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A228274
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a(n) = Sum_{d|n, n/d odd} n * d.
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2
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1, 4, 12, 16, 30, 48, 56, 64, 117, 120, 132, 192, 182, 224, 360, 256, 306, 468, 380, 480, 672, 528, 552, 768, 775, 728, 1080, 896, 870, 1440, 992, 1024, 1584, 1224, 1680, 1872, 1406, 1520, 2184, 1920, 1722, 2688, 1892, 2112, 3510, 2208, 2256, 3072, 2793, 3100
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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(2^e) = 4^e, a(p^e) = p^e * (p^(e+1) - 1) / (p - 1) if p>2.
G.f.: Sum_{k>0} k^2 * (x^k + x^(3*k)) / (1 - x^(2*k))^2. [see Basoco (1943) bottom page 305]
G.f.: Sum_{k>0} k^2 * (3 - (-1)^k)/4 * x^k / (1 - x^k)^2.
G.f.: Sum_{k>0 odd} k * (x^k + x^(2*k)) / (1 - x^k)^3.
a(n) = n * A002131(n). a(2*n) = 4 * a(n).
Sum_{k=1..n} a(k) ~ c * n^3, where c = Pi^2/24 = 0.411233... (A222171). - Amiram Eldar, Nov 30 2022
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EXAMPLE
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G.f. = x + 4*x^2 + 12*x^3 + 16*x^4 + 30*x^5 + 48*x^6 + 56*x^7 + 64*x^8 + ...
a(6) = 48 = 6 * (2 + 6). a(9) = 117 = 9 * (1 + 3 + 9). a(10) = 120 = 10 * (2 + 10).
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MATHEMATICA
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A228274[n_] := If[ n < 1, 0, n Sum[ d Mod[n / d, 2], {d, Divisors @ n}]]; Table[A228274[n], {n, 50}]
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PROG
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(PARI) {a(n) = if( n<1, 0, n * sumdiv(n, d, d * (n/d % 2)))};
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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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