A324900 Fully multiplicative with a(prime(k)) = Lucas(2*(k+1)) for k-th prime p, where Lucas(n) = A000032(n).

1, 7, 18, 49, 47, 126, 123, 343, 324, 329, 322, 882, 843, 861, 846, 2401, 2207, 2268, 5778, 2303, 2214, 2254, 15127, 6174, 2209, 5901, 5832, 6027, 39603, 5922, 103682, 16807, 5796, 15449, 5781, 15876, 271443, 40446, 15174, 16121, 710647, 15498, 1860498, 15778, 15228, 105889, 4870847, 43218, 15129, 15463, 39726, 41307, 12752043

Offset: 1

Author

Antti Karttunen, Apr 15 2019

Keywords

• nonn
• easy
• mult

Links

• Antti Karttunen, Table of n, a(n) for n = 1..4096
• Index entries for sequences computed from indices in prime factorization.

Crossrefs

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

Programs

Mathematica

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 *)

PARI

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

Formula

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

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