A321456 - OEIS

%I #31 Feb 28 2024 10:49:56

%S 16,192,288,704,1470,2112,2160,3168,3240,3872,4096,4608,4752,4860,

%T 5400,6912,7128,7245,8100,9295,10368,11616,13500,15552,15900,17424,

%U 21296,23328,23850,26136,27720,32830,34992,35960,39600,39750,41536,45584,52250,52488,59400,62920,63888,67200,78732,81920,86430

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

%C Numbers k that are divisible by A001222(k)+A235323(k).

%e 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 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 (Python)

%o from sympy.ntheory import factorint, isprime

%o n=100000

%o r=""

%o def calc(n):

%o     global r

%o     a=factorint(n)

%o     lp=[]

%o     for p in a.keys():

%o         lp.append(p)

%o     lexp=[]

%o     for exp in a.values():

%o         lexp.append(exp)

%o     if n%((sum(lp))**2+sum(lexp))==0:

%o        r += ","

%o        r += str(n)

%o     return

%o for i in range(4,n):

%o     calc(i)

%o print(r[1:])

%o (PARI) ok(k)={my(f=factor(k)); k > 1 && k % (vecsum(f[,2]) + vecsum(f[,1])^2) == 0} \\ _Andrew Howroyd_, Nov 19 2018

%Y Cf. A001222, A235323.

%K nonn

%O 1,1

%A _Pierandrea Formusa_, Nov 18 2018