A360160 a(n) is the sum of unitary divisors of n that are odd squares.
1, 1, 1, 1, 1, 1, 1, 1, 10, 1, 1, 1, 1, 1, 1, 1, 1, 10, 1, 1, 1, 1, 1, 1, 26, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 10, 1, 1, 1, 1, 1, 1, 1, 1, 10, 1, 1, 1, 50, 26, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 10, 1, 1, 1, 1, 1, 1, 1, 1, 10, 1, 1, 26, 1, 1, 1, 1, 1, 82, 1, 1, 1
Offset: 1
Links
- Amiram Eldar, Table of n, a(n) for n = 1..10000
Programs
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Mathematica
f[p_, e_] := If[OddQ[e], 1, p^e + 1]; f[2, e_] := 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
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PARI
a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i, 1] == 2, 1, if(f[i, 2]%2, 1, f[i, 1]^f[i, 2] + 1))); }
Formula
a(n) = Sum_{d|n, gcd(d, n/d)=1, d odd square} d.
a(n) = A358347(n) if n is not of the form (2*m - 1)*4^k where m >= 1, k >= 1 (A108269), and otherwise it equals A358347(n)/(A006519(n)+1).
Multiplicative with a(2^e) = 1, and for p > 2, a(p^e) = p^e + 1 if e is even and 1 if e is odd.
Dirichlet g.f.: (zeta(s)*zeta(2*s-2)/zeta(3*s-2))*(2^(3*s)-2^(s+2))/(2^(3*s)-4).
Sum_{k=1..n} a(k) ~ c * n^(3/2), where c = (2*sqrt(2)/(4*sqrt(2)-1)) * zeta(3/2)/(3*zeta(5/2)) = 0.3942576405... .
Comments