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 1-5 of 5 results.

A108825 Numbers j such that j divides the sum of the digits of j!.

Original entry on oeis.org

1, 2, 3, 9, 15, 18, 21, 27, 72, 81, 234, 462, 502, 522, 1314, 1323, 3789, 3897, 6462, 10470, 17532, 17820, 28503, 48248, 48254, 48303, 48644, 48856, 223551, 226149, 227406, 625986, 4838918, 4848004
Offset: 1

Views

Author

Ryan Propper, Jul 11 2005

Keywords

Comments

Especially for larger values, terms can be expected to occur near values of k such that 4.5 times the number of digits in k!, not counting the digits in the string of trailing zeros, is approximately an integer multiple (m) of k. As m increases, such values of k approach k = e * 100^(m/9 + 1/8). - Jon E. Schoenfield, Jun 08 2007
a(35) > 13800000. - Giovanni Resta, Dec 21 2012

Examples

			3! = 6; 6 mod 3 = 0.
9! = 362880; 3 + 6 + 2 + 8 + 8 = 27; 27 mod 9 = 0.
522 is a term because the digit sum of 522!, 4698, is divisible by 522.

Crossrefs

Cf. A125526, A125724.

Programs

  • Mathematica
    Do[If[Mod[Plus @@ IntegerDigits[n! ], n] == 0, Print[n]], {n, 1, 10000}]

Extensions

a(20)-a(28) from Ray Chandler, Jul 25 2005
a(29)-a(31) from Jon E. Schoenfield, Jun 08 2007
Edited by N. J. A. Sloane, Jul 02 2008 at the suggestion of R. J. Mathar
a(32) from Donovan Johnson, Dec 08 2010
a(33)-a(34) from Giovanni Resta, Dec 21 2012

A108827 Numbers k that divide the sum of the digits of k^k.

Original entry on oeis.org

1, 2, 3, 9, 18, 27, 54, 90, 108, 163, 197, 254, 432, 1292, 2202, 9648, 10347, 16596, 17203, 46188, 46992, 77121, 130082, 167410, 216546, 596277
Offset: 1

Views

Author

Ryan Propper, Jul 11 2005

Keywords

Comments

Especially for larger terms k not divisible by 10, we can expect 4.5 times the number of digits in k^k to be close to some integer multiple (m) of k, so k should occur near 100^(m/9). E.g., for m = 10, 11, ..., 16, approximate (and corresponding actual) values would be 167 (163, 197), 278 (254), 464 (432), 774 (none), 1292 (1292), 2154 (2022) and 3594 (none). Larger terms k ending with exactly j zeros would be expected to occur near k = 10^j * 100^(m/9) for some integer m. - Jon E. Schoenfield, Jun 09 2007
The quotients are 1, 2, 3, 5, 6, 7, 7, 4, 9, 10, 10, 11, 12, 14, 15, 18, 18, 19, 19, 21, 21, 22, 23, 19, 24, 26.

Examples

			3^3 = 27; 2 + 7 = 9; 9 mod 3 = 0.
9^9 = 387420489; 3 + 8 + 7 + 4 + 2 + 4 + 8 + 9 = 45; 45 mod 9 = 0.
432 is a term because the sum of the digits of 432^432 = 5184 is divisible by 432.

Crossrefs

Cf. A125526, A125724, A108825.

Programs

  • Maple
    a:=proc(n) local nn: nn:=convert(n^n,base,10): if type(add(nn[j],j=1..nops(nn))/n, integer)=true then n else fi end: seq(a(n),n=1..2000); # Emeric Deutsch
  • Mathematica
    Do[If[Mod[Plus @@ IntegerDigits[n^n], n] == 0, Print[n]], {n, 1, 10000}]
Select[Range[600000],Divisible[Total[IntegerDigits[#^#]],#]&] (* Harvey P. Dale, Jan 28 2017 *)

Extensions

a(16)-a(19) from Simon Nickerson (simonn(AT)maths.bham.ac.uk) and Emeric Deutsch, Jul 15 2005
a(20)-a(22) from Ray Chandler, Jul 25 2005
Edited by N. J. A. Sloane, Apr 27 2008 at the suggestion of Stefan Steinerberger
a(23) from Robert G. Wilson v, May 17 2008
a(24) from Robert G. Wilson v, May 19 2008
a(25)-a(26) from Lars Blomberg, Jul 09 2011

A125724 If "sumdigit" denotes the sum of the digits of a number then these are the numbers n such that n=sumdigit(sumdigit(n)^sumdigit(n)).

