A166590 Totally multiplicative sequence with a(p) = p+2 for prime p.

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

Offset: 1

Author

Jaroslav Krizek, Oct 17 2009

Keywords

• nonn
• mult
• easy

Comments

From Peter Munn, Feb 16 2024: (Start)

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)

Examples

			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.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A001620, A003959.

Programs

Mathematica

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

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

Formula

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

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

From Vaclav Kotesovec, Feb 26 2023: (Start)

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)

Extensions

More terms from Michel Marcus, Jun 09 2014