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1, 5, 1, 21, 1, 5, 1, 85, 1, 5, 1, 21, 1, 5, 1, 341, 1, 5, 1, 21, 1, 5, 1, 85, 1, 5, 1, 21, 1, 5, 1, 1365, 1, 5, 1, 21, 1, 5, 1, 85, 1, 5, 1, 21, 1, 5, 1, 341, 1, 5, 1, 21, 1, 5, 1, 85, 1, 5, 1, 21, 1, 5, 1, 5461, 1, 5, 1, 21, 1, 5, 1, 85, 1, 5, 1, 21, 1, 5, 1, 341, 1, 5, 1, 21, 1, 5, 1, 85, 1, 5, 1, 21, 1, 5, 1, 1365
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OFFSET
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1,2
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COMMENTS
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a(n) is the sum of the second powers of the divisors of n divided by the sum of the second powers of the odd divisors of n.
Conjecture 1: For any nonnegative integer k and positive integer n, the sum of the k-th powers of the divisors of n is divisible by the sum of the k-th powers of the odd divisors of n.
Conjecture 2: Distinct terms form A002450 without A002450(0). In other words, a(2^(n-1)) = A002450(n), for n > 0.
Conjecture 3: For n > 0, the list of the first 2^n - 1 terms is palindromic.
Conjecture 4: For n > 0, the sum of the first 2^n - 1 terms equals A006095(n+1).
To prove Conjecture 1 all one needs to do is to prove that the sum of the k-th powers of the divisors of n divided by the sum of the k-th powers of the odd divisors of n equals: a) A001511(n), for k = 0, and b) ((2^k)^A001511(n) - 1)/(2^k - 1), for k > 0. - Ivan N. Ianakiev, Jan 29 2020
Conjecture 1 indeed follows from multiplicativity of sigma_k, in particular sigma_k(2^j (2m+1)) = sigma_k(2^j) sigma_k(2m+1). Conjecture 3 follows from Radcliffe's formula, since A007814 has this property. - M. F. Hasler, Jan 31 2020
a(n) is the sum of squares of powers of 2 that divide n. - Amiram Eldar, Nov 12 2020
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LINKS
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FORMULA
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Multiplicative with a(2^e) = (4^(e+1)-1)/3, and a(p^e) = 1 for odd primes p. - Amiram Eldar, Nov 12 2020
G.f.: Sum_{k>=0} 4^k * x^(2^k) / (1 - x^(2^k)). - Ilya Gutkovskiy, Dec 14 2020
Dirichlet g.f.: zeta(s)/(1-4/2^s). - Amiram Eldar, Dec 31 2022
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EXAMPLE
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MATHEMATICA
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f[n_]:=DivisorSigma[2, n]/Total[Select[Divisors[n], OddQ]^2]; f/@Range[100]
Table[(4^(IntegerExponent[n, 2] + 1) - 1)/3, {n, 1, 100}] (* Amiram Eldar, Nov 12 2020 *)
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PROG
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(PARI) a(n) = sigma(n, 2)/sumdiv(n, d, d^2*(d % 2)); \\ Michel Marcus, Dec 11 2017
(Magma) [DivisorSigma(2, n)/&+[d^2:d in Divisors(n)|IsOdd(d)]:n in [1..100]]; // Marius A. Burtea, Jan 29 2020
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CROSSREFS
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KEYWORD
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easy,nonn,mult
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AUTHOR
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STATUS
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approved
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