A120390 -id:A120390 - cp's OEIS frontend

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

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

Paolo P. Lava and Giorgio Balzarotti, Nov 30 2007

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

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

G. C. Greubel, Table of n, a(n) for n = 1..1000

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

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);

Select[Range[100], Divisible[Total[IntegerDigits[#!, 10]], Total[IntegerDigits[#, 10]]] &] (* G. C. Greubel, Sep 30 2016 *)

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

Paolo P. Lava, Nov 30 2007

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

G. C. Greubel, Table of n, a(n) for n = 1..1000

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

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);

Select[Range[100], Divisible[Total[IntegerDigits[#!!, 10]], Total[IntegerDigits[#, 10]]] &] (* G. C. Greubel, Sep 30 2016 *)

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

Paolo P. Lava, Nov 30 2007

			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.

Michel Marcus, Table of n, a(n) for n = 1..90

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

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);

Select[Range[1000], Divisible[Total[IntegerDigits[#!, 10]], Total[IntegerDigits[#!!, 10]]] &] (* G. C. Greubel, Sep 30 2016 *)

df(n) = prod(i=0, (n-1)\2, n - 2*i ); \\ A006882
isok(m) = !(sumdigits(m!) % sumdigits(df(m))); \\ Michel Marcus, Jun 18 2024

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

A202707 Numbers k such that (sum of digits of k!!) / 9 is prime.

Original entry on oeis.org

9, 11, 12, 13, 14, 15, 16, 18, 19, 21, 22, 25, 26, 28, 48, 51, 57, 60, 70, 72, 74, 78, 80, 81, 82, 86, 89, 92, 103, 109, 111, 114, 120, 125, 128, 130, 131, 142, 145, 146, 151, 155, 157, 159, 164, 168, 169, 179, 183, 185, 186, 191, 195, 197, 200, 205, 210, 211

Offset: 1

Michel Lagneau, Dec 23 2011

Numbers k such that A120390(k) is prime.

If k = 9 or k > 10, then (sum of digits of k!!) / 9 is an integer (see A120390).

			For k = 9, 9!! = 945, and (9+4+5)/9 = 2 is prime.

Cf. A006882, A120390.

lst={}; Do[If[PrimeQ[Sum[DigitCount[n!!][[i]]*i/9,{i,1,9}]], AppendTo[lst, n]], {n,1,300}]; lst

A202709 (Sum of digits of n!!) / 9.

Original entry on oeis.org

2, 2, 2, 2, 2, 2, 4, 5, 5, 4, 5, 5, 4, 6, 7, 5, 8, 5, 8, 8, 10, 10, 10, 9, 12, 9, 12, 9, 14, 10, 12, 12, 16, 9, 16, 10, 15, 13, 16, 14, 17, 12, 20, 15, 20, 15, 19, 20, 20, 19, 22, 20, 22, 18, 21, 21, 24, 25, 27, 19, 27, 23, 26, 23, 28, 27, 25, 23, 27, 23, 29, 31

Offset: 11

Michel Lagneau, Dec 23 2011

(sum of digits of n!!) / 9 is an integer for n = 9 and n > 10.

Cf. A120390.

Table[Sum[DigitCount[n!!][[i]]*i/9,{i,1,9}],{n,11,100}]

a(n)=my(v=eval(Vec(Str(prod(k=1,n\2,2*k+n%2)))));sum(i=1,#v,v[i])/9 \\ Charles R Greathouse IV, Dec 23 2011

a(n) = A120390(n)/9 for n > 10.

a(n) << n log n. Presumably a(n) ~ n log n but proving this requires showing that not too many digits are 0. (The trailing 0's in even terms are not a problem, being only about n/8.) The expected constant is 1 / (4 log 10) = 0.10857.... [Charles R Greathouse IV, Dec 23 2011]

a(37), a(41), a(45), a(46) corrected by Georg Fischer, Jul 15 2024

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

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

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

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A202707 Numbers k such that (sum of digits of k!!) / 9 is prime.

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A202709 (Sum of digits of n!!) / 9.

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