A175525 Numbers k that divide the sum of digits of 13^k.

1, 2, 5, 140, 158, 428, 788, 887, 914, 1814, 1895, 1976, 2579, 2732, 3074, 3299, 3641, 4658, 4874, 5378, 5423, 5504, 6170, 6440, 6944, 8060, 8249, 8915, 9041, 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

Offset: 1

Author

T. D. Noe, Dec 03 2010

Keywords

• nonn
• base

Comments

Almost certainly there are no further terms.

Comments from Donovan Johnson on the computation of this sequence, Dec 05 2010 (Start):

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.

It's easier to see how the program works if I start with k = 2.

For k = 1, d(2) would have been set to 1 and d(1) would have been set to 3.

k = 2:

x = 13*d(1) = 13*3 = 39

y = 39\10 = 3 (integer division)

x-y*10 = 39-30 = 9, d(1) is set to 9

x = 13*d(2)+y = 13*1+3 = 16, y is the carry from previous digit

y = 16\10 = 1

x-y*10 = 16-10 = 6, d(2) is set to 6

x = 13*d(3)+y = 13*0+1 = 1, y is the carry from previous digit

y = 1\10 = 0

x-y*10 = 1-0 = 1, d(3) is set to 1

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.

(End)

No other terms < 3*10^6. - Donovan Johnson, Dec 07 2010

Crossrefs

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.

Programs

Mathematica

Select[Range[1000], Mod[Total[IntegerDigits[13^#]], #] == 0 &]

Extensions

a(47)-a(79) from N. J. A. Sloane, Dec 04 2010

a(80)-a(85) from Donovan Johnson, Dec 05 2010