A166590 - OEIS

%I #33 Feb 16 2024 15:27:40

%S 1,4,5,16,7,20,9,64,25,28,13,80,15,36,35,256,19,100,21,112,45,52,25,

%T 320,49,60,125,144,31,140,33,1024,65,76,63,400,39,84,75,448,43,180,45,

%U 208,175,100,49,1280,81,196,95,240,55,500,91,576,105,124,61,560

%N Totally multiplicative sequence with a(p) = p+2 for prime p.

%C From _Peter Munn_, Feb 16 2024: (Start)

%C 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.

%C If, instead, we identify congruent parallel elements (i.e., we use only one element with a given dimension and orientation) we get A003959.

%C (End)

%H Michael De Vlieger, <a href="/A166590/b166590.txt">Table of n, a(n) for n = 1..10000</a>

%F Multiplicative with a(p^e) = (p+2)^e.

%F If n = Product p(k)^e(k) then a(n) = Product (p(k)+2)^e(k).

%F From _Vaclav Kotesovec_, Feb 26 2023: (Start)

%F Dirichlet g.f.: Product_{primes p} 1 / (1 - p^(1-s) - 2*p^(-s)).

%F Dirichlet g.f.: zeta(s-1) * (1 + 2/(2^s - 4)) * Product_{primes p, p>2} (1 + 2/(p^s - p - 2)).

%F 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)

%e 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.

%e   Vertices: 8 of nominal size 1                 8

%e   Vertical edges: 4 of length 3         12

%e   Horizontal edges: 8 of length 2       16

%e    Total edge length:                  ---     28

%e   Vertical faces: 4 of area 2 * 3       24

%e   Horizontal faces: 2 of area 2 * 2      8

%e     Total surface area:                ---     32

%e   Volume: n = 2 * 2 * 3                        12

%e                                               ---

%e   Vertices + lengths + areas + volume:         80

%e So a(12) = 80.

%t a166590[n_] := {1}~Join~Rest[Times @@ Power @@@ Transpose[{Plus[First /@ FactorInteger@ #, 2], Last /@ FactorInteger@ #}] & /@ Range@n]; a166590[60] (* _Michael De Vlieger_, Jan 07 2015 *)

%o (PARI) a(n) = my(f=factor(n)); for (i=1, #f~, f[i,1] += 2); factorback(f); \\ _Michel Marcus_, Jun 09 2014

%o (PARI) for(n=1, 100, print1(direuler(p=2, n, 1/(1-p*X-2*X))[n], ", ")) \\ _Vaclav Kotesovec_, Feb 10 2023

%o (Python)

%o from math import prod

%o from sympy import factorint

%o def A166590(n): return prod((p+2)**e for p, e in factorint(n).items()) # _Chai Wah Wu_, Dec 26 2022

%Y Cf. A001620, A003959.

%K nonn,mult,easy

%O 1,2

%A _Jaroslav Krizek_, Oct 17 2009

%E More terms from _Michel Marcus_, Jun 09 2014