A259313 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.

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

Offset

1,2

Comments

Digitsum = (A007953).

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

Links

Table of n, a(n) for n=1..30.

Example

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.

Mathematica

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 *)

Prog

(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)

Crossrefs

Cf. A007953, A061910, A061209, A061210.

Sequence in context: A295486 A032687 A351042 * A170951 A044859 A336754

Adjacent sequences: A259310 A259311 A259312 * A259314 A259315 A259316

Keyword

nonn,base,more

Author

Pieter Post, Jun 24 2015

Extensions

a(21)-a(28) from Giovanni Resta, Jun 24 2015

a(1)-a(28) checked by Robert G. Wilson v, Jul 30 2015

a(29)-a(30) from Robert G. Wilson v, Jul 30 2015

Status

approved