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A351569
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Sum of divisors of the largest unitary divisor of n that is an exponentially odd number.
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7
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1, 3, 4, 1, 6, 12, 8, 15, 1, 18, 12, 4, 14, 24, 24, 1, 18, 3, 20, 6, 32, 36, 24, 60, 1, 42, 40, 8, 30, 72, 32, 63, 48, 54, 48, 1, 38, 60, 56, 90, 42, 96, 44, 12, 6, 72, 48, 4, 1, 3, 72, 14, 54, 120, 72, 120, 80, 90, 60, 24, 62, 96, 8, 1, 84, 144, 68, 18, 96, 144, 72, 15, 74, 114, 4, 20, 96, 168, 80, 6, 1, 126, 84
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OFFSET
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1,2
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LINKS
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FORMULA
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Multiplicative with a(p^e) = (p^(e+1)-1)/(p-1) if e is odd and 1 otherwise.
Sum_{k=1..n} a(k) ~ c * n^2, where c = zeta(4)/2 = Pi^4/180 = 0.541161... . - Amiram Eldar, Nov 20 2022
Dirichlet g.f.: zeta(2*s) * zeta(2*s-2) * Product_{p prime} (1 + 1/p^(s-1) + 1/p^s - 1/p^(2*s-2)). - Amiram Eldar, Sep 03 2023
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MATHEMATICA
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f[p_, e_] := If[OddQ[e], (p^(e + 1) - 1)/(p - 1), 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Feb 23 2022 *)
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PROG
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(PARI)
A350389(n) = { my(m=1, f=factor(n)); for(k=1, #f~, if(1==(f[k, 2]%2), m *= (f[k, 1]^f[k, 2]))); (m); };
(Python)
from math import prod
from sympy import factorint
def A351569(n): return prod((p**(e+1)-1)//(p-1) if e % 2 else 1 for p, e in factorint(n).items()) # Chai Wah Wu, Feb 24 2022
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CROSSREFS
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KEYWORD
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nonn,mult
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AUTHOR
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STATUS
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approved
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