A211218 -id:A211218 - cp's OEIS frontend

A211219 Maximum value of sigma(x) * sigma(y) * sigma(z), where x + y + z = n.

1, 3, 9, 27, 36, 63, 84, 147, 196, 343, 336, 588, 576, 1008, 864, 1728, 1152, 2160, 1872, 2700, 2340, 4032, 2925, 5040, 4368, 6300, 5460, 9408, 6552, 11760, 10192, 14112, 10080, 21952, 12096, 18816, 18816, 24304, 16128, 30576, 20832, 32928, 26208, 33852

Offset: 3

Paolo P. Lava, Apr 05 2012

nonn

			For n=79 the maximum product is reached when 24, 27 and 28 (24+27+28=79) are considered: sigma(24)*sigma(27)*sigma(28) = 60*40*56 = 134400.
For n=83 the maximum product is reached when 24, 24 and 35 (24+24+35=83) are considered: sigma(24)*sigma(24)*sigma(35) = 60*60*48 = 172800.

Paolo P. Lava, Table of n, a(n) for n = 3..500

Cf. A211216-A211218.

with(numtheory);
A211219:=proc(i)
local a,b,c,d,m,n,s;
for n from 3 to i do
  s:=0; a:=0; b:=0; c:=n;
  while a<=floor(n/3) do
    while b<=floor((n-a)/2) do
      for m from 1 to 3 do d:=sigma(a)*sigma(b)*sigma(c); od;
      if d>s then s:=d; fi; b:=b+1; c:=c-1;
    od;
    a:=a+1; b:=a; c:=n-a-b;
  od;
  print(s);
od; end:
A211219(1000);

a[n_] := Max[Times @@ DivisorSigma[1, #]& /@ IntegerPartitions[n, {3}]]; Table[a[n], {n, 3, 100}] (* Jean-François Alcover, Dec 26 2013 *)

A211217 Maximum of sigma(x) * sigma(y), where x + y = n.

Original entry on oeis.org

1, 3, 9, 12, 21, 28, 49, 48, 84, 72, 144, 96, 180, 156, 225, 195, 336, 234, 420, 364, 504, 360, 784, 432, 672, 672, 868, 576, 1092, 744, 1176, 936, 1209, 1008, 1680, 992, 1638, 1440, 1860, 1344, 2340, 1344, 2520, 1920, 2232, 1680, 3600, 1860, 3024, 2400

Offset: 2

Paolo P. Lava, Apr 05 2012

nonn

			For n=83 the maximum product is reached when 35 and 48 (35+48=83) are considered: sigma(35)*sigma(48) = 48*124 = 5952.
For n=99 the maximum product is reached when 36 and 63 (36+63=99) are considered: sigma(36)*sigma(63) = 91*104 = 9464.

Paolo P. Lava, Table of n, a(n) for n = 2..10000
Paolo P. Lava, Plot of the first 10000 terms of the sequence

Cf. A211216, A211218, A211219.

with(combstruct); with(numtheory);
A211217:=proc(i)
local a,b,j,n,t;
for n from 2 to i do
  t:=0;
  for j from 0 to floor(n/2) do
    a:=n-j; b:=sigma(j)*sigma(a); if b>t then t:=b; fi;
  od;
  print(t);
od; end:
A211217(1000);

a[n_] := Max[Times @@ DivisorSigma[1, #]& /@ IntegerPartitions[n, {2}]]; Table[a[n], {n, 2, 100}] (* Jean-François Alcover, Dec 26 2013 *)

A211221 For any partition of n consider the product of the sigma of each element. Sequence gives the maximum of such values.

Original entry on oeis.org

1, 3, 4, 9, 12, 27, 36, 81, 108, 243, 324, 729, 972, 2187, 2916, 6561, 8748, 19683, 26244, 59049, 78732, 177147, 236196, 531441, 708588, 1594323, 2125764, 4782969, 6377292, 14348907, 19131876, 43046721, 57395628, 129140163, 172186884, 387420489, 516560652

Offset: 1

Paolo P. Lava, Apr 13 2012

			For n=21 the partition (2,2,2,2,2,2,2,2,2,3) gives sigma(2)^9*sigma(3)=3^9*4=78732 that is the maximum value that can be reached.

Index entries for linear recurrences with constant coefficients, signature (0,3).

Cf. A000203, A085884, A211217, A211218, A211219, A211220.

with(numtheory); with(combinat);
A211221:=proc(q)
local b,c,i,j,k,m,n,t;
for n from 1 to q do
  k:=partition(n); b:=numbpart(n); m:=0;
  for i from 1 to b do
    c:=nops(k[i]); t:=1;
    for j from 1 to c do t:=t*sigma(k[i][j]); od; if t>m then m:=t; fi; od;
  print(m);
od; end:
A211221(100)

LinearRecurrence[{0,3},{1,3,4},40] (* Harvey P. Dale, Jun 06 2015 *)

For n>1, a(n) = 3^n/2 for n even and a(n) = 4*3^(n-3)/2 for n odd.
For n>3, a(n) = 3*a(n-2). G.f.: x*(1+3*x+x^2)/(1-3*x^2). [Colin Barker, Apr 18 2012]
Closed form: a(1)=1, then a(n) = 1/6*(7-(-1)^(n-2))*3^(1/4*(-1)^(n-2))*3^(1/2*(n-2))*27^(1/4) = 3^((2*n+(-1)^n-5)/4)*(7-(-1)^n)/2. [Paolo P. Lava, Apr 20 2012]

cp's OEIS Frontend

A211219 Maximum value of sigma(x) * sigma(y) * sigma(z), where x + y + z = n.

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A211217 Maximum of sigma(x) * sigma(y), where x + y = n.

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Author

Keywords

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Maple

Mathematica

A211221 For any partition of n consider the product of the sigma of each element. Sequence gives the maximum of such values.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

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