A004166 -id:A004166 - cp's OEIS frontend

A292931 Numbers n such that the sum of digits of 3^n (A004166) is divisible by 7.

25, 26, 30, 32, 47, 58, 79, 81, 87, 89, 102, 123, 141, 144, 145, 151, 164, 176, 178, 193, 201, 227, 239, 242, 257, 264, 282, 289, 300, 306, 319, 324, 329, 335, 336, 338, 348, 351, 358, 365, 395, 403, 437, 441, 450, 460, 468, 484, 489, 492, 495, 517, 518, 541, 542, 544, 554, 555, 563, 565, 570

Offset: 1

Robert Israel, Sep 27 2017

			a(3) = 30 is in the sequence because 3^30 = 205891132094649 has sum of digits 63, which is divisible by 7.

Chai Wah Wu, Table of n, a(n) for n = 1..10000
Mathematics StackExchange, Is sum of digits of 3^1000 divisible by 7?

Cf. A004166.

select(n -> convert(convert(3^n,base,10),`+`) mod 7 = 0, [$1..1000]);

Select[Range[600],Divisible[Total[IntegerDigits[3^#]],7]&] (* Harvey P. Dale, Mar 01 2018 *)

isok(n) = !(sumdigits(3^n) % 7); \\ Michel Marcus, Sep 27 2017

from _future_ import division
A292931_list = [n for n in range(1000) if not sum(int(d) for d in str(3**n)) % 7] # Chai Wah Wu, Sep 28 2017

A292995 Sum of digits of 3^n (A004166) divided by 9.

Original entry on oeis.org

0, 0, 1, 1, 1, 1, 2, 2, 2, 3, 3, 3, 2, 3, 5, 4, 3, 3, 5, 4, 5, 3, 5, 6, 6, 7, 7, 9, 8, 8, 7, 9, 7, 8, 11, 9, 9, 10, 10, 9, 10, 11, 10, 12, 10, 11, 12, 14, 13, 12, 16, 13, 13, 15, 12, 10, 10, 12, 14, 13, 11, 15, 17, 17, 16, 15, 13, 18, 17, 17, 16, 20, 18, 17, 19, 20, 17, 18

Offset: 0

M. F. Hasler, Sep 27 2017

All terms A004166(n), n >= 2, are multiples of 9.
For the first two terms, the (zero) integer part of the fractional values (1/9 and 3/9) is taken: This seems to be the most natural extension of the maybe more natural variant of this sequence which would start only at offset n = 2.
Divisibility of A004166(n) by any prime different from 3 is equivalent to divisibility of a(n) by that prime. For example, indices of terms of A004166 divisible by 7, listed in A292931, are also exactly the indices > 1 of terms a(n) divisible by 7.

Chai Wah Wu, Table of n, a(n) for n = 0..10000

Cf. A004166, A292931.

[n lt 2 select 0 else &+Intseq(3^n)/9: n in [0..100]]; // Vincenzo Librandi, Sep 28 2017

0,0,seq(convert(convert(3^n,base,10),`+`)/9, n=2..100); # Robert Israel, Sep 28 2017

Rest[Table[Sum[DigitCount[(3^n)][[i]] i, {i, 9}] / 9, {n, 100}]] (* Vincenzo Librandi, Sep 28 2017 *)

PARI
```
a(n)=sumdigits(3^n)\9
```

from _future_ import division
def A292995(n):
    return sum(int(d) for d in str(3**n))//9 # Chai Wah Wu, Sep 28 2017

A001370 Sum of digits of 2^n.

Original entry on oeis.org

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

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

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

cp's OEIS Frontend

A292931 Numbers n such that the sum of digits of 3^n (A004166) is divisible by 7.

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A292995 Sum of digits of 3^n (A004166) divided by 9.

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A001370 Sum of digits of 2^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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A175527 Digit sum of 13^n.

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