cp's OEIS Frontend

A259313 Numbers m for which there exists a k>=2 such that m equals the average of digitsum(m^p) for p from 1 to k.

%I A259313 #40 Jun 06 2025 00:24:28
%S A259313 1,9,12,13,16,19,21,49,61,67,84,106,160,191,207,250,268,373,436,783,
%T A259313 2321,3133,3786,3805,4842,5128,8167,13599,29431,35308
%N A259313 Numbers m for which there exists a k>=2 such that m equals the average of digitsum(m^p) for p from 1 to k.
%C A259313 Digitsum = (A007953).
%C A259313 The 'k's are 2, 2, 4, 3, 4, 5, 7, 12, 15, 16, 19, 21, 57, 37, 38, 79, 48, 63, 72, 119, 306, 397, 469, 472, 582, 613, 927, 1461, 2926, 3449, ..., . - _Robert G. Wilson v_, Jul 30 2015
%e A259313 Digitsum(9) is 9, digitsum(9^2) is 9. (9+9)/2 = 9. So 9 is in this sequence.
%e A259313 12^1 = 12, 12^2 = 144, 12^3 = 1728 and 12^4 = 20736. Digitsum(12) = 3, digitsum(144) = 9, digitsum(1728) = 18, digitsum(20736) = 18, (3+9+18+18)/4 = 12. So 12 is in this sequence.
%t A259313 fQ[n_] := If[ IntegerQ@ Log10@ n, False, Block[{pwr = 2, s = Plus @@ IntegerDigits@ n}, While[s = s + Plus @@ IntegerDigits[n^pwr]; s < n*pwr, pwr++]; If[s == n*pwr, True, False]]]; k = 1; lst = {1}; While[k < 100001, If[fQ@ k, AppendTo[lst, k]]; k++]; lst (* _Robert G. Wilson v_, Jul 30 2015 *)
%o A259313 (Python)
%o A259313 def sod(n):
%o A259313     kk = 0
%o A259313     while n > 0:
%o A259313         kk= kk+(n%10)
%o A259313         n =int(n//10)
%o A259313     return kk
%o A259313 for c in range (2, 10**3):
%o A259313     bb=0
%o A259313     for a in range(1,200):
%o A259313         bb=bb+sod(c**a)
%o A259313         if bb==c*a:
%o A259313             print (c,a)
%Y A259313 Cf. A007953, A061910, A061209, A061210.
%K A259313 nonn,base,more
%O A259313 1,2
%A A259313 _Pieter Post_, Jun 24 2015
%E A259313 a(21)-a(28) from _Giovanni Resta_, Jun 24 2015
%E A259313 a(1)-a(28) checked by _Robert G. Wilson v_, Jul 30 2015
%E A259313 a(29)-a(30) from _Robert G. Wilson v_, Jul 30 2015