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A369759
The sum of unitary divisors of the smallest cubefull exponentially odd number that is divisible by n.
3
1, 9, 28, 9, 126, 252, 344, 9, 28, 1134, 1332, 252, 2198, 3096, 3528, 33, 4914, 252, 6860, 1134, 9632, 11988, 12168, 252, 126, 19782, 28, 3096, 24390, 31752, 29792, 33, 37296, 44226, 43344, 252, 50654, 61740, 61544, 1134, 68922, 86688, 79508, 11988, 3528, 109512
OFFSET
1,2
LINKS
FORMULA
a(n) = A034448(A356192(n)).
Multiplicative with a(p) = p^3 + 1, a(p^e) = p^e + 1 for an odd e >= 3, and a(p^e) = p^(e+1) + 1 for an even e.
a(n) >= A034448(n), with equality if and only if n is cubefull exponentially odd number (A335988).
Dirichlet g.f.: zeta(s) * zeta(2*s-2) * Product_{p prime} (1 + 1/p^(s-3) - 1/p^(2*s-2) - 1/p^(3*s-5) + 1/p^(4*s-5) - 1/p^(4*s-3)).
Sum_{k=1..n} a(k) ~ c * n^4 / 4, where c = (zeta(4)*zeta(6)/zeta(2)) * Product_{p prime} (1 - 1/p^6 + 1/p^11 - 1/p^12) = 0.65813930591740259189... .
MATHEMATICA
f[p_, e_] := p^If[OddQ[e], Max[e, 3], e+1] + 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
PROG
(PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, 1 + f[i, 1]^if(f[i, 2]%2, max(f[i, 2], 3), f[i, 2] + 1)); }
KEYWORD
nonn,easy,mult
AUTHOR
Amiram Eldar, Jan 31 2024
STATUS
approved