A372379 The largest divisor of n whose number of divisors is a power of 2.

1, 2, 3, 2, 5, 6, 7, 8, 3, 10, 11, 6, 13, 14, 15, 8, 17, 6, 19, 10, 21, 22, 23, 24, 5, 26, 27, 14, 29, 30, 31, 8, 33, 34, 35, 6, 37, 38, 39, 40, 41, 42, 43, 22, 15, 46, 47, 24, 7, 10, 51, 26, 53, 54, 55, 56, 57, 58, 59, 30, 61, 62, 21, 8, 65, 66, 67, 34, 69, 70

Offset

1,2

Comments

First differs from A350390 at n = 32.

The largest term of A036537 dividing n.

The largest divisor of n whose exponents in its prime factorization are all of the form 2^k-1 (A000225).

Links

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

Formula

Multiplicative with a(p^e) = p^(2^floor(log_2(e+1)) - 1).

a(n) = n if and only if n is in A036537.

a(A162643(n)) = A282940(n).

Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = 0.7907361848... = 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))-1 for k >= 1, and f(0) = 0.

Mathematica

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

Prog

(PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i, 1]^(2^exponent(f[i, 2]+1)-1)); }
(Python)
from math import prod
from sympy import factorint
def A372379(n): return prod(p**((1<<(e+1).bit_length()-1)-1) for p, e in factorint(n).items()) # Chai Wah Wu, Apr 30 2024

Crossrefs

Cf. A000225, A000523, A036537, A138302, A162643, A282940, A350390, A353897, A372380, A372381.

Sequence in context: A161871 A331737 A135875 * A350390 A366765 A361253

Adjacent sequences: A372376 A372377 A372378 * A372380 A372381 A372382

Keyword

nonn,easy,mult

Author

Amiram Eldar, Apr 29 2024

Status

approved