A066001 -id:A066001 - cp's OEIS frontend

A001370 Sum of digits of 2^n.

1, 2, 4, 8, 7, 5, 10, 11, 13, 8, 7, 14, 19, 20, 22, 26, 25, 14, 19, 29, 31, 26, 25, 41, 37, 29, 40, 35, 43, 41, 37, 47, 58, 62, 61, 59, 64, 56, 67, 71, 61, 50, 46, 56, 58, 62, 70, 68, 73, 65, 76, 80, 79, 77, 82, 92, 85, 80, 70, 77

Offset: 0

N. J. A. Sloane, Simon Plouffe

Same digital roots as A065075 (sum of digits of the sum of the preceding numbers) and A004207 (sum of digits of all previous terms); they enter into the cycle {1 2 4 8 7 5}. - Alexandre Wajnberg, Dec 11 2005
It is believed that a(n) ~ n*9*log_10(2)/2, but this is an open problem. - N. J. A. Sloane, Apr 21 2013
The Radcliffe preprint shows that a(n) > log_4(n). - M. F. Hasler, May 18 2017
Sierpiński shows that if n >= A137284(k-1) then a(n) >= k (Problem 209). - David Radcliffe, Dec 26 2022

Archimedeans Problems Drive, Eureka, 26 (1963), 12.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Zak Seidov, Table of n, a(n) for n = 0..10000
David Radcliffe, The growth of digital sums of powers of two. Preprint, 2015.
David G. Radcliffe, The growth of digital sums of powers of two, arXiv:1605.02839 [math.NT], 2016.
W. Sierpiński, 250 Problems in Elementary Number Theory, 1970.
C. L. Stewart, On the representation of an integer in two different bases, Journal für die reine und angewandte Mathematik 319 (1980): 63-72.

Cf. sum of digits of k^n: A004166 (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).

Cf. A007953, A000079, A261009, A011754, A137284.

a001370 = a007953 . a000079  -- Reinhard Zumkeller, Aug 14 2015

seq(convert(convert(2^n,base,10),`+`),n=0..1000); # Robert Israel, Mar 29 2015

Table[Total[IntegerDigits[2^n]], {n, 0, 55}] (* Vincenzo Librandi, Oct 08 2013 *)

a(n)=sumdigits(2^n); \\ Michel Marcus, Nov 01 2013

[sum(map(int, str(2**n))) for n in range(56)] # David Radcliffe, Mar 29 2015

a(n) = A007953(A000079(n)). - Michel Marcus, Nov 01 2013

A004166 Sum of digits of 3^n.

Original entry on oeis.org

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

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

A175527 Digit sum of 13^n.

Original entry on oeis.org

1, 4, 16, 19, 22, 25, 37, 40, 34, 46, 67, 52, 55, 58, 97, 73, 85, 88, 91, 85, 115, 91, 121, 106, 109, 121, 133, 118, 121, 133, 163, 184, 169, 181, 193, 169, 172, 175, 178, 199, 193, 214, 226, 238, 169, 190, 247, 241, 208, 247, 232, 253, 292, 241, 316, 292, 268, 271, 301, 286, 298, 337, 304, 325

Offset: 0

N. J. A. Sloane, Dec 03 2010

It is surprising that many values repeat twice (for 85, 91, 121, 133, 169 this happens at a(n) = a(n+3) (but 169 occurs later for a third time), for 193, 241, 292, ... the second occurrence comes later) while many other values never occur. Is there a simple explanation? - M. F. Hasler, May 18 2017

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A001022, A175528, A175525.

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), A066006 (k=12), this sequence (k=13).

Table[Total[IntegerDigits[13^k]], {k,0,1000}]

a(n)=sumdigits(13^n); \\ Michel Marcus, Nov 01 2013

a(n) = A007953(A001022(n)). - Michel Marcus, Nov 01 2013

a(n) ~ 4.5*log_10(13)*n ~ 5.0127*n (conjectured). - M. F. Hasler, May 18 2017

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

cp's OEIS Frontend

A001370 Sum of digits of 2^n.

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A004166 Sum of digits of 3^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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A175527 Digit sum of 13^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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