login

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).

A376720
Product of numbers m that are neither squarefree nor prime powers and rad(m), where rad = A007947.
1
72, 108, 200, 144, 392, 216, 400, 968, 675, 288, 500, 1352, 324, 784, 1800, 1323, 2312, 432, 1125, 2888, 800, 3528, 1936, 2700, 4232, 576, 1372, 3267, 1000, 2704, 648, 1568, 6728, 4563, 3600, 7688, 5292, 8712, 2025, 4624, 9800, 864, 3087, 10952, 4500, 5776, 7803
OFFSET
1,1
COMMENTS
Term a(n) = k is powerful but not a prime power (i.e., in A286708) such that k/rad(k) is not squarefree, where rad = A007947 and k/rad(k) = A003557(k).
Permutation of A372404.
LINKS
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