This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A372841 #17 Sep 10 2024 14:21:10
%S A372841 1296,2592,3888,5184,7776,10000,10368,11664,15552,20000,20736,23328,
%T A372841 31104,34992,38416,40000,41472,46656,50000,50625,62208,69984,76832,
%U A372841 80000,82944,93312,100000,104976,124416,139968,151875,153664,160000,165888,186624,194481,200000
%N A372841 4-full numbers that are not prime powers.
%C A372841 Numbers k such that rad(k)^4 | k and omega(k) > 1. In other words, numbers with at least 2 distinct prime factors whose prime power factors have exponents that exceed 3.
%C A372841 Proper subset of the following sequences: A001694, A036966, A036967, A126706, A286708, A372695.
%C A372841 Smallest term k with omega(k) = m is k = A002110(m)^4.
%H A372841 Michael De Vlieger, <a href="/A372841/b372841.txt">Table of n, a(n) for n = 1..10000</a>
%F A372841 Intersection of A036967 and A024619.
%F A372841 Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + 1/(p^3*(p-1))) - Sum_{p prime} 1/(p^3*(p-1)) - 1 = 0.0026996042121456100761... . - _Amiram Eldar_, May 17 2024
%e A372841 Table of smallest 12 terms:
%e A372841 n a(n)
%e A372841 -----------------------
%e A372841 1 1296 = 2^4 * 3^4
%e A372841 2 2592 = 2^5 * 3^4
%e A372841 3 3888 = 2^4 * 3^5
%e A372841 4 5184 = 2^6 * 3^4
%e A372841 5 7776 = 2^5 * 3^5
%e A372841 6 10000 = 2^4 * 5^4
%e A372841 7 10368 = 2^7 * 3^4
%e A372841 8 11664 = 2^4 * 3^6
%e A372841 9 15552 = 2^6 * 3^5
%e A372841 10 20000 = 2^5 * 5^4
%e A372841 11 20736 = 2^8 * 3^4
%e A372841 12 23328 = 2^5 * 3^6
%t A372841 With[{nn = 200000}, Rest@ Select[Union@ Flatten@ Table[a^7 * b^6 * c^5 * d^4, {d, Surd[nn, 4]}, {c, Surd[nn/(d^4), 5]}, {b, Surd[nn/(c^5 * d^4), 6]}, {a, Surd[nn/(b^6 * c^5 * d^4), 7]}], Not@*PrimePowerQ]]
%o A372841 (Python)
%o A372841 from math import gcd
%o A372841 from sympy import primepi, integer_nthroot, factorint
%o A372841 def A372841(n):
%o A372841 def f(x):
%o A372841 c = n+x+1+sum(primepi(integer_nthroot(x, k)[0]) for k in range(4, x.bit_length()))
%o A372841 for u in range(1,integer_nthroot(x,7)[0]+1):
%o A372841 if all(d<=1 for d in factorint(u).values()):
%o A372841 for w in range(1,integer_nthroot(a:=x//u**7,6)[0]+1):
%o A372841 if gcd(w,u)==1 and all(d<=1 for d in factorint(w).values()):
%o A372841 for y in range(1,integer_nthroot(z:=a//w**6,5)[0]+1):
%o A372841 if gcd(w,y)==1 and gcd(u,y)==1 and all(d<=1 for d in factorint(y).values()):
%o A372841 c -= integer_nthroot(z//y**5,4)[0]
%o A372841 return c
%o A372841 def bisection(f,kmin=0,kmax=1):
%o A372841 while f(kmax) > kmax: kmax <<= 1
%o A372841 while kmax-kmin > 1:
%o A372841 kmid = kmax+kmin>>1
%o A372841 if f(kmid) <= kmid:
%o A372841 kmax = kmid
%o A372841 else:
%o A372841 kmin = kmid
%o A372841 return kmax
%o A372841 return bisection(f,n,n) # _Chai Wah Wu_, Sep 10 2024
%Y A372841 Cf. A001694, A007947, A024619, A036966, A036967, A126706, A286708, A372695.
%K A372841 nonn
%O A372841 1,1
%A A372841 _Michael De Vlieger_, May 14 2024