A113177 - OEIS

A113177

Fully additive with a(p) = Fibonacci(p); If, for p prime, p^(m_{n,p}) is the highest power of p dividing n with m>=0, then a(n) = Sum_{p prime} F(p)*(m_{n,p}), where F(p) = p-th Fibonacci number.

0, 1, 2, 2, 5, 3, 13, 3, 4, 6, 89, 4, 233, 14, 7, 4, 1597, 5, 4181, 7, 15, 90, 28657, 5, 10, 234, 6, 15, 514229, 8, 1346269, 5, 91, 1598, 18, 6, 24157817, 4182, 235, 8, 165580141, 16, 433494437, 91, 9, 28658, 2971215073, 6, 26, 11, 1599, 235, 53316291173, 7, 94

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OFFSET

1,3

LINKS

Danny Rorabaugh, Table of n, a(n) for n = 1..4000

FORMULA

Totally additive with a(p) = A000045(p).

EXAMPLE

12 = 2^2 * 3^1, so a(12) = F(2)*2 + F(3)*1 = 2 + 2 = 4.

MATHEMATICA

b[t_]:=Fibonacci[First[t]]Last[t] a[n_]:=Apply[Plus, Map[b, FactorInteger[n]]] (* Esa Peuha, Oct 26 2005 *)

PROG

(PARI) { for(n=1, 100, f=factor(n); s=0; for(i=1, matsize(f)[1], s+=fibonacci(f[i, 1])*f[i, 2]); print1(s, ", ")) } \\ Lambert Klasen, Oct 26 2005

(Sage) [0]+[sum([fibonacci(x[0])*x[1] for x in factor(n)]) for n in range(2, 56)] # Danny Rorabaugh, Apr 03 2015

CROSSREFS

Cf. A000045, A113178.

Cf. A373586 (indices of even terms), A373587 (of odd terms), A374052 (of multiples of 3), A374206 (2-adic valuation), A374207 (3-adic valuation), A374208 (5-adic valuation), A374209 [A007895(a(n))], A374124 [a(n) mod 360].

Cf. A374106 [gcd(a(n), A328845(n))], A374112 [gcd(a(n), A276085(n))].

For other completely additive sequences see the cross-references in A001414.

Sequence in context: A190170 A147524 A374124 * A344507 A322786 A184243

Adjacent sequences: A113174 A113175 A113176 * A113178 A113179 A113180

KEYWORD

nonn,easy

AUTHOR

Leroy Quet, Oct 16 2005

EXTENSIONS

More terms from Esa Peuha (esa.peuha(AT)helsinki.fi) and Lambert Klasen (lambert.klasen(AT)gmx.net), Oct 26 2005

Prefixed the name with a more succinct form of the definition given in comments. - Antti Karttunen, Jul 08 2024

STATUS

approved