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A321456
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Numbers k that are divisible by sum(pi)^2+sum(ei) where k=p1^e1*...*pj^ej with pi primes.
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0
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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
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OFFSET
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1,1
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COMMENTS
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LINKS
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EXAMPLE
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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.
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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