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a(n) = Sum_{d|Fibonacci(n)} d^2.
2

%I #31 Jan 05 2025 19:51:36

%S 1,1,5,10,26,85,170,500,1450,3172,7922,31031,54290,143140,483860,

%T 1105000,2550410,8923300,17494900,53353040,150655700,313727044,

%U 821223650,3291697500,5862907302,14736586180,48413209000,113026206800

%N a(n) = Sum_{d|Fibonacci(n)} d^2.

%H Amiram Eldar, <a href="/A063478/b063478.txt">Table of n, a(n) for n = 1..1422</a> (terms 1..200 from Harry J. Smith)

%H Florian Luca, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/37-3/luca.pdf">Arithmetic Functions of Fibonacci Numbers</a>, The Fibonacci Quarterly, Vol. 37, No. 3 (1999), pp. 265-268.

%F From _Amiram Eldar_, Jan 12 2022: (Start)

%F a(n) = A001157(A000045(n)).

%F a(n) <= A000045(A001157(n)), with equality if and only if n = 1 (Luca, 1999). (End)

%t DivisorSigma[2,Fibonacci[Range[30]]] (* _Harvey P. Dale_, Jul 05 2021 *)

%o (PARI) j=[]; for(n=1,40,j=concat(j,sumdiv(fibonacci(n),d,d^2))); j

%o (PARI) { allocatemem(20000000); h=1; g=f=0; for (n=1, 200, f+=h; h=g; g=f; write("b063478.txt", n, " ", sumdiv(f, d, d^2)) ) } \\ _Harry J. Smith_, Aug 23 2009

%o (Sage) [sigma(fibonacci(n),2)for n in range(1,29)] # _Zerinvary Lajos_, Jun 04 2009

%Y Cf. A000045, A001157.

%K easy,nonn

%O 1,3

%A _Jason Earls_, Jul 28 2001