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A062367 Multiplicative with a(p^e) = (e+1)*(e+2)*(2*e+3)/6.

%I A062367 #48 May 15 2025 08:18:21
%S A062367 1,5,5,14,5,25,5,30,14,25,5,70,5,25,25,55,5,70,5,70,25,25,5,150,14,25,
%T A062367 30,70,5,125,5,91,25,25,25,196,5,25,25,150,5,125,5,70,70,25,5,275,14,
%U A062367 70,25,70,5,150,25,150,25,25,5,350,5,25,70,140,25,125,5,70,25,125,5
%N A062367 Multiplicative with a(p^e) = (e+1)*(e+2)*(2*e+3)/6.
%H A062367 Charles R Greathouse IV, <a href="/A062367/b062367.txt">Table of n, a(n) for n = 1..10000</a>
%F A062367 a(n) = Sum_{i|n, j|n} tau(gcd(i, j)) = Sum_{d|n} tau(d)^2.
%F A062367 a(n) = Sum_{i|n, j|n} tau(i)*tau(j)/tau(lcm(i, j)), where tau(n) = number of divisors of n, cf. A000005.
%F A062367 Dirichlet convolution of A035116 and A000012 (i.e., inverse Mobius transform of A035116). Dirichlet g.f.: zeta^5(s)/zeta(2s). - _R. J. Mathar_, Feb 03 2011
%F A062367 G.f.: Sum_{n>=1} A000005(n)^2*x^n/(1-x^n). - _Mircea Merca_, Feb 26 2014
%F A062367 L.g.f.: -log(Product_{k>=1} (1 - x^k)^(tau(k)^2/k)) = Sum_{n>=1} a(n)*x^n/n. - _Ilya Gutkovskiy_, May 23 2018
%F A062367 Dirichlet convolution of A007426 and A008966. Dirichlet convolution of A007425 and A034444. - _R. J. Mathar_, Jun 05 2020
%F A062367 Let b(n), n > 0, be Dirichlet inverse of a(n). Then b(n) is multiplicative with b(p^e) = (-1)^e*(Sum_{i=0..e} binomial(4,i)) for prime p and e >= 0, where binomial(n,k)=0 if n < k; abs(b(n)) is multiplicative and has the Dirichlet g.f.: (zeta(s))^5/(zeta(2*s))^4. - _Werner Schulte_, Feb 07 2021
%F A062367 a(n) = Sum_{d divides n} tau(d^2)*tau(n/d), Dirichlet convolution of A048691 and A000005. - _Peter Bala_, Jan 26 2024
%p A062367 A062367 := proc(n)
%p A062367     add(numtheory[tau](d)^2,d=numtheory[divisors](n)) ;
%p A062367 end proc:
%p A062367 seq(A062367(n),n=1..40) ; # _R. J. Mathar_, May 15 2025
%t A062367 {1}~Join~Array[Times @@ Map[((# + 1) (# + 2) (2 # + 3))/6 &, FactorInteger[#][[All, -1]] ] &, 70, 2] (* or *)
%t A062367 Array[DivisorSum[#, DivisorSigma[0, #]^2 &] &, 71] (* _Michael De Vlieger_, Mar 05 2021 *)
%o A062367 (PARI) a(n) = sumdiv(n, d, numdiv(d)^2) \\ _Michel Marcus_, Jun 17 2013
%Y A062367 Cf. A000005, A000012, A000330, A029939, A035116, A048691, A060648, A062368.
%K A062367 nonn,mult
%O A062367 1,2
%A A062367 _Vladeta Jovovic_, Jul 07 2001