A001370 -id:A001370 - cp's OEIS frontend

A066004 Sum of digits of 8^n.

1, 8, 10, 8, 19, 26, 19, 26, 37, 35, 37, 62, 64, 71, 46, 62, 73, 80, 82, 80, 82, 89, 109, 89, 109, 125, 100, 107, 118, 107, 118, 125, 127, 107, 118, 125, 145, 143, 145, 152, 172, 170, 172, 188, 181, 170, 190, 215, 172, 215, 235, 233, 217, 215

Offset: 0

N. J. A. Sloane, Dec 11 2001

Harry J. Smith, Table of n, a(n) for n = 0..1000

Cf. sum of digits of k^n: A001370 (k=2), A004166 (k=3), A065713 (k=4), A066001 (k=5), A066002 (k=6), A066003(k=7), this sequence (k=8), A065999 (k=9), A066005 (k=11), A066006 (k=12), A175527 (k=13).

Table[Total[IntegerDigits[8^n]], {n, 0, 60}] (* Vincenzo Librandi, Oct 08 2013 *)

a(n) = sumdigits(8^n); \\ Michel Marcus, Nov 01 2013

a(n) = A007953(A001018(n)). - Michel Marcus, Nov 01 2013

A066005 Sum of digits of 11^n.

Original entry on oeis.org

1, 2, 4, 8, 16, 14, 28, 38, 40, 53, 43, 41, 55, 47, 76, 71, 88, 86, 82, 83, 94, 71, 97, 95, 118, 101, 112, 125, 124, 140, 145, 137, 139, 143, 178, 140, 172, 200, 184, 188, 205, 203, 190, 164, 175, 215, 196, 248, 190, 218, 265, 251, 223, 230

Offset: 0

N. J. A. Sloane, Dec 11 2001

Harry J. Smith, Table of n, a(n) for n = 0..1000

Cf. sum of digits of k^n: A001370 (k=2), A004166 (k=3), A065713 (k=4), A066001 (k=5), A066002 (k=6), A066003 (k=7), A066004 (k=8), A065999 (k=9), this sequence (k=11), A066006 (k=12), A175527 (k=13).

Total/@(IntegerDigits/@(11^Range[0,60])) (* Harvey P. Dale, Nov 02 2011 *)

a(n) = sumdigits(11^n); \\ Michel Marcus, Nov 01 2013

a(n) = A007953(A001020(n)). - Michel Marcus, Nov 01 2013

A065713 Sum of digits of 4^n.

Original entry on oeis.org

1, 4, 7, 10, 13, 7, 19, 22, 25, 19, 31, 25, 37, 40, 43, 37, 58, 61, 64, 67, 61, 46, 58, 70, 73, 76, 79, 82, 85, 70, 82, 85, 88, 109, 103, 70, 109, 130, 106, 100, 112, 124, 118, 112, 115, 118, 139, 151, 127, 112, 115, 118, 121, 142, 145, 121, 160

Offset: 0

N. J. A. Sloane, Dec 11 2001

Harry J. Smith, Table of n, a(n) for n = 0..1000

Cf. A000302, A007953.

Cf. sum of digits of k^n: A001370 (k=2), A004166 (k=3), this sequence (k=4), A066001 (k=5), A066002 (k=6), A066003(k=7), A066004 (k=8), A065999 (k=9), A066005 (k=11), A066006 (k=12), A175527 (k=13).

Table[Total[IntegerDigits[4^n]], {n, 0, 60}] (* Vincenzo Librandi, Oct 08 2013 *)

a065713(n)=sumdigits(4^n); \\ Michel Marcus, Nov 01 2013

a(n) = A007953(A000302(n)). - Michel Marcus, Nov 01 2013 [corrected by Georg Fischer, Dec 19 2020]

a(n) = A001370(2n). Results given there imply a(n) > log_4(n) + 1/2, n > 0, but we can conjecture & expect a(n) ~ 9*log_10(2)*n. - M. F. Hasler, May 18 2017

A066006 Sum of digits of 12^n.

