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A321456 Numbers k that are divisible by sum(pi)^2+sum(ei) where k=p1^e1*...*pj^ej with pi primes.

%I A321456 #31 Feb 28 2024 10:49:56
%S A321456 16,192,288,704,1470,2112,2160,3168,3240,3872,4096,4608,4752,4860,
%T A321456 5400,6912,7128,7245,8100,9295,10368,11616,13500,15552,15900,17424,
%U A321456 21296,23328,23850,26136,27720,32830,34992,35960,39600,39750,41536,45584,52250,52488,59400,62920,63888,67200,78732,81920,86430
%N A321456 Numbers k that are divisible by sum(pi)^2+sum(ei) where k=p1^e1*...*pj^ej with pi primes.
%C A321456 Numbers k that are divisible by A001222(k)+A235323(k).
%e A321456 704 is an item as its prime factorization is 2^6+11^1, sum(pi)=2+11=13, sum(e1)=6+1=7, sum(pi)^2+sum(e1)=13^2+7=169+7=176, finally 704=c*176 for c=4.
%t A321456 fun[n_] := Module[{f = FactorInteger[n]}, Total@f[[;;, 1]]^2 + Total@f[[;;, 2]]]; aQ[n_] := Divisible[n, fun[n]]; Select[Range[100000], aQ] (* _Amiram Eldar_, Nov 18 2018 *)
%o A321456 (Python)
%o A321456 from sympy.ntheory import factorint, isprime
%o A321456 n=100000
%o A321456 r=""
%o A321456 def calc(n):
%o A321456     global r
%o A321456     a=factorint(n)
%o A321456     lp=[]
%o A321456     for p in a.keys():
%o A321456         lp.append(p)
%o A321456     lexp=[]
%o A321456     for exp in a.values():
%o A321456         lexp.append(exp)
%o A321456     if n%((sum(lp))**2+sum(lexp))==0:
%o A321456        r += ","
%o A321456        r += str(n)
%o A321456     return
%o A321456 for i in range(4,n):
%o A321456     calc(i)
%o A321456 print(r[1:])
%o A321456 (PARI) ok(k)={my(f=factor(k)); k > 1 && k % (vecsum(f[,2]) + vecsum(f[,1])^2) == 0} \\ _Andrew Howroyd_, Nov 19 2018
%Y A321456 Cf. A001222, A235323.
%K A321456 nonn
%O A321456 1,1
%A A321456 _Pierandrea Formusa_, Nov 18 2018