A175527 -id:A175527 - cp's OEIS frontend

A004166 Sum of digits of 3^n.

1, 3, 9, 9, 9, 9, 18, 18, 18, 27, 27, 27, 18, 27, 45, 36, 27, 27, 45, 36, 45, 27, 45, 54, 54, 63, 63, 81, 72, 72, 63, 81, 63, 72, 99, 81, 81, 90, 90, 81, 90, 99, 90, 108, 90, 99, 108, 126, 117, 108, 144, 117, 117, 135, 108, 90, 90, 108, 126, 117, 99

Offset: 0

N. J. A. Sloane

All terms a(n), n > 1, are divisible by 9. - M. F. Hasler, Sep 27 2017

Michel Marcus, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Vincenzo Librandi)
Albert Frank, Solutions of International Contest Of Logical Sequences, 2002 - 2003. (The original Contest page without solutions was removed but remains available on web.archive.org.)

Cf. A067500, A118872, A175435.

Cf. sum of digits of k^n: A001370 (k=2), this sequence (k=3), A065713 (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).

Total[IntegerDigits[#]]&/@(3^Range[0,60]) (* Harvey P. Dale, Mar 03 2013 *)
Table[Total[IntegerDigits[3^n]], {n, 0, 60}] (* Vincenzo Librandi, Oct 08 2013 *)

a(n)=sumdigits(3^n); \\ Michel Marcus, Nov 01 2013

def a(n): return sum(map(int, str(3**n)))
print([a(n) for n in range(61)]) # Michael S. Branicky, Apr 25 2022

a(n) = A007953(A000244(n)). - Michel Marcus, Nov 01 2013

Edited by M. F. Hasler, May 18 2017

A066001 Sum of digits of 5^n.

Original entry on oeis.org

1, 5, 7, 8, 13, 11, 19, 23, 25, 26, 40, 38, 28, 23, 34, 44, 58, 56, 64, 59, 61, 62, 67, 74, 82, 77, 79, 89, 85, 83, 91, 104, 106, 89, 103, 92, 109, 104, 124, 134, 130, 137, 145, 149, 151, 116, 112, 128, 145, 158, 151, 152, 130, 119, 127, 167, 196, 215, 211, 191

Offset: 0

N. J. A. Sloane, Dec 11 2001

We can expect and conjecture that a(n) ~ 4.5*log_10(5)*n, but for n ~ 10^3..10^4 there are still fluctuations of +- 1%, e.g., a(10^3)/log_10(5) ~ 4538, a(10^4)/log_10(5) ~ 44518. Modulo 9, the sequence is periodic with period (1, 5, 7, 8, 4, 2) of length 6. No term is divisible by 3, a(n) = (-1)^n (mod 3). - M. F. Hasler, May 18 2017

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

Cf. A007953, A000351.

Cf. sum of digits of k^n: A001370 (k=2), A004166 (k=3), A065713 (k=4), this sequence (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[5^n], {n, 0, 60}] (* Robert G. Wilson v, Oct 25 2006 *)
Table[Total[IntegerDigits[5^n]], {n, 0, 60}] (* Vincenzo Librandi, Oct 08 2013 *)

a(n)=sumdigits(5^n); \\ Michel Marcus, Sep 04 2014

a(n) = A007953(A000351(n)). - Michel Marcus, Aug 05 2025

A066002 Sum of digits of 6^n.

Original entry on oeis.org

1, 6, 9, 9, 18, 27, 27, 36, 36, 36, 36, 45, 45, 36, 54, 63, 54, 72, 72, 63, 72, 81, 63, 72, 90, 90, 99, 99, 90, 135, 117, 99, 126, 126, 135, 135, 126, 135, 135, 162, 171, 126, 153, 153, 153, 162, 180, 162, 153, 162, 171, 216, 171, 216, 171, 162

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), this sequence (k=6), A066003(k=7), A066004 (k=8), A065999 (k=9), A066005 (k=11), A066006 (k=12), A175527 (k=13).

Cf. A007953.

Table[Total[IntegerDigits[6^n]], {n, 0, 60}] (* Vincenzo Librandi, Oct 08 2013 *)

a(n) = sumdigits(6^n); \\ Michel Marcus, Nov 01 2013

a(n) = A007953(A000400(n)). - Michel Marcus, Nov 01 2013 [corrected by Georg Fischer, Dec 28 2020]

A066003 Sum of digits of 7^n.

Original entry on oeis.org

1, 7, 13, 10, 7, 22, 28, 25, 31, 28, 43, 49, 37, 52, 58, 64, 52, 58, 73, 79, 76, 82, 97, 85, 73, 97, 112, 91, 133, 121, 118, 115, 103, 127, 142, 157, 136, 115, 130, 136, 142, 148, 136, 169, 175, 163, 187, 175, 136, 178, 184, 217, 196, 220, 217

Offset: 0

N. J. A. Sloane, Dec 11 2001

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

Cf. A000420 (7^n), A007953 (sum of digits).

Cf. sum of digits of k^n: A001370 (k=2), A004166 (k=3), A065713 (k=4), A066001 (k=5), A066002 (k=6), this sequence (k=7), A066004 (k=8), A065999 (k=9), A066005 (k=11), A066006 (k=12), A175527 (k=13).

Magma
```
[ &+Intseq(7^n): n in [0..60] ];
```

Table[Total[IntegerDigits[7^n]],{n,55}] (* Harvey P. Dale, Nov 22 2010 *)

a(n) = sumdigits(7^n); \\ Michel Marcus, Nov 01 2013

a(n) = A007953(A000420(n)). - Michel Marcus, Nov 01 2013

A065999 Sum of digits of 9^n.

Original entry on oeis.org

1, 9, 9, 18, 18, 27, 18, 45, 27, 45, 45, 45, 54, 63, 72, 63, 63, 99, 81, 90, 90, 90, 90, 108, 117, 144, 117, 108, 90, 126, 99, 153, 144, 117, 153, 144, 162, 171, 153, 153, 153, 198, 162, 171, 198, 216, 171, 198, 198, 225, 153, 252, 216, 234, 207

Offset: 0

N. J. A. Sloane, Dec 11 2001

a(n) mod 9 = 0 for n > 0. - Reinhard Zumkeller, May 14 2011

N. J. A. Sloane, Table of n, a(n) for n = 0..10000
M. Sapir et al., The Decimal Expansions of Powers of 9: Problem 10758, Amer. Math. Monthly, 108 (Dec., 2001), 977-978.
H. G. Senge and E. G. Straus, PV-numbers and sets of multiplicity, Periodica Math. Hungar., 3 (1971), 93-100.
C. L. Stewart, On the representation of an integer in two different bases, J. Reine Angew. Math., 319 (1980), 63-72.

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), this sequence (k=9), A066005 (k=11), A066006 (k=12), A175527 (k=13).

Cf. also A056888, A001019.

Table[Total[IntegerDigits[9^n]], {n, 0, 60}] (* Vincenzo Librandi, Oct 08 2013 *)

a(n) = sumdigits(9^n); \\ Michel Marcus, Nov 01 2013

a(n) = A007953(A001019(n)). - Michel Marcus, Nov 01 2013

A066004 Sum of digits of 8^n.

Original entry on oeis.org

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

cp's OEIS Frontend

A004166 Sum of digits of 3^n.

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A066001 Sum of digits of 5^n.

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A066002 Sum of digits of 6^n.

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A066003 Sum of digits of 7^n.

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A065999 Sum of digits of 9^n.

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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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