Original entry on oeis.org

1, 3, 9, 18, 18, 27, 45, 36, 54, 45, 45, 54, 54, 63, 81, 72, 90, 72, 81, 117, 108, 90, 99, 99, 117, 117, 135, 153, 135, 135, 153, 180, 153, 117, 117, 180, 171, 171, 189, 198, 216, 198, 225, 225, 216, 198, 225, 234, 252, 234, 216, 234, 279, 243

Offset: 0

N. J. A. Sloane, Dec 11 2001

Harry J. Smith, Table of n, a(n) for n = 0..1000

Cf. sum of digits of k^n: A001370 (k=2), A004166 (k=3), A065713 (k=4), A066001 (k=5), A066002 (k=6), A066003(k=7), A066004 (k=8), A065999 (k=9), A066005 (k=11), this sequence (k=12), A175527 (k=13).

Table[Total[IntegerDigits[12^n]], {n, 0, 60}] (* Vincenzo Librandi, Oct 08 2013 *)

a(n) = sumdigits(12^n); \\ Michel Marcus, Nov 01 2013

a(n) = A007953(A001021(n)). - Michel Marcus, Nov 01 2013

A175169 Numbers k that divide the sum of digits of 2^k.

Original entry on oeis.org

1, 2, 5, 70

Offset: 1

N. J. A. Sloane, Dec 03 2010

No other terms <= 200000. - Harvey P. Dale, Dec 16 2010
No other terms <= 1320000. - Robert G. Wilson v, Dec 18 2010
There are almost certainly no further terms.

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.

Select[Range[200000],Divisible[Total[IntegerDigits[2^#]],#]&]
(* Harvey P. Dale, Dec 16 2010 *)

is(n)=sumdigits(2^n)%n==0 \\ Charles R Greathouse IV, Sep 06 2016

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

Original entry on oeis.org

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

T. D. Noe, Dec 03 2010

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

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.

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

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

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

A175552 Numbers k such that the digit sum of 167^k is divisible by k.

Original entry on oeis.org

1, 2, 5, 7, 22, 490, 724, 778, 868, 994, 1109, 1390, 1415, 1462, 1642, 1739, 1829, 2146, 2362, 3136, 4954, 6437, 6628, 7103, 11200, 12424, 12863, 14242, 14249, 15059, 15203, 16222, 17140, 18353, 19192, 21233, 22853, 24106, 24574, 24833, 26896, 27652, 28253, 30323, 31306, 31594, 32386, 33790, 34985, 36184, 36310, 40673, 42196, 43931, 45911, 45983

Offset: 1

N. J. A. Sloane, Dec 03 2010

From Donovan Johnson, Dec 03 2010: (Start)
To generate the additional terms I used PFGW.exe to get the decimal expansion for each number of the form 167^n (n <= 50000). Then I wrote a program in powerbasic to read the pfgw.out file and get the digit sums.
The digit sum is 10 times the n value for terms a(5) to a(56). (End)
I believe that this sequence is finite. - N. J. A. Sloane, Dec 05 2010

Donovan Johnson, Table of n, a(n) for n = 1..129

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.

Select[Range[10000], Mod[Total[IntegerDigits[167^#]], #] == 0 &]

a(25)-a(56) from Donovan Johnson, Dec 03 2010

A175435 (Digit sum of 3^n) mod n.

Original entry on oeis.org

0, 1, 0, 1, 4, 0, 4, 2, 0, 7, 5, 6, 1, 3, 6, 11, 10, 9, 17, 5, 6, 1, 8, 6, 13, 11, 0, 16, 14, 3, 19, 31, 6, 31, 11, 9, 16, 14, 3, 10, 17, 6, 22, 2, 9, 16, 32, 21, 10, 44, 15, 13, 29, 0, 35, 34, 51, 10, 58, 39, 13, 29, 27, 16, 5, 51, 28, 17, 15, 4, 38, 18, 7

Offset: 1

N. J. A. Sloane, Dec 03 2010

			a(5) = 4 because digsum(3^5) mod 5 = digsum(243) mod 5 = 9 mod 5 = 4.

