n.

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A355367 Maximal LCM of six positive integers with sum n.

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