

A068916


Smallest positive integer that is equal to the sum of the nth powers of its prime factors (counted with multiplicity).


5




OFFSET

1,1


COMMENTS

Does a(n) exist for all n?
a(12)=65536, a(27)=4294967296. a(n) exists for all n of the form n=p^ii, where p is prime and i > 0, since p^p^i is an example (see A067688 and A081177).  Jud McCranie, Mar 16 2003
a(23) <= 298023223876953125. a(24) <= 7625597484987.  Jud McCranie, Jan 18 2016


LINKS



EXAMPLE

a(3) = 1096744 = 2^3*11^3*103; the sum of the cubes of the prime factors is 3*2^3 + 3*11^3 + 103^3 = 1096744.


MATHEMATICA

a[n_] := For[x=1, True, x++, If[x==Plus@@(#[[2]]#[[1]]^n&/@FactorInteger[x]), Return[x]]]


PROG

(PARI) isok(k, n) = {my(f=factor(k)); sum(j=1, #f~, f[j, 2]*f[j, 1]^n) == k; }
a(n) = {my(k = 1); while(! isok(k, n), k++); k; } \\ Michel Marcus, Jan 25 2016
(Python)
from sympy import factorint
def a(n):
k = 1
while True:
f = factorint(k)
if k == sum(f[d]*d**n for d in f): return k
k += 1
for n in range(1, 8):


CROSSREFS



KEYWORD

nonn,hard,more


AUTHOR



STATUS

approved



