A384531 Multiplicative sequence a(n) with a(p^e) = ((2*e+1) * p - 2*e) * p^(e-1) for prime p and e >= 0.

1, 4, 7, 12, 13, 28, 19, 32, 33, 52, 31, 84, 37, 76, 91, 80, 49, 132, 55, 156, 133, 124, 67, 224, 105, 148, 135, 228, 85, 364, 91, 192, 217, 196, 247, 396, 109, 220, 259, 416, 121, 532, 127, 372, 429, 268, 139, 560, 217, 420, 343, 444, 157, 540, 403, 608, 385, 340, 175, 1092

Offset: 1

Author

Werner Schulte, Jun 01 2025

Keywords

• nonn
• easy
• mult

Links

• Vaclav Kotesovec, Graph - the asymptotic ratio (100000 terms)
• David Radcliffe, Notes on two arithmetical functions (Contains a proof that a(n) = Sum_{i=1..n} gcd(i, n) * gcd(i+1, n)).

Crossrefs

Cf. A018804, A173557.

Cf. A065473.

Programs

Maple

A384531 := proc(n)
    local a,pe,p,e;
    a :=1 ;
    for pe in ifactors(n)[2] do
        p := op(1,pe) ;
        e := op(2,pe) ;
        a := a*((2*e+1) * p - 2*e) * p^(e-1) ;
    end do:
    a ;
end proc:
seq(A384531(n),n=1..100) ;# R. J. Mathar, Jun 04 2025

Mathematica

f[p_, e_] := ((2*e+1)*p - 2*e)*p^(e-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jun 01 2025 *)

PARI

a(n)=my(f=factor(n)); prod(k=1,#f[,1],((2*f[k,2]+1)*f[k,1]-2*f[k,2])*f[k,1]^(f[k,2]-1))

Python

from math import prod
from sympy import factorint
def A384531(n): return prod((((m:=e<<1)|1)*p-m)*p**(e-1) for p, e in factorint(n).items()) # Chai Wah Wu, Jun 03 2025

Formula

Dirichlet g.f.: Sum_{n > 0} a(n) / n^s = (zeta(s-1))^2 * Product_{p prime} (1 + (p-2) / p^s).

Dirichlet convolution of A018804 and A173557.

Conjecture: a(n) = Sum_{i=1..n} gcd(i, n) * gcd(i+1, n).

From Vaclav Kotesovec, Jun 04 2025: (Start)

Let f(s) = Product_{primes p} (1 + 2/p^(2*s-1) - 1/p^(2*s-2) - 2/p^s).

Dirichlet g.f.: zeta(s-1)^3 * f(s).

Sum_{k=1..n} a(k) ~ f(2) * n^2 * (log(n)^2 + (6*gamma - 1 + 2*f'(2)/f(2))*log(n) + 1/2 - 3*gamma + 6*gamma^2 - 6*sg1 + (6*gamma - 1)*f'(2)/f(2) + f''(2)/f(2))/4, where

f(2) = Product_{primes p} (1 - 3/p^2 + 2/p^3) = A065473 = 0.2867474284344787341...,

f'(2) = f(2) * Sum_{primes p} 4*log(p)/(p^2 + p - 2) = 0.53488225650873164189786660885838556843579696135554271633442328...,

f''(2) = f'(2)^2/f(2) + f(2) * Sum_{primes p} (-2*p*(3*p+2)*log(p)^2 / (p^2+p-2)^2) = -0.29112624105319980992840485620511000074444413707069816872854442...,

gamma is the Euler-Mascheroni constant A001620 and sg1 is the first Stieltjes constant (see A082633). (End)