A376720 Product of numbers m that are neither squarefree nor prime powers and rad(m), where rad = A007947.
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
Examples
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
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.
Programs
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Mathematica
Map[#*Times @@ FactorInteger[#][[All, 1]] &, Select[Range[12, 160], Nor[PrimePowerQ[#], SquareFreeQ[#]] &] ]
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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
Formula
a(n) = m * rad(m) for m in A126706.
Comments