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Fully multiplicative with a(prime(k)) = Lucas(2*(k+1)) for k-th prime p, where Lucas(n) = A000032(n).
2

%I #13 Aug 28 2023 08:22:53

%S 1,7,18,49,47,126,123,343,324,329,322,882,843,861,846,2401,2207,2268,

%T 5778,2303,2214,2254,15127,6174,2209,5901,5832,6027,39603,5922,103682,

%U 16807,5796,15449,5781,15876,271443,40446,15174,16121,710647,15498,1860498,15778,15228,105889,4870847,43218,15129,15463,39726,41307,12752043

%N Fully multiplicative with a(prime(k)) = Lucas(2*(k+1)) for k-th prime p, where Lucas(n) = A000032(n).

%H Antti Karttunen, <a href="/A324900/b324900.txt">Table of n, a(n) for n = 1..4096</a>

%H <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>.

%F Fully multiplicative with a(prime(k)) = A000032(2*(k+1)) = A000045(2k+1) + A000045(2k+3).

%F Sum_{n>=1} 1/a(n) = 1 / Product_{k>=1} (1 - 1/Lucas(2*k+2)) = 1.278911382005... . - _Amiram Eldar_, Aug 28 2023

%t f[p_, e_] := LucasL[2*(PrimePi[p]+1)]^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 60] (* _Amiram Eldar_, Aug 28 2023 *)

%o (PARI)

%o A000032(n) = (fibonacci(n+1)+fibonacci(n-1));

%o A324900(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = A000032(2*(1+primepi(f[i, 1])))); factorback(f); };

%Y Cf. A000032, A000045, A000720, A003965, A324901.

%K nonn,easy,mult

%O 1,2

%A _Antti Karttunen_, Apr 15 2019