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User:Carlos Eduardo Olivieri/ideas/polyfromdivisors

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A polynomial from divisors


Let { d_1, d_2, ..., d_k } be the divisors of n. And a(n) = d_k^1 + d_(k-1)^2 + ... + d_1^k. For example, for n = 6: a(6) = 6^1 + 3^2 + 2^3 + 1^4 = 24.


So we have the following initial terms for the sequence A264786:

na(n)
11
23
34
49
56
624
78
833
919
1044


Mathematica code:

a[n_] := Sum[Sort[Divisors[n], #1 > #2 &]i^i, {i, DivisorSigma[0, n]}];

Table[{n, a[n]}, {n, 6000}]


Some notes:

When n is a prime power :

Its divisors are:


{ 1, p, ..., , }


Obviously, for k = 1, a(n) = n + 1.


For squares (k = 2), follows:

p^2^1 + p^(2-1)^2 + 1^3 = 2*p^2 + 1


Examples:

npa(n)
429
9319
25551


For cubes (k = 3), follows:

p^3^1 + p^(3-1)^2 + p^(3-2)^3 + 1^4 = p^4 + 2*p^3 + 1


Examples:

npa(n)
8233
273136
1255876


Graph (logarithmic scale):

LogPlot.png

Some initial prime terms:

a(n)n
32
199
233350
1036964
162181
28978171676
1182439729
6502926954781901538281900