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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. 0
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 (list; graph; refs; listen; history; text; internal format)
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.

MAPLE

with(numtheory): P:=proc(n) local a, k; a:=ifactors(n)[2];

if frac(n/(add(a[k][1], k=1..nops(a))^2+add(a[k][2], k=1..nops(a))))=0

then n; fi; end: seq(P(i), i=2..10^5); # Paolo P. Lava, Nov 19 2018

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

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Last modified November 14 12:21 EST 2019. Contains 329114 sequences. (Running on oeis4.)