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 1-4 of 4 results.

A131954 a(n) = sum of digits of (n! + a(n-1)), with a(1)=1.

Original entry on oeis.org

1, 3, 9, 6, 9, 18, 18, 18, 36, 36, 45, 36, 36, 54, 54, 72, 72, 63, 54, 63, 72, 81, 108, 90, 81, 90, 117, 99, 144, 126, 144, 117, 153, 153, 153, 180, 162, 117, 198, 207, 153, 198, 198, 234, 216, 225, 234, 243, 234, 225, 207, 288, 297, 279, 297, 351, 279, 306, 333, 297
Offset: 1

Views

Author

Paolo P. Lava and Giorgio Balzarotti, Jul 31 2007

Keywords

Comments

If n >= 5, then 9 divides a(n); see comment in A004152. - Bernard Schott, Jun 27 2019

Examples

			a(4) = Sum_digits(4!+9) = 6.

Links

Crossrefs

Cf. A004152, A055263, A131955.

Programs

  • Maple
    P:=proc(n) local a,i,k,w; a:=0; for i from 1 by 1 to n do w:=0; k:=a+i!; while k>0 do w:=w+k-(trunc(k/10)*10); k:=trunc(k/10); od; a:=w; print(a); od; end: P(100);
# alternative:
sd:= n-> convert(convert(n,base,10),`+`):
A[1]:= 1:
for n from 2 to 100 do A[n]:= sd(n!+A[n-1]) od:
seq(A[i],i=1..100); # Robert Israel, Jun 26 2019
  • Mathematica
    s={1};Do[AppendTo[s,DigitSum[n!+s[[-1]]]],{n,2,60}];s (* James C. McMahon, Mar 03 2025 *)

Formula

a(n) = Sum_digits(n!+a(n-1)).

Extensions

Offset corrected by Robert Israel, Jun 26 2019

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
Showing 1-4 of 4 results.

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