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A372379
The largest divisor of n whose number of divisors is a power of 2.
5
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
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
KEYWORD
nonn,easy,mult
AUTHOR
Amiram Eldar, Apr 29 2024
STATUS
approved