A299773 a(n) is the index of the partition that contains the divisors of n in the list of colexicographically ordered partitions of the sum of the divisors of n.

1, 2, 3, 9, 7, 48, 15, 119, 72, 269, 56, 2740, 101, 1163, 1208, 5218, 297, 24319, 490, 42150, 6669, 14098, 1255, 792335, 5564, 42501, 30585, 432413, 4565, 4513067, 6842, 1251217, 122818, 317297, 124253, 54782479, 21637, 802541, 445414, 48590725, 44583

Offset: 1

Author

Omar E. Pol, Mar 25 2018

Keywords

• nonn

Comments

If n is a noncomposite number (that is, 1 or prime), then a(n) = A000041(n).

For n >= 3, p(sigma(n-2)) < a(n) <= p(sigma(n-1)), where p(n) = A000041(n) and sigma(n) = A000203(n).

Examples

			For n = 4 the sum of the divisors of 4 is 1 + 2 + 4 = 7. Then we have that, in list of colexicographically ordered partitions of 7, the divisors of 4 are in the 9th partition, so a(4) = 9 (see below):
------------------------------------------------------
   k        Diagram        Partitions of 7
------------------------------------------------------
         _ _ _ _ _ _ _
   1    |_| | | | | | |    [1, 1, 1, 1, 1, 1, 1]
   2    |_ _| | | | | |    [2, 1, 1, 1, 1, 1]
   3    |_ _ _| | | | |    [3, 1, 1, 1, 1]
   4    |_ _|   | | | |    [2, 2, 1, 1, 1]
   5    |_ _ _ _| | | |    [4, 1, 1, 1]
   6    |_ _ _|   | | |    [3, 2, 1, 1]
   7    |_ _ _ _ _| | |    [5, 1, 1]
   8    |_ _|   |   | |    [2, 2, 2, 1]
   9    |_ _ _ _|   | |    [4, 2, 1]       <---- Divisors of 4
  10    |_ _ _|     | |    [3, 3, 1]
  11    |_ _ _ _ _ _| |    [6, 1]
  12    |_ _ _|   |   |    [3, 2, 2]
  13    |_ _ _ _ _|   |    [5, 2]
  14    |_ _ _ _|     |    [4, 3]
  15    |_ _ _ _ _ _ _|    [7]
.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..3200 (terms 1..700 from Andrew Howroyd)

Crossrefs

Cf. A000040, A000041, A000203, A008578, A026807, A027750, A056538, A135010, A141285, A194446, A211992, A272024.

Programs

Mathematica

b[n_, k_] := b[n, k] = If[k < 1 || k > n, 0, If[n == k, 1, b[n, k + 1] + b[n - k, k]]];
PartIndex[v_] := Module[{s = 1, t = 0}, For[i = Length[v], i >= 1, i--, t += v[[i]]; s += b[t, If[i == 1, 1, v[[i - 1]]]] - b[t, v[[i]]]]; s];
a[n_] := PartIndex[Divisors[n]];
a /@ Range[1, 100] (* Jean-François Alcover, Sep 27 2019, after Andrew Howroyd *)

PARI

a(n)={my(d=divisors(n)); vecsearch(vecsort(partitions(vecsum(d))), d)} \\ Andrew Howroyd, Jul 15 2018

PARI

\\ here b(n,k) is A026807.
b(n,k)=polcoeff(1/prod(i=k, n, 1-x^i + O(x*x^n)), n)
PartIndex(v)={my(s=1,t=0); forstep(i=#v, 1, -1, t+=v[i]; s+=b(t, if(i==1, 1, v[i-1])) - b(t, v[i])); s}
a(n)=PartIndex(divisors(n)); \\ Andrew Howroyd, Jul 15 2018

Extensions

Terms a(8) and beyond from Andrew Howroyd, Jul 15 2018