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

16, 192, 288, 704, 1470, 2112, 2160, 3168, 3240, 3872, 4096, 4608, 4752, 4860, 5400, 6912, 7128, 7245, 8100, 9295, 10368, 11616, 13500, 15552, 15900, 17424, 21296, 23328, 23850, 26136, 27720, 32830, 34992, 35960, 39600, 39750, 41536, 45584, 52250, 52488, 59400, 62920, 63888, 67200, 78732, 81920, 86430

Offset: 1

Author

Pierandrea Formusa, Nov 18 2018

Keywords

• nonn

Comments

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

Examples

			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.
		

Crossrefs

Cf. A001222, A235323.

Programs

Mathematica

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 *)

PARI

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

Python

from sympy.ntheory import factorint, isprime
n=100000
r=""
def calc(n):
    global r
    a=factorint(n)
    lp=[]
    for p in a.keys():
        lp.append(p)
    lexp=[]
    for exp in a.values():
        lexp.append(exp)
    if n%((sum(lp))**2+sum(lexp))==0:
       r += ","
       r += str(n)
    return
for i in range(4,n):
    calc(i)
print(r[1:])