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User:Carlos Eduardo Olivieri/ideas/polyfromdivisors
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:
n | a(n) |
1 | 1 |
2 | 3 |
3 | 4 |
4 | 9 |
5 | 6 |
6 | 24 |
7 | 8 |
8 | 33 |
9 | 19 |
10 | 44 |
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:
n | p | a(n) |
4 | 2 | 9 |
9 | 3 | 19 |
25 | 5 | 51 |
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:
n | p | a(n) |
8 | 2 | 33 |
27 | 3 | 136 |
125 | 5 | 876 |
Graph (logarithmic scale):
Some initial prime terms:
a(n) | n |
3 | 2 |
19 | 9 |
2333 | 50 |
10369 | 64 |
1621 | 81 |
28978171 | 676 |
1182439 | 729 |
6502926954781901538281 | 900 |