A324911 -id:A324911 - cp's OEIS frontend

A324912 Binary weight of A324911(n).

1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 4, 1, 2, 1, 2, 1, 1, 1, 3, 2, 1, 3, 2, 1, 1, 1, 5, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 4, 2, 2, 1, 2, 1, 3, 1, 3, 1, 1, 1, 2, 1, 1, 2, 6, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 2, 2, 1, 1, 1, 4, 4, 1, 1, 2, 1, 1, 1, 3, 1, 2, 1, 2, 1, 1, 1, 5, 1, 2, 2, 2, 1, 1, 1, 3, 1

Offset: 1

Antti Karttunen, Apr 14 2019

nonn

After 1 differs from A051903 at n = 900, 1764, 1800, 2700, 3528, 4356, 4500, 4900, 5292, ..., which seems to be a subsequence of A318720.

			For n = 900 = 2^2 * 3^2 * 5^2, A324911(900) = A156552(4) * A156552(9) * A156552(25) = 3*6*12 = 216, which in base-2 is written as "11011000", thus a(900) = 4.

Cf. A000120, A051903, A156552, A318720, A324911.

Differs from A072411 for the first time at n=72, where a(72) = 3, while A072411(72) = 6.

A156552(n) = { my(f = factor(n), p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res }; \\ From A156552
A324911(n) = { my(f=factor(n)); prod(i=1, #f~, A156552(f[i,1]^f[i,2])); };
A324912(n) = hammingweight(A324911(n));

a(n) = A000120(A324911(n)).

A324910 Multiplicative with a(p^e) = (2^e)-1.

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 1, 7, 3, 1, 1, 3, 1, 1, 1, 15, 1, 3, 1, 3, 1, 1, 1, 7, 3, 1, 7, 3, 1, 1, 1, 31, 1, 1, 1, 9, 1, 1, 1, 7, 1, 1, 1, 3, 3, 1, 1, 15, 3, 3, 1, 3, 1, 7, 1, 7, 1, 1, 1, 3, 1, 1, 3, 63, 1, 1, 1, 3, 1, 1, 1, 21, 1, 1, 3, 3, 1, 1, 1, 15, 15, 1, 1, 3, 1, 1, 1, 7, 1, 3, 1, 3, 1, 1, 1, 31, 1, 3, 3, 9, 1, 1, 1, 7, 1

Offset: 1

Antti Karttunen, Apr 14 2019

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences computed from exponents in factorization of n

Cf. A000225, A156552, A246674, A322993, A324911.

Array[Times @@ (2^(FactorInteger[#][[All, -1]]) - 1) &, 105] (* Michael De Vlieger, Apr 14 2019 *)

A324910(n) = factorback(apply(e -> -1+(2^e), factor(n)[,2]~));

Multiplicative with a(p^e) = A000225(e).
Multiplicative with a(p^e) = A322993(p^e).
a(n) = A246674(A156552(n)).

cp's OEIS Frontend

A324912 Binary weight of A324911(n).

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