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%I #63 Apr 29 2021 01:08:23
%S 2,16,1096744,3125,256,823543,19683
%N Smallest positive integer that is equal to the sum of the n-th powers of its prime factors (counted with multiplicity).
%C Does a(n) exist for all n?
%C a(12)=65536, a(27)=4294967296. a(n) exists for all n of the form n=p^i-i, where p is prime and i > 0, since p^p^i is an example (see A067688 and A081177). - _Jud McCranie_, Mar 16 2003
%C a(23) <= 298023223876953125. a(24) <= 7625597484987. - _Jud McCranie_, Jan 18 2016
%C a(10) = 285311670611. - _Jud McCranie_, Jan 25 2016
%C a(24) = 7625597484987. - _Jud McCranie_, Jan 30 2016
%H S. P. Hurd and J. S. McCranie, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL22/Hurd/hurd1.html">Integers that are Sums of Uniform Powers of all their Prime Factors: the sequence A068916</a>, J. of Int. Seq., vol 22, article 19.3.4.
%e 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.
%t a[n_] := For[x=1, True, x++, If[x==Plus@@(#[[2]]#[[1]]^n&/@FactorInteger[x]), Return[x]]]
%o (PARI) isok(k, n) = {my(f=factor(k)); sum(j=1, #f~, f[j,2]*f[j,1]^n) == k;}
%o a(n) = {my(k = 1); while(! isok(k,n), k++); k;} \\ _Michel Marcus_, Jan 25 2016
%o (Python)
%o from sympy import factorint
%o def a(n):
%o k = 1
%o while True:
%o f = factorint(k)
%o if k == sum(f[d]*d**n for d in f): return k
%o k += 1
%o for n in range(1, 8):
%o print(a(n), end=", ") # _Michael S. Branicky_, Feb 16 2021
%Y Cf. A067688, A268036.
%Y Cf. A081177, A000325, A024024, A024050.
%K nonn,hard,more
%O 1,1
%A _Dean Hickerson_, Mar 07 2002
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