cp's OEIS Frontend

A359280 Powerful numbers that are neither prime powers nor powers of squarefree composites.

%I A359280 #54 Feb 09 2025 11:21:08
%S A359280 72,108,144,200,288,324,392,400,432,500,576,648,675,784,800,864,968,
%T A359280 972,1125,1152,1323,1352,1372,1568,1600,1728,1800,1936,1944,2000,2025,
%U A359280 2304,2312,2500,2592,2700,2704,2888,2916,3087,3136,3200,3267,3456,3528,3600,3872,3888,3969
%N A359280 Powerful numbers that are neither prime powers nor powers of squarefree composites.
%C A359280 Numbers k such that omega(k) > 1 and for prime power factors p^e | k, multiplicities e > 1, yet the multiplicities are not equal.
%C A359280 Subset of A286708, which in turn is a subset of A361098, itself a subset of A126706, the sequence of numbers neither squarefree nor prime powers.
%C A359280 Since A001694 = Union({1}, A246547, A286708), this sequence is a subset of A001694.
%H A359280 Michael De Vlieger, <a href="/A359280/b359280.txt">Table of n, a(n) for n = 1..10000</a>
%H A359280 Michael De Vlieger, <a href="/A359280/a359280.png">Plot A001694(ym + x) at (x, y)</a> for n = ym + x, m = 1024 and x = 1..m, y = 0..m-1, showing terms in this sequence in black, and both prime powers and those in A303606 in white.
%H A359280 Michael De Vlieger, <a href="/A359280/a359280_1.png">Plot A286708(n) at (x, y)</a> for n = ym + x, m = 1024 and x = 1..m, y = 0..m-1, showing terms in this sequence in black, and those in A303606 in white.
%F A359280 This sequence is A286708 \ A303606.
%F A359280 Sum_{n>=1} 1/a(n) = zeta(2)*zeta(3)/zeta(6) - Sum_{k>=2} (zeta(k)/zeta(2*k) - 1) - 1 = 0.094962568855... . - _Amiram Eldar_, Dec 09 2023
%e A359280 Let b(n) = A286708(n).
%e A359280 b(1) = 36 is not in the sequence since rad(36) = A007947(36) = 6, and 36 = 6^2.
%e A359280 b(2) = a(1) = 72 since 72 is not a perfect power of rad(72).
%e A359280 b(3) = 100 = rad(100)^2 = 10^2, so it is not in the sequence.
%e A359280 b(4) = a(2) = 108, since 108 is not a perfect power of rad(108) = 6.
%e A359280 b(5) = a(3) = 144, since 144 is not a perfect power of rad(144) = 6.
%e A359280 b(6) = 196 is not in the sequence since 196 = rad(196)^2 = 14^2, etc.
%t A359280 nn = 5000; s = Rest@ Select[Union@ Flatten@Table[a^2*b^3, {b, nn^(1/3)}, {a, Sqrt[nn/b^3]}], Not@*PrimePowerQ]; Select[s, !SameQ @@ FactorInteger[#][[All, -1]] &]
%o A359280 (Python)
%o A359280 from math import isqrt
%o A359280 from sympy import mobius, integer_nthroot
%o A359280 def A359280(n):
%o A359280     def squarefreepi(n): return int(sum(mobius(k)*(n//k**2) for k in range(1, isqrt(n)+1)))
%o A359280     def bisection(f,kmin=0,kmax=1):
%o A359280         while f(kmax) > kmax: kmax <<= 1
%o A359280         kmin = kmax >> 1
%o A359280         while kmax-kmin > 1:
%o A359280             kmid = kmax+kmin>>1
%o A359280             if f(kmid) <= kmid:
%o A359280                 kmax = kmid
%o A359280             else:
%o A359280                 kmin = kmid
%o A359280         return kmax
%o A359280     def f(x):
%o A359280         j = isqrt(x)
%o A359280         c, l = n+x+3-(y:=x.bit_length())+squarefreepi(j)+sum(squarefreepi(integer_nthroot(x, k)[0]) for k in range(4, y)), 0
%o A359280         while j>1:
%o A359280             k2 = integer_nthroot(x//j**2,3)[0]+1
%o A359280             w = squarefreepi(k2-1)
%o A359280             c -= j*(w-l)
%o A359280             l, j = w, isqrt(x//k2**3)
%o A359280         return c+l
%o A359280     return bisection(f,n,n) # _Chai Wah Wu_, Feb 09 2025
%Y A359280 Cf. A001694, A007947, A082695, A126706, A286708, A303606, A361098.
%K A359280 nonn
%O A359280 1,1
%A A359280 _Michael De Vlieger_, Aug 01 2023