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A100107
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Inverse Moebius transform of Lucas numbers (A000032) 1,3,4,7,11,..
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5
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1, 4, 5, 11, 12, 26, 30, 58, 81, 138, 200, 355, 522, 876, 1380, 2265, 3572, 5880, 9350, 15272, 24510, 39806, 64080, 104084, 167773, 271968, 439285, 711530, 1149852, 1862022, 3010350, 4873112, 7881400, 12755618, 20633280, 33391491, 54018522, 87413156
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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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a(n) = Sum_{d|n} Lucas(d) = Sum_{d|n} A000032(d).
G.f.: Sum_{k>=1} Lucas(k) * x^k/(1 - x^k) = Sum_{k>=1} x^k * (1 + 2*x^k)/(1 - x^k - x^(2*k)). - Ilya Gutkovskiy, Aug 14 2019
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EXAMPLE
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a(2) = 4 because the prime 2 is divisible only by 1 and 2, so L(1) + L(2) = 1 + 3 = 4.
a(3) = 5 because the prime 3 is divisible only by 1 and 3, so L(1) + L(3) = 1 + 4 = 5.
a(4) = 11 because the semiprime 4 is divisible only by 1, 2, 4, so L(1) + L(2) + L(4) = 1 + 3 + 7 = 11.
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MAPLE
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with(numtheory): with(combinat): a:=proc(n) local div: div:=divisors(n): sum(2*fibonacci(div[j]+1)-fibonacci(div[j]), j=1..tau(n)) end: seq(a(n), n=1..42); # Emeric Deutsch, Jul 31 2005
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MATHEMATICA
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Table[Plus @@ Map[Function[d, LucasL[d]], Divisors[n]], {n, 100}] (* T. D. Noe, Aug 14 2012 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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