A070288 -id:A070288 - cp's OEIS frontend

A318366 a(n) = Sum_{d|n} bigomega(d)*bigomega(n/d).

0, 0, 0, 1, 0, 2, 0, 4, 1, 2, 0, 8, 0, 2, 2, 10, 0, 8, 0, 8, 2, 2, 0, 20, 1, 2, 4, 8, 0, 12, 0, 20, 2, 2, 2, 24, 0, 2, 2, 20, 0, 12, 0, 8, 8, 2, 0, 40, 1, 8, 2, 8, 0, 20, 2, 20, 2, 2, 0, 34, 0, 2, 8, 35, 2, 12, 0, 8, 2, 12, 0, 52, 0, 2, 8, 8, 2, 12, 0, 40, 10, 2, 0, 34, 2, 2, 2, 20, 0, 34, 2, 8, 2, 2, 2

Offset: 1

Ilya Gutkovskiy, Aug 24 2018

nonn

Dirichlet convolution of A001222 with itself.

			24 has 8 divisors, namely 1, 2, 3, 4, 6, 8, 12, 24, and four prime factors counted with multiplicity. The divisors have 0, 1, 1, 2, 2, 3, 3, 4 divisors respectively. So a(24) = 0 * (4 - 0) + 1 * (4 - 1) + 1 * (4 - 1) + 2 * (4 - 2) + 2 * (4 - 2) + 3 * (4 - 3) + 4 * (4 - 4) = 0 + 3 + 3 + 4 + 4 + 3 + 3 + 0 = 20. - _David A. Corneth_, Jan 12 2019

Antti Karttunen, Table of n, a(n) for n = 1..20160
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences computed from exponents in factorization of n

Cf. A000005, A001222, A008578 (positions of 0's), A069264, A070288, A112967, A317938, A322375.

f:= proc(n) local F,G,t,x;
   F:= map(t -> t[2], ifactors(n)[2]);
   G:= unapply(normal(mul((1-x^(t+1))/(1-x), t = F)),x);
  (convert(F,`+`)-1)*D(G)(1) - (D@@2)(G)(1);
end proc:
map(f, [$1..100]); # Robert Israel, Jan 17 2019

Table[Sum[PrimeOmega[d] PrimeOmega[n/d], {d, Divisors[n]}], {n, 95}]

a(n) = sumdiv(n, d, bigomega(d)*bigomega(n/d)); \\ Michel Marcus, Aug 25 2018

a(n) = bn = bigomega(n); sumdiv(n, d, bd = bigomega(d); bd * (bn - bd)) \\ David A. Corneth, Jan 12 2019

a(A025487(n)) = A322375(n). - David A. Corneth, Jan 12 2019
From Robert Israel, Jan 17 2019: (Start)
If x and y are coprime, a(x*y) = a(x)*A000005(y) + A000005(x)*a(y) + A000005(x*y)*A001222(x)*A001222(y).
If p is prime, a(p^k) = (k^3-k)/6 = A000292(k-1). (End)

A328486 Dirichlet g.f.: zeta(s)^4 * (1 - 2^(-s))^2.

Original entry on oeis.org

1, 2, 4, 3, 4, 8, 4, 4, 10, 8, 4, 12, 4, 8, 16, 5, 4, 20, 4, 12, 16, 8, 4, 16, 10, 8, 20, 12, 4, 32, 4, 6, 16, 8, 16, 30, 4, 8, 16, 16, 4, 32, 4, 12, 40, 8, 4, 20, 10, 20, 16, 12, 4, 40, 16, 16, 16, 8, 4, 48, 4, 8, 40, 7, 16, 32, 4, 12, 16, 32, 4, 40, 4, 8, 40, 12, 16, 32, 4, 20

Offset: 1

Ilya Gutkovskiy, Oct 16 2019

Dirichlet convolution of A001227 with itself.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000005, A001227, A007426, A070288, A318366, A328487.

with(numtheory):
b:= proc(n) option remember; tau(2*n)-tau(n) end:
a:= n-> add(b(d)*b(n/d), d=divisors(n)):
seq(a(n), n=1..100);  # Alois P. Heinz, Oct 16 2019

nmax = 80; A001227 = Table[DivisorSum[n, Mod[#, 2] &], {n, 1, nmax}]; Table[DivisorSum[n, A001227[[#]] A001227[[n/#]] &], {n, 1, nmax}]
f[2, e_] := e + 1; f[p_, e_] := (e + 1)*(e + 2)*(e + 3)/6; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Nov 30 2020 *)

a(n) = Sum_{d|n} A001227(d) * A001227(n/d).
Sum_{k=1..n} a(k) ~ n * (log(n)^3/24 + (g/2 + log(2)/4 - 1/8)* log(n)^2 + (1/4 - g + 3*g^2/2 - log(2)/2 + 2*g*log(2) - sg1)* log(n) - 1/4 + (1 - 2*log(2))*g + (3*log(2) - 3/2)*g^2 + g^3 + log(2)/2 - log(2)^3/6 + (1 - 3*g - 2*log(2))* sg1 + sg2/2), where g is the Euler-Mascheroni constant A001620 and sg1, sg2 are the Stieltjes constants, see A082633 and A086279. - Vaclav Kotesovec, Oct 17 2019
Multiplicative with a(2^e) = e + 1, and a(p^e) = (e + 1)*(e + 2)*(e + 3)/6 for odd primes p. - Amiram Eldar, Nov 30 2020

A349712 a(n) = Sum_{d|n} sopf(d) * sopf(n/d).

Original entry on oeis.org

0, 0, 0, 4, 0, 12, 0, 8, 9, 20, 0, 32, 0, 28, 30, 12, 0, 42, 0, 48, 42, 44, 0, 52, 25, 52, 18, 64, 0, 124, 0, 16, 66, 68, 70, 87, 0, 76, 78, 76, 0, 164, 0, 96, 78, 92, 0, 72, 49, 90, 102, 112, 0, 72, 110, 100, 114, 116, 0, 234, 0, 124, 102, 20, 130, 244, 0, 144, 138, 236, 0, 132, 0, 148, 110

Offset: 1

Ilya Gutkovskiy, Nov 26 2021

nonn

Dirichlet convolution of A008472 with itself.

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A008472, A034761, A061397, A070288, A319131, A349711.

sopf[n_] := DivisorSum[n, # &, PrimeQ[#] &]; a[n_] := Sum[sopf[d] sopf[n/d], {d, Divisors[n]}]; Table[a[n], {n, 1, 75}]

sopf(n) = vecsum(factor(n)[, 1]); \\ A008472
a(n) = sumdiv(n, d, sopf(d)*sopf(n/d)); \\ Michel Marcus, Nov 26 2021

from sympy import divisors, factorint
def sopf(n): return sum(factorint(n))
def a(n): return sum(sopf(d)*sopf(n//d) for d in divisors(n))
print([a(n) for n in range(1, 76)]) # Michael S. Branicky, Nov 26 2021

Dirichlet g.f.: ( zeta(s) * primezeta(s-1) )^2.
a(n) = Sum_{d|n} A061397(d) * A319131(n/d).
a(p) = 0 for p prime. - Michael S. Branicky, Nov 26 2021
a(p^k) = (k-1)*p^2 for p prime and k > 0. - Chai Wah Wu, Nov 28 2021

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A318366 a(n) = Sum_{d|n} bigomega(d)*bigomega(n/d).

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A328486 Dirichlet g.f.: zeta(s)^4 * (1 - 2^(-s))^2.

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A349712 a(n) = Sum_{d|n} sopf(d) * sopf(n/d).

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