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

1,2

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)

Links

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

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)

Example

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.

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 *)

Prog

(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

Crossrefs

Cf. A001620, A003959.

Sequence in context: A289742 A340850 A249113 * A368739 A244643 A085768

Adjacent sequences: A166587 A166588 A166589 * A166591 A166592 A166593

Keyword

nonn,mult,easy

Author

Jaroslav Krizek, Oct 17 2009

Extensions

More terms from Michel Marcus, Jun 09 2014

Status

approved