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.

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

A135204 Numbers n for which Sum_digits(n!) is a multiple of Sum_digits(n).

Original entry on oeis.org

1, 2, 3, 9, 10, 11, 12, 14, 16, 18, 20, 21, 22, 27, 28, 30, 33, 35, 36, 44, 45, 51, 54, 60, 61, 63, 72, 75, 81, 87, 90, 99, 100, 102, 105, 108, 111, 114, 117, 120, 126, 130, 135, 143, 144, 153, 158, 162, 165, 171, 180, 182, 185, 189, 190, 192, 200, 201, 202, 204, 206
Offset: 1

Views

Author

Paolo P. Lava and Giorgio Balzarotti, Nov 30 2007

Keywords

Comments

I expect a(n) to be around kn log n for some constant k. - Charles R Greathouse IV, Apr 24 2013

Examples

			11 -> 11*10*9*8*7*6*5*4*3*2*1=39916800 -> (3+9+9+1+6+8+0+0)/(1+1)=18.

Links

Crossrefs

Cf. A004152, A120390, A108825, A129980, A131954, A131955, A135205, A135206.

Programs

  • Maple
    P:=proc(n) local i,k,w,x; 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; x:=0; k:=i!; while k>0 do x:=x+k-(trunc(k/10)*10); k:=trunc(k/10); od; if trunc(x/w)=x/w then print(i); fi; od; end: P(1000);
  • Mathematica
    Select[Range[100], Divisible[Total[IntegerDigits[#!, 10]], Total[IntegerDigits[#, 10]]] &] (* G. C. Greubel, Sep 30 2016 *)
  • PARI
    is(n)=sumdigits(n!)%sumdigits(n)==0 \\ Charles R Greathouse IV, Apr 24 2013

A135205 Numbers m for which Sum_digits(m!!) is a multiple of Sum_digits(m).

Original entry on oeis.org

1, 2, 3, 4, 6, 9, 10, 11, 12, 15, 18, 20, 21, 24, 25, 27, 30, 32, 33, 36, 42, 45, 46, 54, 55, 63, 72, 75, 81, 88, 90, 91, 93, 100, 101, 102, 105, 108, 111, 112, 117, 120, 121, 122, 123, 124, 126, 127, 135, 141, 144, 153, 154, 156, 162, 171, 176, 180, 182, 189, 198
Offset: 1

Views

Author

Paolo P. Lava, Nov 30 2007

Keywords

Examples

			11 -> 11*9*7*5*3*1=10395 -> (1+0+3+9+5)/(1+1) = 9.

Links

Crossrefs

Cf. A004152, A120390, A108825, A129980, A131954, A131955, A135204, A135206.

Programs

  • Maple
    P:=proc(n) local i,j,k,w,x; 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; x:=i; j:=i-2; while j >0 do x:=x*j; j:=j-2; od: k:=x; x:=0; while k>0 do x:=x+k-(trunc(k/10)*10); k:=trunc(k/10); od; if trunc(x/w)=x/w then print(i); fi; od; end: P(1000);
  • Mathematica
    Select[Range[100], Divisible[Total[IntegerDigits[#!!, 10]], Total[IntegerDigits[#, 10]]] &] (* G. C. Greubel, Sep 30 2016 *)

Extensions

Offset 1 and b-file adapted by Paolo P. Lava, Jun 17 2024

A135206 Numbers m for which Sum_digits(m!) is a multiple of Sum_digits(m!!).

Original entry on oeis.org

1, 2, 3, 11, 19, 28, 48, 64, 158, 164, 190, 308, 324, 602, 782, 926, 1202, 1540, 1568, 1614, 2076, 2122, 2340, 2546, 2818, 2858, 2866, 3334, 3582, 3714, 4120, 4266, 4794, 5084, 5432, 5454, 5696, 6112, 6250, 6276, 6358, 6760, 7368, 8218, 8970, 9004, 9088
Offset: 1

Views

Author

Paolo P. Lava, Nov 30 2007

Keywords

Examples

			11!=11*10*9*8*7*6*5*4*3*2*1=39916800 -> (3+9+9+1+6+8+0+0)=36,
11!!=11*9*7*5*3*1=10395 -> (1+0+3+9+5)=18,
36/18=2.

Links

Crossrefs

Cf. A004152, A120390, A108825, A129980, A131954, A131955, A135204, A135205.

Programs

  • Maple
    P:=proc(n) local i,j,k,w,x; 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; x:=i; j:=i-2; while j >0 do x:=x*j; j:=j-2; od: k:=x; x:=0; while k>0 do x:=x+k-(trunc(k/10)*10); k:=trunc(k/10); od; if trunc(w/x)=w/x then print(i); fi; od; end: P(1000);
  • Mathematica
    Select[Range[1000], Divisible[Total[IntegerDigits[#!, 10]], Total[IntegerDigits[#!!, 10]]] &] (* G. C. Greubel, Sep 30 2016 *)
  • PARI
    df(n) = prod(i=0, (n-1)\2, n - 2*i ); \\ A006882
isok(m) = !(sumdigits(m!) % sumdigits(df(m))); \\ Michel Marcus, Jun 18 2024

Extensions

Changed offset to 1 by Paolo P. Lava, Jun 17 2024

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}]
Showing 1-5 of 5 results.

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