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A166590
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Totally multiplicative sequence with a(p) = p+2 for prime p.
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23
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1, 4, 5, 16, 7, 20, 9, 64, 25, 28, 13, 80, 15, 36, 35, 256, 19, 100, 21, 112, 45, 52, 25, 320, 49, 60, 125, 144, 31, 140, 33, 1024, 65, 76, 63, 400, 39, 84, 75, 448, 43, 180, 45, 208, 175, 100, 49, 1280, 81, 196, 95, 240, 55, 500, 91, 576, 105, 124, 61, 560
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OFFSET
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1,2
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COMMENTS
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Consider the orthotope with sides given by the prime factors of n (including repetitions). a(n) is the sum of the sizes of all the orthotope's elements (vertices, edges, faces, ..., whole orthotope) with the size of a vertex taken to be 1. See the example.
If, instead, we identify congruent parallel elements (i.e., we use only one element with a given dimension and orientation) we get A003959.
(End)
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LINKS
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FORMULA
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Multiplicative with a(p^e) = (p+2)^e.
If n = Product p(k)^e(k) then a(n) = Product (p(k)+2)^e(k).
Dirichlet g.f.: Product_{primes p} 1 / (1 - p^(1-s) - 2*p^(-s)).
Dirichlet g.f.: zeta(s-1) * (1 + 2/(2^s - 4)) * Product_{primes p, p>2} (1 + 2/(p^s - p - 2)).
Let f(s) = Product_{primes p, p>2} (1 + 2/(p^s - p - 2)), then Sum_{k=1..n} a(k) has an average value n^2*(f(2)*(2*log(n) + 3*log(2) + 2*gamma - 1)/(8*log(2)) + f'(2)/(4*log(2))), where f(2) = Product_{primes p, p>2} (1 + 2/(p^2 - p - 2)) = 1.8687850774185607888850727174873699009051478019094666888484965828668606561..., f'(2) = f(2) * Sum_{primes p, p>2} (2*p*log(p) / (-p^3 + 2*p^2 + p - 2)) = -2.563594878667999839768204519417845474796924720924625514292420625983768019... and gamma is the Euler-Mascheroni constant A001620. (End)
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EXAMPLE
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For n = 12. 12 = 2 * 2 * 3, so we sum the sizes of the elements of a cuboid with base 2 X 2 and height 3.
Vertices: 8 of nominal size 1 8
Vertical edges: 4 of length 3 12
Horizontal edges: 8 of length 2 16
Total edge length: --- 28
Vertical faces: 4 of area 2 * 3 24
Horizontal faces: 2 of area 2 * 2 8
Total surface area: --- 32
Volume: n = 2 * 2 * 3 12
---
Vertices + lengths + areas + volume: 80
So a(12) = 80.
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MATHEMATICA
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a166590[n_] := {1}~Join~Rest[Times @@ Power @@@ Transpose[{Plus[First /@ FactorInteger@ #, 2], Last /@ FactorInteger@ #}] & /@ Range@n]; a166590[60] (* Michael De Vlieger, Jan 07 2015 *)
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PROG
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(PARI) a(n) = my(f=factor(n)); for (i=1, #f~, f[i, 1] += 2); factorback(f); \\ Michel Marcus, Jun 09 2014
(PARI) for(n=1, 100, print1(direuler(p=2, n, 1/(1-p*X-2*X))[n], ", ")) \\ Vaclav Kotesovec, Feb 10 2023
(Python)
from math import prod
from sympy import factorint
def A166590(n): return prod((p+2)**e for p, e in factorint(n).items()) # Chai Wah Wu, Dec 26 2022
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CROSSREFS
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KEYWORD
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nonn,mult,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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