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%I A175525 #51 Aug 29 2021 12:00:33
%S A175525 1,2,5,140,158,428,788,887,914,1814,1895,1976,2579,2732,3074,3299,
%T A175525 3641,4658,4874,5378,5423,5504,6170,6440,6944,8060,8249,8915,9041,
%U A175525 9158,9725,9824,10661,11291,13820,15305,17051,17393,18716,19589,20876,21641,23756,24188,25961,28409,30632,31307,32387,33215,34970,35240,36653,36977,41558,43970,44951,47444,51764,52655,53375,53852,54104,56831,57506,59153,66479,68063,73562,78485,79286,87908,92093,102029,106934,114854,116321,134051,139397,184037,192353,256469,281381,301118,469004
%N A175525 Numbers k that divide the sum of digits of 13^k.
%C A175525 Almost certainly there are no further terms.
%C A175525 Comments from _Donovan Johnson_ on the computation of this sequence, Dec 05 2010 (Start):
%C A175525 The number of digits of 13^k is approximately 1.114*k, so I defined an array d() that is a little bigger than 1.114 times the maximum k value to be checked. The elements of d() each are the value of a single digit of the decimal expansion of 13^k with d(1) being the least significant digit.
%C A175525 It's easier to see how the program works if I start with k = 2.
%C A175525 For k = 1, d(2) would have been set to 1 and d(1) would have been set to 3.
%C A175525 k = 2:
%C A175525 x = 13*d(1) = 13*3 = 39
%C A175525 y = 39\10 = 3 (integer division)
%C A175525 x-y*10 = 39-30 = 9, d(1) is set to 9
%C A175525 x = 13*d(2)+y = 13*1+3 = 16, y is the carry from previous digit
%C A175525 y = 16\10 = 1
%C A175525 x-y*10 = 16-10 = 6, d(2) is set to 6
%C A175525 x = 13*d(3)+y = 13*0+1 = 1, y is the carry from previous digit
%C A175525 y = 1\10 = 0
%C A175525 x-y*10 = 1-0 = 1, d(3) is set to 1
%C A175525 These steps would of course be inside a loop and that loop would be inside a k loop. A pointer to the most significant digit increases usually by one and sometimes by two for each successive k value checked. The number of steps of the inner loop is the size of the pointer. A scan is done from the first element to the pointer element to get the digit sum.
%C A175525 (End)
%C A175525 No other terms < 3*10^6. - _Donovan Johnson_, Dec 07 2010
%t A175525 Select[Range[1000], Mod[Total[IntegerDigits[13^#]], #] == 0 &]
%Y A175525 Sum of digits of k^n mod n: (k=2) A000079, A001370, A175434, A175169; (k=3) A000244, A004166, A175435, A067862; (k=5) A000351, A066001, A175456; (k=6) A000400, A066002, A175457, A067864; (k=7) A000420, A066003, A175512, A067863; (k=8) A062933; (k=13) A001022, A175527, A175528, A175525; (k=21) A175589; (k=167) A175558, A175559, A175560, A175552.
%K A175525 nonn,base
%O A175525 1,2
%A A175525 _T. D. Noe_, Dec 03 2010
%E A175525 a(47)-a(79) from _N. J. A. Sloane_, Dec 04 2010
%E A175525 a(80)-a(85) from _Donovan Johnson_, Dec 05 2010

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