Original entry on oeis.org

1, 13, 25, 45, 58, 88, 98
Offset: 1

Views

Author

Paolo P. Lava and Giorgio Balzarotti, Feb 02 2007

Keywords

Comments

Sequence is complete. - Giovanni Resta, Mar 21 2013

Examples

			sumdigit(13)=4; 4^4 = 256; sumdigit(256)=13

Crossrefs

Cf. A125526.

Programs

  • Maple
    P:=proc(n) local i,j,k,w; for i from 1 by 1 to n do w:=0; k:=i; while k>0 do w:=w+k-trunc(k/10)*10; k:=trunc(k/10); od; k:=w^w; w:=0; while k>0 do w:=w+k-trunc(k/10)*10; k:=trunc(k/10); od; if (i=w) then print(i,w); fi; od; end: P(100);
  • Mathematica
    sdQ[n_]:=Module[{sd=Total[IntegerDigits[n]]},n==Total[ IntegerDigits[ sd^sd]]]; Select[Range[100],sdQ] (* Harvey P. Dale, Oct 22 2013 *)

A140604 Least nontrivial number k such that the sum of the digits of k^k (mod k) == n.

Original entry on oeis.org

1, 4, 55, 6, 7, 8, 12, 2236, 11, 15, 14, 20, 21, 17, 274, 35, 22, 44, 36, 82, 73, 41, 29, 28, 26, 115, 85, 98, 2054, 31, 46, 502, 40, 39, 79, 3248, 45, 38, 128, 64, 511, 80, 183, 83, 76, 47, 127, 176, 52, 70, 190, 57, 65, 425, 63, 56, 95, 59, 10327, 794, 1248, 89, 410, 69
Offset: 0

Views

Author

Robert G. Wilson v, May 17 2008

Keywords

Examples

			1^1 (mod 1)==0; 4^4=256 so 13 (mod 4)==1; 55^55=... so 442 (mod 55)==2, 6^6=46656 so 27 (mod 6)==3; etc.

Links

Crossrefs

Cf. A108827, A125526, A125724, A108825.

Programs

  • Mathematica
    t = Table[0, {101}]; Do[ a = Mod[Plus @@ IntegerDigits[n^n], n]; If[a < 101 && t[[a + 1]] == 0, t[[a + 1]] = n; Print[{a, n}]], {n, 10000}]

A334249 Numbers k such that (k - digitsum(k))(k + digitsum(k)) contains k as a substring.

Original entry on oeis.org

11, 88, 101, 448, 673, 776, 1001, 2879, 3553, 9537, 10001, 14651, 36559, 49056, 51073, 54116, 59600, 100001, 505025, 998999, 1000001, 4115964, 5050250, 5133355, 10000001, 10050125, 19349727, 26550976, 33726078, 35792647, 42349456, 43605459, 50050025, 66952741, 88027284, 88819024, 100000001, 105124922
Offset: 1

Views

Author

Scott R. Shannon, May 05 2020

Keywords

Comments

All numbers of the form 10^n + 1 (for n > 0) are in the sequence.

Examples

			11 is a term as digitsum(11) = 2 and (11 - 2)(11 + 2) = 117, which contains '11' as a substring.
9537 is a term as digitsum(9537) = 24 and (9537 - 24)(9537 + 24) = 90953793, which contains '9537' as a substring.

Links

Crossrefs

Cf. A007953, A257784, A125526, A082208, A225780.
Showing 1-5 of 5 results.

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