A175527 -id:A175527

A004166

Sum of digits of 3^n.

+10
25

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

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

LINKS

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.)

FORMULA

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

MATHEMATICA

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 *)

PROG

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

(Python)

def a(n): return sum(map(int, str(3**n)))

print([a(n) for n in range(61)]) # Michael S. Branicky, Apr 25 2022

CROSSREFS

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).

KEYWORD

nonn,base

AUTHOR

N. J. A. Sloane

EXTENSIONS

Edited by M. F. Hasler, May 18 2017

STATUS

approved

A066001

Sum of digits of 5^n.

+10
23

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

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

LINKS

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

MATHEMATICA

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 *)

PROG

(PARI) SumD(x)= { local(s=0); while (x>9, s+=x%10; x\=10); return(s + x) } { for (n=0, 1000, a=SumD(5^n); write("b066001.txt", n, " ", a) ) } \\ Harry J. Smith, Nov 06 2009

(PARI) A066001=a(n)=sumdigits(5^n); \\ Michel Marcus, Sep 04 2014

CROSSREFS

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).

KEYWORD

nonn,base

AUTHOR

N. J. A. Sloane, Dec 11 2001

STATUS

approved

A066002

Sum of digits of 6^n.

+10
20

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

LINKS

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

FORMULA

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

MATHEMATICA

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

PROG

(PARI) SumD(x)= { local(s=0); while (x>9, s+=x%10; x\=10); return(s + x) } { for (n=0, 1000, a=SumD(6^n); write("b066002.txt", n, " ", a) ) } \\ Harry J. Smith, Nov 06 2009

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

CROSSREFS

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.

KEYWORD

nonn,base

AUTHOR

N. J. A. Sloane, Dec 11 2001

STATUS

approved

A066003

Sum of digits of 7^n.

+10
20

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

LINKS

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

FORMULA

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

MATHEMATICA

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

PROG

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

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

CROSSREFS

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).

KEYWORD

nonn,base

AUTHOR

N. J. A. Sloane, Dec 11 2001

STATUS

approved

A065999

Sum of digits of 9^n.

+10
15

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

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

LINKS

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.

FORMULA

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

MATHEMATICA

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

PROG

(PARI) SumD(x)= { local(s=0); while (x>9, s+=x%10; x\=10); return(s + x) }

{ for (n=0, 1000, a=SumD(9^n); write("b065999.txt", n, " ", a) ) } \\ Harry J. Smith, Nov 06 2009

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

CROSSREFS

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.

KEYWORD

nonn,easy,base

AUTHOR

N. J. A. Sloane, Dec 11 2001

STATUS

approved

A066004

Sum of digits of 8^n.

+10
14

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

LINKS

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

FORMULA

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

MATHEMATICA

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

PROG

(PARI) SumD(x)= { local(s=0); while (x>9, s+=x%10; x\=10); return(s + x) } { for (n=0, 1000, a=SumD(8^n); write("b066004.txt", n, " ", a) ) } \\ Harry J. Smith, Nov 06 2009

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

CROSSREFS

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).

KEYWORD

nonn,base

AUTHOR

N. J. A. Sloane, Dec 11 2001

STATUS

approved

A066005

Sum of digits of 11^n.

+10
14

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

LINKS

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

FORMULA

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

MATHEMATICA

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

PROG

(PARI) SumD(x)= { local(s=0); while (x>9, s+=x%10; x\=10); return(s + x) } { f=1; for (n=0, 1000, a=SumD(f); f*=11; write("b066005.txt", n, " ", a) ) } [Harry J. Smith, Nov 06 2009]

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

CROSSREFS

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).

KEYWORD

nonn,base

AUTHOR

N. J. A. Sloane, Dec 11 2001

STATUS

approved

A065713

Sum of digits of 4^n.

+10
13

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

LINKS

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

FORMULA

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

MATHEMATICA

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

PROG

(PARI) SumD(x)= { local(s=0); while (x>9, s+=x-10*(x\10); x\=10); return(s + x) } { for (n=0, 1000, a=SumD(4^n); write("b065713.txt", n, " ", a) ) } \\ Harry J. Smith, Oct 27 2009

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

CROSSREFS

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).

KEYWORD

nonn,base

AUTHOR

N. J. A. Sloane, Dec 11 2001

STATUS

approved

A066006

Sum of digits of 12^n.

+10
13

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

LINKS

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

FORMULA

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

MATHEMATICA

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

PROG

(PARI) SumD(x)= { local(s=0); while (x>9, s+=x%10; x\=10); return(s + x) } { for (n=0, 1000, a=SumD(12^n); write("b066006.txt", n, " ", a) ) } \\ Harry J. Smith, Nov 06 2009

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

CROSSREFS

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).

KEYWORD

nonn,base

AUTHOR

N. J. A. Sloane, Dec 11 2001

STATUS

approved

A175169

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

+10
9

1, 2, 5, 70

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

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.

LINKS

Table of n, a(n) for n=1..4.

MATHEMATICA

Select[Range[200000], Divisible[Total[IntegerDigits[2^#]], #]&]

(* Harvey P. Dale, Dec 16 2010 *)

PROG

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

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.

KEYWORD

nonn,base

AUTHOR

N. J. A. Sloane, Dec 03 2010

STATUS

approved