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

%I A384531 #24 Jun 10 2025 01:15:22
%S A384531 1,4,7,12,13,28,19,32,33,52,31,84,37,76,91,80,49,132,55,156,133,124,
%T A384531 67,224,105,148,135,228,85,364,91,192,217,196,247,396,109,220,259,416,
%U A384531 121,532,127,372,429,268,139,560,217,420,343,444,157,540,403,608,385,340,175,1092
%N A384531 Multiplicative sequence a(n) with a(p^e) = ((2*e+1) * p - 2*e) * p^(e-1) for prime p and e >= 0.
%H A384531 Vaclav Kotesovec, <a href="/A384531/a384531.jpg">Graph - the asymptotic ratio (100000 terms)</a>
%H A384531 David Radcliffe, <a href="/A384531/a384531.pdf">Notes on two arithmetical functions</a> (Contains a proof that a(n) = Sum_{i=1..n} gcd(i, n) * gcd(i+1, n)).
%F A384531 Dirichlet g.f.: Sum_{n > 0} a(n) / n^s = (zeta(s-1))^2 * Product_{p prime} (1 + (p-2) / p^s).
%F A384531 Dirichlet convolution of A018804 and A173557.
%F A384531 Conjecture: a(n) = Sum_{i=1..n} gcd(i, n) * gcd(i+1, n).
%F A384531 From _Vaclav Kotesovec_, Jun 04 2025: (Start)
%F A384531 Let f(s) = Product_{primes p} (1 + 2/p^(2*s-1) - 1/p^(2*s-2) - 2/p^s).
%F A384531 Dirichlet g.f.: zeta(s-1)^3 * f(s).
%F A384531 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 A384531 f(2) = Product_{primes p} (1 - 3/p^2 + 2/p^3) = A065473 = 0.2867474284344787341...,
%F A384531 f'(2) = f(2) * Sum_{primes p} 4*log(p)/(p^2 + p - 2) = 0.53488225650873164189786660885838556843579696135554271633442328...,
%F A384531 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...,
%F A384531 gamma is the Euler-Mascheroni constant A001620 and sg1 is the first Stieltjes constant (see A082633). (End)
%p A384531 A384531 := proc(n)
%p A384531     local a,pe,p,e;
%p A384531     a :=1 ;
%p A384531     for pe in ifactors(n)[2] do
%p A384531         p := op(1,pe) ;
%p A384531         e := op(2,pe) ;
%p A384531         a := a*((2*e+1) * p - 2*e) * p^(e-1) ;
%p A384531     end do:
%p A384531     a ;
%p A384531 end proc:
%p A384531 seq(A384531(n),n=1..100) ;# _R. J. Mathar_, Jun 04 2025
%t A384531 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 *)
%o A384531 (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))
%o A384531 (Python)
%o A384531 from math import prod
%o A384531 from sympy import factorint
%o A384531 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
%Y A384531 Cf. A018804, A173557.
%Y A384531 Cf. A065473.
%K A384531 nonn,easy,mult
%O A384531 1,2
%A A384531 _Werner Schulte_, Jun 01 2025