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A069648 a(1) = 1, otherwise smallest m > 1 such that the sum of digits of m^n is k^n for some k > 1.

%I A069648 #29 May 22 2025 10:21:34
%S A069648 1,2,2,11,47,46,983,193534,676644395
%N A069648 a(1) = 1, otherwise smallest m > 1 such that the sum of digits of m^n is k^n for some k > 1.
%C A069648 n-th root of A069647(n).
%C A069648 Probably k = 2 in all cases. - _Charles R Greathouse IV_, Feb 26 2014
%F A069648 a(n) > c^2^n for n > 1 where c = 10^(1/81) = 1.0288.... - _Charles R Greathouse IV_, Feb 26 2014
%o A069648 (Python)
%o A069648 import sympy
%o A069648 from sympy import factorint
%o A069648 def DigitSum(x):
%o A069648     return sum(int(i) for i in str(x))
%o A069648 def PowExp(p):
%o A069648     n = 2
%o A069648     while n < 10000*(10**(int(2**p/9)/p)):
%o A069648         if DigitSum(n**p) != 1:
%o A069648             count = 0
%o A069648             for i in list(factorint(DigitSum(n**p)).values()):
%o A069648                 if (int(i)/p) % 1 == 0:
%o A069648                     count += 1
%o A069648             if count == len(list(factorint(DigitSum(n**p)).values())):
%o A069648                 return n
%o A069648             else:
%o A069648                 n += 1
%o A069648         else:
%o A069648             n += 1
%o A069648 print(1)
%o A069648 x = 2
%o A069648 while x < 20:
%o A069648     print(PowExp(x))
%o A069648     x += 1
%o A069648 # _Derek Orr_, Feb 16 2014
%o A069648 (PARI)
%o A069648 a237992(maxn, maxm) = {
%o A069648   print1("1, ");
%o A069648   for(n=1, maxn,
%o A069648     for(m=2, maxm,
%o A069648       t=eval(Vec(Str(m^n)));
%o A069648       d=sum(i=1, #t, t[i]);
%o A069648       if(d>1 && ispower(d, n), print1(m, ", "); break())
%o A069648     )
%o A069648   )
%o A069648 }
%o A069648 a237992(8, 1000000) \\ _Colin Barker_, Feb 23 2014
%o A069648 (PARI) a(n)=if(n==1,return(1)); my(t=2^n,t3=3^n,k,s); while((s=sumdigits(k++^n))<t || (s<t3 && s!=t) || (s>=t3 && ispower(s,n)),); k \\ _Charles R Greathouse IV_, Feb 26 2014
%o A069648 (Python)
%o A069648 from sympy import factorint
%o A069648 def A069648(n):
%o A069648     if (n == 1):
%o A069648         return 1
%o A069648     else:
%o A069648         m = 2
%o A069648         while True:
%o A069648             x = sum(int(d) for d in str(m**n))
%o A069648             if x > 1 and not any(map(lambda x:x%n,factorint(x).values())):
%o A069648                 return m
%o A069648             m += 1 # _Chai Wah Wu_, Aug 11 2014
%Y A069648 Cf. A069647.
%K A069648 more,nonn,base
%O A069648 1,2
%A A069648 _Amarnath Murthy_, Apr 04 2002
%E A069648 Corrected and extended by _David Wasserman_, Apr 23 2003