cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-8 of 8 results.

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

Views

Author

N. J. A. Sloane, Dec 03 2010

Keywords

Comments

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

Links

Crossrefs

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).

Programs

  • Mathematica
    Table[Total[IntegerDigits[13^k]], {k,0,1000}]
  • PARI
    a(n)=sumdigits(13^n); \\ Michel Marcus, Nov 01 2013

Formula

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

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

Original entry on oeis.org

1, 2, 5, 70
Offset: 1

Views

Author

N. J. A. Sloane, Dec 03 2010

Keywords

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.

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.

Programs

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

Views

Author

N. J. A. Sloane, Dec 03 2010

Keywords

Comments

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

Links

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.

Programs

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

Extensions

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

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

Views

Author

N. J. A. Sloane, Dec 03 2010

Keywords

Examples

			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

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.

Programs

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

Extensions

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

A175528 (Digit sum of 13^n) mod n.

Original entry on oeis.org

0, 0, 1, 2, 0, 1, 5, 2, 1, 7, 8, 7, 6, 13, 13, 5, 3, 1, 9, 15, 7, 11, 14, 13, 21, 3, 10, 9, 17, 13, 29, 9, 16, 23, 29, 28, 27, 26, 4, 33, 9, 16, 23, 37, 10, 17, 6, 16, 2, 32, 49, 32, 29, 46, 17, 44, 43, 11, 50, 58, 32, 56, 10, 45, 33, 61, 60, 18, 67, 66, 47, 1, 17, 15, 22, 69, 18, 61, 5, 11, 73, 63, 42, 40, 29, 18, 7, 57, 12, 46, 53, 53, 49, 11, 18, 40, 84, 39, 37, 35
Offset: 1

Views

Author

N. J. A. Sloane, Dec 03 2010

Keywords

Links

Crossrefs

Cf. A001022, A175527, A175525.

Programs

  • Mathematica
    Array[Mod[Total[IntegerDigits[13^#]],#]&, 1000]

A062933 Numbers k such that k divides the sum of digits of 8^k.

Original entry on oeis.org

1, 2, 25, 70, 106, 268, 304, 358, 1559, 2369, 2824, 2855, 3616, 5218
Offset: 1

Views

Author

Shyam Sunder Gupta and Amarnath Murthy, Feb 16 2002

Keywords

Comments

The next term, if it exists, is greater than 100000. - Ryan Propper, Aug 31 2005
No further terms less than 1000000 using the same method Donovan Johnson explains in A175525.

Examples

			25 divides the sum of digits of 8^25 (i.e., 3+7+7+7+8+9+3+1+8+6+2+9+5+7+1+6+1+7+0+9+5+6+8 = 125), so 25 is in the sequence.

Programs

  • Mathematica
    k = 1; Do[k *= 8; s = Plus @@ IntegerDigits[k]; If[Mod[s, n] == 0, Print[n]], {n, 1, 10^5}] (* Ryan Propper, Aug 31 2005 *)
Select[Range[6000],Mod[Total[IntegerDigits[8^#]],#]==0&] (* Harvey P. Dale, Dec 26 2024 *)

Extensions

Corrected and extended by Ryan Propper, Aug 31 2005
Edited by Jon E. Schoenfield, May 29 2010

A175589 Numbers k that divide the sum of digits of 21^k.

Original entry on oeis.org

1, 3, 6, 9, 12, 18, 57, 84, 87, 102, 117, 216, 234, 288, 360, 468, 477, 681, 741, 798, 987, 1029, 1161, 1245, 1251, 1266, 1449, 1458, 1500, 1677, 2214, 2232, 2703, 2880, 3090, 3117, 3333, 3345, 3351, 3789, 4176, 4512, 4779, 4932, 4980, 5763, 6213, 6846, 6903, 6918, 8097, 8712, 9285, 10404, 11085, 11274, 11532, 11922, 12369, 12378, 14871, 16710
Offset: 1

Views

Author

T. D. Noe and N. J. A. Sloane, Dec 03 2010

Keywords

Comments

No more terms < 10^6 using the same method used for A175525. - Donovan Johnson, Dec 08 2010

Programs

  • Mathematica
    Select[Range[10000], Mod[Total[IntegerDigits[21^#]], #] == 0 &]

Extensions

a(54)-a(62) from Donovan Johnson, Dec 08 2010

A220365 a(n) is conjectured to be the largest power k for which k divides the sum of digits of n^k.

Original entry on oeis.org

1, 70, 486, 35, 10, 90, 805, 5218, 243, 1, 35, 1494, 469004, 1045, 288, 116, 7, 195, 29, 70, 16710, 23, 2, 1017, 28, 58, 162, 166, 209, 486, 205, 106, 1206, 2053, 37120
Offset: 1

Views

Author

Robert G. Wilson v, Dec 12 2012

Keywords

Comments

a(36) >= 423378.
Please consult the argument in A067863 for the reason that it is believed that all individual such sequences (all k's which divide b^k) terminate.

Examples

			a(2) = 70 since the sum of digits of 2^70 is divisible by 70 and it is believed that there does not exist any larger exponent which satisfies this criterion.

Crossrefs

Numbers n such that n divides the sum of digits of k^n: A175169 (k=2), A067862 (k=3), A067864 (k=6), A067863 (k=7), A062933 (k=8), A062927 (k=9), A175525 (k=13), A175589 (k=21), A220364 (k=36), A175552 (k=167).

Programs

  • Mathematica
    For any individual base, b, fQ[n_] := Mod[Plus @@ IntegerDigits[b^n], n] == 0; k = 1; lst = {}; While[k < 100001, If[ fQ@ k, AppendTo[lst, k]; Print[k]]; k++]; lst

Formula

If a(n) = k, then a(10*n) = k.

Extensions

Definition and example corrected by Giovanni Resta, Dec 14 2012
Showing 1-8 of 8 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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