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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.

%I A299773 #54 Sep 27 2019 07:58:58
%S A299773 1,2,3,9,7,48,15,119,72,269,56,2740,101,1163,1208,5218,297,24319,490,
%T A299773 42150,6669,14098,1255,792335,5564,42501,30585,432413,4565,4513067,
%U A299773 6842,1251217,122818,317297,124253,54782479,21637,802541,445414,48590725,44583
%N 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.
%C A299773 If n is a noncomposite number (that is, 1 or prime), then a(n) = A000041(n).
%C A299773 For n >= 3, p(sigma(n-2)) < a(n) <= p(sigma(n-1)), where p(n) = A000041(n) and sigma(n) = A000203(n).
%H A299773 Amiram Eldar, <a href="/A299773/b299773.txt">Table of n, a(n) for n = 1..3200</a> (terms 1..700 from Andrew Howroyd)
%e A299773 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):
%e A299773 ------------------------------------------------------
%e A299773    k        Diagram        Partitions of 7
%e A299773 ------------------------------------------------------
%e A299773          _ _ _ _ _ _ _
%e A299773    1    |_| | | | | | |    [1, 1, 1, 1, 1, 1, 1]
%e A299773    2    |_ _| | | | | |    [2, 1, 1, 1, 1, 1]
%e A299773    3    |_ _ _| | | | |    [3, 1, 1, 1, 1]
%e A299773    4    |_ _|   | | | |    [2, 2, 1, 1, 1]
%e A299773    5    |_ _ _ _| | | |    [4, 1, 1, 1]
%e A299773    6    |_ _ _|   | | |    [3, 2, 1, 1]
%e A299773    7    |_ _ _ _ _| | |    [5, 1, 1]
%e A299773    8    |_ _|   |   | |    [2, 2, 2, 1]
%e A299773    9    |_ _ _ _|   | |    [4, 2, 1]       <---- Divisors of 4
%e A299773   10    |_ _ _|     | |    [3, 3, 1]
%e A299773   11    |_ _ _ _ _ _| |    [6, 1]
%e A299773   12    |_ _ _|   |   |    [3, 2, 2]
%e A299773   13    |_ _ _ _ _|   |    [5, 2]
%e A299773   14    |_ _ _ _|     |    [4, 3]
%e A299773   15    |_ _ _ _ _ _ _|    [7]
%e A299773 .
%t A299773 b[n_, k_] := b[n, k] = If[k < 1 || k > n, 0, If[n == k, 1, b[n, k + 1] + b[n - k, k]]];
%t A299773 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];
%t A299773 a[n_] := PartIndex[Divisors[n]];
%t A299773 a /@ Range[1, 100] (* _Jean-François Alcover_, Sep 27 2019, after _Andrew Howroyd_ *)
%o A299773 (PARI) a(n)={my(d=divisors(n)); vecsearch(vecsort(partitions(vecsum(d))), d)} \\ _Andrew Howroyd_, Jul 15 2018
%o A299773 (PARI) \\ here b(n,k) is A026807.
%o A299773 b(n,k)=polcoeff(1/prod(i=k, n, 1-x^i + O(x*x^n)), n)
%o A299773 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}
%o A299773 a(n)=PartIndex(divisors(n)); \\ _Andrew Howroyd_, Jul 15 2018
%Y A299773 Cf. A000040, A000041, A000203, A008578, A026807, A027750, A056538, A135010, A141285, A194446, A211992, A272024.
%K A299773 nonn
%O A299773 1,2
%A A299773 _Omar E. Pol_, Mar 25 2018
%E A299773 Terms a(8) and beyond from _Andrew Howroyd_, Jul 15 2018