A353897 a(n) is the largest divisor of n whose exponents in its prime factorization are all powers of 2 (A138302).

1, 2, 3, 4, 5, 6, 7, 4, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 12, 25, 26, 9, 28, 29, 30, 31, 16, 33, 34, 35, 36, 37, 38, 39, 20, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 18, 55, 28, 57, 58, 59, 60, 61, 62, 63, 16, 65, 66, 67, 68

Offset

1,2

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Formula

Multiplicative with a(p^e) = p^(2^floor(log_2(e))).

a(n) = n if and only if n is in A138302.

Sum_{k=1..n} a(k) ~ c*n^2, where c = 0.4616988732... = (1/2) * Product_{p prime} (1 + Sum_{k>=1} (p^f(k) - p^(f(k-1)+1))/p^(2*k)), f(k) = 2^floor(log_2(k)) and f(0) = 0.

Example

a(27) = 9 since 9 = 3^2 is the largest divisor of 27 with an exponent in its prime factorization, 2, that is a power of 2.

Mathematica

f[p_, e_] := p^(2^Floor[Log2[e]]); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

Crossrefs

Cf. A138302, A353898.

Similar sequences: A000265, A007947, A008834, A055071, A350390.

Sequence in context: A007948 A377518 A355261 * A366905 A348262 A038389

Adjacent sequences: A353894 A353895 A353896 * A353898 A353899 A353900

Keyword

nonn,mult

Author

Amiram Eldar, May 10 2022

Status

approved