A374207 -id:A374207 - cp's OEIS frontend

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

Offset: 1

Leroy Quet, Oct 16 2005

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

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

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.

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

{ 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

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

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

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

A374208 The 5-adic valuation of A113177(n), where A113177 is fully additive with a(p) = Fibonacci(p).

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1

Offset: 2

Antti Karttunen, Jul 01 2024

nonn

Antti Karttunen, Table of n, a(n) for n = 2..100000

Cf. A000045, A112765, A113177.

Cf. also A374205, A374206, A374207.

A113177(n) = if(n<=1, 0, my(f=factor(n)); sum(i=1, #f~, f[i, 2]*fibonacci(f[i,1])));
A374208(n) = valuation(A113177(n), 5);

a(n) = A112765(A113177(n)).

A374206 The 2-adic valuation of A113177(n), where A113177 is fully additive with a(p) = Fibonacci(p).

Original entry on oeis.org

0, 1, 1, 0, 0, 0, 0, 2, 1, 0, 2, 0, 1, 0, 2, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 1, 0, 0, 3, 0, 0, 0, 1, 1, 1, 0, 1, 0, 3, 0, 4, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 1, 4, 0, 1, 0, 0, 0, 1, 0, 1, 1, 2, 0, 0, 0, 0, 0, 0, 0, 1, 2, 0, 1, 2, 0, 0, 3, 1, 0, 0, 1, 1, 0, 2, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 2, 0, 6, 0, 2, 2, 1, 0, 3

Offset: 2

Antti Karttunen, Jul 01 2024

nonn

Antti Karttunen, Table of n, a(n) for n = 2..100000

Cf. A007814, A113177, A373586 (gives the indices of nonzero terms here, after its initial 1), A373587 (gives the indices of 0's).

Cf. also A374132, A374212, A374202, A374207, A374208.

A113177(n) = if(n<=1, 0, my(f=factor(n)); sum(i=1, #f~, f[i, 2]*fibonacci(f[i,1])));
A374206(n) = valuation(A113177(n), 2);

a(n) = A007814(A113177(n)).

A374217 The 3-adic valuation of sopfr(n), where sopfr is the sum of prime factors with repetition, A001414.

Original entry on oeis.org

0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 2, 0, 1, 2, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 2, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 2, 2, 0, 0, 1, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 3, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 2, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 2

Offset: 2

Antti Karttunen, Jul 02 2024

nonn

Antti Karttunen, Table of n, a(n) for n = 2..100000

Cf. A001414, A007949, A289142 (gives the indices of nonzero terms here, after its initial 1), A373371

Cf. also A374207, A374216.

A001414(n) = ((n=factor(n))[, 1]~*n[, 2]);
A374217(n) = valuation(A001414(n), 3);

a(n) = A007949(A001414(n)).

cp's OEIS Frontend

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.

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A374208 The 5-adic valuation of A113177(n), where A113177 is fully additive with a(p) = Fibonacci(p).

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A374206 The 2-adic valuation of A113177(n), where A113177 is fully additive with a(p) = Fibonacci(p).

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A374217 The 3-adic valuation of sopfr(n), where sopfr is the sum of prime factors with repetition, A001414.

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