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A259313
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Numbers n for which there exists a k>=2 such that n equals the average of digitsum(n^p) for p from 1 to k.
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1
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1, 9, 12, 13, 16, 19, 21, 49, 61, 67, 84, 106, 160, 191, 207, 250, 268, 373, 436, 783, 2321, 3133, 3786, 3805, 4842, 5128, 8167, 13599, 29431, 35308
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OFFSET
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1,2
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COMMENTS
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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
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LINKS
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EXAMPLE
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Digitsum(9) is 9, digitsum(9^2) is 9. (9+9)/2 = 9. So 9 is in this sequence.
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.
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MATHEMATICA
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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 *)
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PROG
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(Python)
def sod(n):
....kk = 0
....while n > 0:
........kk= kk+(n%10)
........n =int(n//10)
....return kk
for c in range (2, 10**4):
....bb=0
....for a in range(1, 200):
........bb=bb+sod(c**a, 10)
........if bb==c*a:
............print (c, a)
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CROSSREFS
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KEYWORD
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nonn,base,more
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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