OFFSET
1,1
COMMENTS
LINKS
Michael De Vlieger, Table of n, a(n) for n = 1..65536
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^16.
FORMULA
a(n) = m * rad(m) for m in A126706.
EXAMPLE
Let b(n) = A126706(n).
Table of b(n) and a(n) for n <= 12:
n b(n) a(n)
-----------------------------------------
1 12 = 2^2 * 3 72 = 2^3 * 3^2
2 18 = 2 * 3^2 108 = 2^2 * 3^3
3 20 = 2^2 * 5 200 = 2^3 * 5^2
4 24 = 2^3 * 3 144 = 2^4 * 3^2
5 28 = 2^2 * 7 392 = 2^3 * 7^2
6 36 = 2^2 * 3^2 216 = 2^3 * 3^3
7 40 = 2^3 * 5 400 = 2^4 * 5^2
8 44 = 2^2 * 11 968 = 2^3 * 11^2
9 45 = 3^2 * 5 675 = 3^3 * 5^2
10 48 = 2^4 * 3 288 = 2^5 * 3^2
11 50 = 2 * 5^2 500 = 2^2 * 5^3
12 52 = 2^2 * 13 1352 = 2^3 * 13^2
MATHEMATICA
Map[#*Times @@ FactorInteger[#][[All, 1]] &, Select[Range[12, 160], Nor[PrimePowerQ[#], SquareFreeQ[#]] &] ]
PROG
(Python)
from math import prod, isqrt
from sympy import primepi, integer_nthroot, mobius, primefactors
def A376720(n):
def bisection(f, kmin=0, kmax=1):
while f(kmax) > kmax: kmax <<= 1
while kmax-kmin > 1:
kmid = kmax+kmin>>1
if f(kmid) <= kmid:
kmax = kmid
else:
kmin = kmid
return kmax
def f(x): return int(n+sum(primepi(integer_nthroot(x, k)[0]) for k in range(2, x.bit_length()))+sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1)))
return (m:=bisection(f, n, n))*prod(primefactors(m)) # Chai Wah Wu, Oct 05 2024
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Michael De Vlieger, Oct 05 2024
STATUS
approved