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

1,1

Comments

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

Links

Table of n, a(n) for n=1..47.

Example

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.

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 *)

Prog

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

Crossrefs

Cf. A001222, A235323.

Sequence in context: A004333 A016237 A036735 * A304307 A316206 A302298

Adjacent sequences: A321453 A321454 A321455 * A321457 A321458 A321459

Keyword

nonn

Author

Pierandrea Formusa, Nov 18 2018

Status

approved