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Additive with a(p^e) = A000045(e+1).
+10
4
0, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 3, 1, 2, 2, 5, 1, 3, 1, 3, 2, 2, 1, 4, 2, 2, 3, 3, 1, 3, 1, 8, 2, 2, 2, 4, 1, 2, 2, 4, 1, 3, 1, 3, 3, 2, 1, 6, 2, 3, 2, 3, 1, 4, 2, 4, 2, 2, 1, 4, 1, 2, 3, 13, 2, 3, 1, 3, 2, 3, 1, 5, 1, 2, 3, 3, 2, 3, 1, 6, 5, 2, 1, 4, 2, 2, 2, 4, 1, 4, 2, 3, 2, 2, 2, 9, 1, 3, 3, 4, 1, 3, 1, 4, 3
FORMULA
Sum_{k=1..n} a(k) ~ n * (log(log(n)) + B + C), where B is Mertens's constant ( A077761) and C = Sum_{k>=2} Fibonacci(k-1) * P(k) = 1.30985781707683753402..., where P(s) is the prime zeta function. - Amiram Eldar, Oct 09 2023
MATHEMATICA
a[n_] := Total@ Fibonacci[FactorInteger[n][[;; , 2]] + 1]; a[1] = 0; Array[a, 100] (* Amiram Eldar, May 15 2023 *)
PROG
(PARI) A318473(n) = vecsum(apply(e -> fibonacci(1+e), factor(n)[, 2]));
CROSSREFS
Differs from A008481 for the first time at n=32, where a(32)=8, while A008481(32)=7.
Multiplicative with a(p^e) = 2^ A000041(e).
+10
2
1, 2, 2, 4, 2, 4, 2, 8, 4, 4, 2, 8, 2, 4, 4, 32, 2, 8, 2, 8, 4, 4, 2, 16, 4, 4, 8, 8, 2, 8, 2, 128, 4, 4, 4, 16, 2, 4, 4, 16, 2, 8, 2, 8, 8, 4, 2, 64, 4, 8, 4, 8, 2, 16, 4, 16, 4, 4, 2, 16, 2, 4, 8, 2048, 4, 8, 2, 8, 4, 8, 2, 32, 2, 4, 8, 8, 4, 8, 2, 64, 32, 4, 2, 16, 4, 4, 4, 16, 2, 16, 4, 8, 4, 4, 4, 256, 2, 8, 8, 16, 2, 8, 2, 16, 8, 4, 2, 32, 2, 8, 4, 64
PROG
(PARI) A318312(n) = factorback(apply(e -> 2^numbpart(e), factor(n)[, 2]));
CROSSREFS
Differs from A318474 for the first time at n=32, where a(32) = 128, while A318474(32) = 256.
Number of numbers mapped to A127668(n) with the map described there.
+10
1
1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 3, 1, 2, 2, 5, 1, 3, 1, 3, 2, 2, 1, 5, 2, 2, 3, 3, 1, 3, 1, 7, 2, 2, 2, 5, 1, 2, 2, 5, 1, 3, 1, 3, 3, 2, 1, 7, 2, 3, 2, 3, 1, 5, 2, 5, 2, 2, 1, 5, 1, 2, 3, 11, 2, 3, 1, 3, 2, 3, 1, 7, 1, 2, 3, 3, 2, 3, 1, 7, 5, 2, 1, 5, 2, 2, 2, 5, 1, 5, 2, 3, 2, 2, 2, 11, 1, 3, 3, 5
COMMENTS
This is not A008481(n), n>=2, which starts similarly, but differs, beginning with n=24.
FORMULA
a(n)<=pa(Length( A127668(n))), n>=2. Length gives the number of digits and pa(k):= A000041(k) (partition numbers). (It was originally claimed that this is equality, but that is not correct. - Franklin T. Adams-Watters, May 21 2014)
EXAMPLE
a(4)=2 because two numbers are mapped to 11= A127668(4), namely n=p(1)*p(1)=4 and n=p(11)=31. p(n)= A000041(n) (partition numbers).
CROSSREFS
The five numbers mapped to A127668(24)= 2111 are: 18433, 2594, 2263, 292, 24.
If n = Product (p_j^k_j) then a(n) = Sum (2^(k_j - 1)).
+10
0
0, 1, 1, 2, 1, 2, 1, 4, 2, 2, 1, 3, 1, 2, 2, 8, 1, 3, 1, 3, 2, 2, 1, 5, 2, 2, 4, 3, 1, 3, 1, 16, 2, 2, 2, 4, 1, 2, 2, 5, 1, 3, 1, 3, 3, 2, 1, 9, 2, 3, 2, 3, 1, 5, 2, 5, 2, 2, 1, 4, 1, 2, 3, 32, 2, 3, 1, 3, 2, 3, 1, 6, 1, 2, 3, 3, 2, 3, 1, 9, 8, 2, 1, 4, 2, 2, 2, 5, 1, 4
FORMULA
If n = Product (p_j^k_j) then a(n) = Sum ordered partition(k_j).
Additive with a(p^e) = 2^(e-1).
EXAMPLE
a(72) = 6 because 72 = 2^3 * 3^2 and 2^(3 - 1) + 2^(2 - 1) = 6.
MAPLE
a:= n-> add(2^(i[2]-1), i=ifactors(n)[2]):
MATHEMATICA
a[1] = 0; a[n_] := Plus @@ (2^(#[[2]] - 1) & /@ FactorInteger[n]); Table[a[n], {n, 1, 90}]
PROG
(PARI) a(n)={vecsum([2^(k-1) | k<-factor(n)[, 2]])} \\ Andrew Howroyd, Oct 29 2019
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