A124316 - OEIS

%I #41 Mar 28 2024 09:03:33

%S 1,2,2,5,2,4,2,8,6,4,2,10,2,4,4,15,2,12,2,10,4,4,2,16,8,4,10,10,2,8,2,

%T 22,4,4,4,30,2,4,4,16,2,8,2,10,12,4,2,30,10,16,4,10,2,20,4,16,4,4,2,

%U 20,2,4,12,37,4,8,2,10,4,8,2,48,2,4,16,10,4,8,2,30,23,4,2,20,4,4,4,16,2,24

%N a(n) = Sum_{d|n} sigma(gcd(d,n/d)), where sigma is the sum of divisors function, A000203.

%C Apparently multiplicative and the inverse Mobius transform of A069290. - _R. J. Mathar_, Feb 07 2011

%H Antti Karttunen, <a href="/A124316/b124316.txt">Table of n, a(n) for n = 1..10000</a>

%H Ekkehard Krätzel, Werner Georg Nowak, and László Tóth, <a href="https://eudml.org/doc/269485">On certain arithmetic functions involving the greatest common divisor</a>, Cent. Eur. J. Math., 10 (2012), 761-774.

%H Manfred Kühleitner and Werner Georg Nowak, <a href="http://dx.doi.org/10.2478/s11533-012-0143-2">On a question of A. Schinzel: Omega estimates for a special type of arithmetic functions</a>, Cent. Eur. J. Math., Vol. 11, No. 3 (2013), pp. 477-486; <a href="http://arxiv.org/abs/1204.1146">arXiv preprint</a>, arXiv:1204.1146 [math.NT], 2012.

%H László Tóth, <a href="https://doi.org/10.1007/978-1-4939-1106-6_19">Multiplicative arithmetic functions of several variables: a survey</a>, in: T. Rassias, P. Pardalos (eds.) Mathematics Without Boundaries, Springer, New York, NY, 2024; <a href="https://arxiv.org/abs/1310.7053">arXiv preprint</a>, arXiv:1310.7053 [math.NT], 2013-2014.

%F From _Amiram Eldar_, Mar 28 2024: (Start)

%F Multiplicative with a(p^e) = (p^(e/2 + 1)*(p+1) - (e+3)*p + e + 1)/(p-1)^2, if e is even, and (2*p^((e+3)/2) - (e+3)*p + e + 1)/(p-1)^2 if e is odd.

%F Dirichlet g.f.: zeta(s)^2 * zeta(2*s-1).

%F Sum_{k=1..n} a(k) = (log(n)^2/4 + (2*gamma - 1/2)*log(n) + 5*gamma^2/2 - 2*gamma - 3*gamma_1 + 1/2) * n + O(n^(2/3)*log(n)^(16/9)), where gamma is Euler's constant (A001620) and gamma_1 is the first Stieltjes constant (A082633) (Krätzel et al., 2012). (End)

%p A124316 := proc(n) local a,d;

%p   a := 0 ;

%p   for d in numtheory[divisors](n) do

%p      igcd(d,n/d) ;

%p      a := a+numtheory[sigma](%) ;

%p    end do:

%p    a;

%p end proc: # _R. J. Mathar_, Apr 14 2011

%t Table[Plus @@ Map[DivisorSigma[1, GCD[ #, n/# ]] &, Divisors@n], {n, 90}]

%t f[p_, e_] := (If[OddQ[e], 2*p^((e+3)/2), p^(e/2 + 1)*(p+1)] - (e+3)*p + e + 1)/(p-1)^2; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Mar 28 2024 *)

%o (PARI) a(n) = sumdiv(n, d, sigma(gcd(d, n/d))); \\ _Michel Marcus_, Feb 13 2016

%o (Python)

%o from sympy import divisors, divisor_sigma, gcd

%o def a(n): return sum([divisor_sigma(gcd(d, n/d)) for d in divisors(n)]) # _Indranil Ghosh_, May 25 2017

%Y Cf. A000203, A001616, A001620, A069290, A061884, A082633, A124315.

%K nonn,easy,mult

%O 1,2

%A _Robert G. Wilson v_, Sep 30 2006