Nathaniel Johnston, Table of n, a(n) for n = 1..5000

Cf. A000244, A004166, A067862; A000079, A001370, A175434, A175169.

read(transforms): A175435 := proc(n) return digsum(3^n) mod n: end proc: seq(A175435(n), n=1..50); # Nathaniel Johnston, Oct 09 2013

a(n) = sumdigits(3^n) % n; \\ Michel Marcus, Oct 09 2013

A095412 Exponents k such that the sum of decimal digits of 2^k is also a power of 2.

Original entry on oeis.org

0, 1, 2, 3, 9, 36, 85, 176, 194, 200, 375, 1517, 1523, 3042, 5953, 6043, 6109, 12068, 12104, 96251, 193734, 386797, 387589, 1545477, 3092224, 3098800, 6188717, 6191693, 6199469, 24753865, 99084345

Offset: 1

Labos Elemer, Jun 21 2004

			2^9 = 512 with digit sum = 8;
2^36 = 68719476736 with digit sum = 64;
2^85 = 38685626227668133590597632 with digit sum = 128;
2^96251 has a decimal digit sum of 131072.

Cf. A001370 (sum of digits of 2^n).

Do[If[IntegerQ[Log[2, Plus@@IntegerDigits[2^n]]], Print[n] ], {n, 0, 10^6}];

isp(n) = (n==1) || (n==2) || (ispower(n,,&k) && (k==2));
isok(n) = isp(sumdigits(2^n)); \\ Michel Marcus, Apr 25 2017

More terms from Ryan Propper, Jun 13 2006
a(21)-a(23) from Ray Chandler, Jun 16 2006
a(24)-a(29) from Jon E. Schoenfield, Jul 22 2006
a(30) from Giovanni Resta, Apr 24 2017
a(31) from Bert Dobbelaere, Feb 22 2019
Offset corrected by Jon E. Schoenfield, Nov 25 2022

A175434 (Digit sum of 2^n) mod n.

Original entry on oeis.org

0, 0, 2, 3, 0, 4, 4, 5, 8, 7, 3, 7, 7, 8, 11, 9, 14, 1, 10, 11, 5, 3, 18, 13, 4, 14, 8, 15, 12, 7, 16, 26, 29, 27, 24, 28, 19, 29, 32, 21, 9, 4, 13, 14, 17, 24, 21, 25, 16, 26, 29, 27, 24, 28, 37, 29, 23, 12, 18, 22, 13, 23, 26, 24, 21, 43, 43, 35, 20, 0, 15, 37, 37, 56, 50, 30, 27, 22, 31, 32, 26, 42, 39, 34, 43, 26, 20, 27, 24, 28, 55, 47, 32, 57, 45, 31, 40, 14, 8, 15

Offset: 1

N. J. A. Sloane, Dec 03 2010

			For n = 1,2,3,4,5,6, the digit-sum of 2^n is 2,4,8,7,5,10, so
a(1) through a(6) are 0,0,2,3,0,4. - _N. J. A. Sloane_, Aug 12 2014

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.

Table[Mod[Total[IntegerDigits[2^n]],n],{n,100}] (* Harvey P. Dale, Aug 12 2014 *)

Offset changed to 1 at the suggestion of Harvey P. Dale, Aug 12 2014

cp's OEIS Frontend

A066004 Sum of digits of 8^n.

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A066005 Sum of digits of 11^n.

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A065713 Sum of digits of 4^n.

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A066006 Sum of digits of 12^n.

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A175169 Numbers k that divide the sum of digits of 2^k.

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

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A175552 Numbers k such that the digit sum of 167^k is divisible by k.

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A175435 (Digit sum of 3^n) mod n.

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A095412 Exponents k such that the sum of decimal digits of 2^k is also a power of 2.