A052489 -id:A052489 - cp's OEIS frontend

A277223 a(n) = A052489(n)/n.

9, 9, 9, 12, 9, 9, 12, 9, 9, 9, 18, 9, 15, 9, 9, 18, 9, 9, 21, 9, 18, 18, 9, 9, 15, 18, 18, 21, 9, 9, 18, 18, 18, 12, 9, 18, 27, 18, 9, 12, 18, 18, 18, 18, 9, 21, 18, 18, 18, 9, 18, 18, 18, 18, 18, 9, 9, 15, 9, 9, 18, 0, 0, 17, 0, 18, 12, 9, 9, 12, 18, 18, 26, 27, 0

Offset: 1

Michel Marcus, Oct 06 2016

a(n) is the largest multiplier k such that m = k*n is n times the sum of its decimal digits.

a(n) is never 1, 2, 3, 4, 5 or 6. Conjecture: if a(n) < 12 then a(n) = 0 or 9. - Robert Israel, Oct 06 2016

			a(2)=9 because m=2*9=18 is the largest m that is twice the sum of its decimal digits.
a(4)=12 because m=4*12=48 is the largest m that is four times the sum of its decimal digits.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A003635, A052489.

N:= 200: # to get a(1) .. a(N)
A:= Vector(N):
for t from 1 while 9*(1+ilog10(t))*N >= t do
   k:= convert(convert(t,base,10),`+`);
   if t mod k = 0 and t <= N*k then
      A[t/k]:= max(A[t/k],k)
   fi
od:
convert(A,list); # Robert Israel, Oct 06 2016

Table[Last[Select[Range[10^(IntegerLength@ n + 2)], n Total@ IntegerDigits@ # == # &] /. {} -> {0}]/n, {n, 75}] (* Michael De Vlieger, Oct 06 2016 *)

a(n) = {nbd = 1; while (9*nbd*n > 10^nbd, nbd++); forstep(k=9*nbd*n, 1, -1, if (sumdigits(k)*n == k, return(k/n));); 0;}

a(n) = 0 for n in A003635.

a(n) = A007953(A052489(n)). - Altug Alkan, Oct 06 2016

A057147 a(n) = n times sum of digits of n.

Original entry on oeis.org

0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 10, 22, 36, 52, 70, 90, 112, 136, 162, 190, 40, 63, 88, 115, 144, 175, 208, 243, 280, 319, 90, 124, 160, 198, 238, 280, 324, 370, 418, 468, 160, 205, 252, 301, 352, 405, 460, 517, 576, 637, 250, 306, 364, 424, 486, 550, 616

Offset: 0

N. J. A. Sloane, Sep 13 2000

A056992(n) = A010888(a(n)). - Reinhard Zumkeller, Mar 19 2014

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
F. B. Diniz, About a new family of sequences, arXiv:1607.06082 [math.GM], 2016.

Cf. A007953, A003634, A005349, A052489, A052490, A003635, A245788.

Iterations: A047892 (start=2), A047912 (start=3), A047897 (start=5), A047898 (start=6), A047899 (start=7), A047900 (start=8), A047901 (start=9), A047902 (start=11).

a057147 n = a007953 n * n  -- Reinhard Zumkeller, Mar 19 2014

for n from 0 to 150 do printf(`%d,`,n*add(convert(n, base, 10)[i], i=1..nops(convert(n,base, 10)))) od:

Table[n*Total[IntegerDigits[n]], {n, 0, 100}]

a(n) = n*sumdigits(n) \\ Franklin T. Adams-Watters, Aug 03 2014

[n*sum([int(d) for d in str(n)]) for n in range(10**5)] # Chai Wah Wu, Aug 05 2014

a(n) = n*A007953(n). - Michel Marcus, Aug 10 2014

G.f.: x * (d/dx) (1/(1 - x))*Sum_{k>=1} (x^k - x^(10^k+k) - 9*x^(10^k))/(1 - x^(10^k)). - Ilya Gutkovskiy, Mar 27 2018

More terms from James Sellers and Larry Reeves (larryr(AT)acm.org), Sep 13 2000

A003635 Inconsummate numbers in base 10: no number is this multiple of the sum of its digits (in base 10).

Original entry on oeis.org

62, 63, 65, 75, 84, 95, 161, 173, 195, 216, 261, 266, 272, 276, 326, 371, 372, 377, 381, 383, 386, 387, 395, 411, 416, 422, 426, 431, 432, 438, 441, 443, 461, 466, 471, 476, 482, 483, 486, 488, 491, 492, 493, 494, 497, 498, 516, 521, 522, 527, 531, 533, 536

Offset: 1

N. J. A. Sloane and Mira Bernstein

J. H. Conway, personal communication.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Daniel Mondot, Table of n, a(n) for n = 1..10867
Giovanni Resta, Numbers Aplenty: Inconsummate numbers

Cf. A003634, A052489, A052490, A052491, A058898-A058917.

Maple
```
For Maple code see A058906.
```

nmax = 1000; Reap[ Do[k = n; kmax = 100*n; While[ Tr[ IntegerDigits[k]]*n != k && k < kmax, k = k + n]; If[k == kmax, Sow[n]], {n, 1, nmax}]][[2, 1]] (* Jean-François Alcover, Jul 12 2012 *)

from itertools import count, islice, combinations_with_replacement
def A003635_gen(startvalue=1): # generator of terms >= startvalue
    for n in count(max(startvalue,1)):
        for l in count(1):
            if 9*l*n < 10**(l-1):
                yield n
                break
            for d in combinations_with_replacement(range(10),l):
                if (s:=sum(d))>0 and sorted(str(s*n)) == [str(e) for e in d]:
                    break
            else:
                continue
            break
A003635_list = list(islice(A003635_gen(),20)) # Chai Wah Wu, May 09 2023

A065880 Largest positive number that is n times the number of 1's in its binary expansion, or 0 if no such number exists.

Original entry on oeis.org

0, 1, 2, 6, 4, 10, 12, 21, 8, 18, 20, 55, 24, 0, 42, 60, 16, 34, 36, 0, 40, 126, 110, 115, 48, 0, 0, 108, 84, 116, 120, 155, 32, 66, 68, 0, 72, 222, 0, 156, 80, 246, 252, 172, 220, 180, 230, 0, 96, 0, 0, 204, 0, 318, 216, 0, 168, 285, 232, 295, 240, 366, 310, 378, 64, 130

Offset: 0

Henry Bottomley, Nov 26 2001

a(n) is bounded above by n*A272756(n), so a program only has to check values up to that point to see if a(n) is zero. - Peter Kagey, May 05 2016

			a(23)=115 since 115 is written in binary as 1110011 and 115/(1+1+1+0+0+1+1)=23 and there is no higher possibility (if k is more than 127 then k divided by its number of binary 1's is more than 26).

Peter Kagey, Table of n, a(n) for n = 0..10000

A052489 is the base 10 equivalent.

Cf. A000120, A049445, A058898, A065413, A065878, A065879, A272756.

Table[SelectFirst[Reverse@ Range@ #, First@ DigitCount[#, 2] == #/n &] &[n SelectFirst[Range[2^12], # > IntegerLength[n #, 2] &]], {n, 80}] /. k_ /; MissingQ@ k -> 0 (* Michael De Vlieger, May 05 2016, Version 10.2 *)

A037478 Number of positive solutions to "numbers that are n times sum of their digits".

Original entry on oeis.org

9, 1, 1, 4, 1, 1, 4, 1, 1, 9, 1, 1, 3, 1, 1, 3, 1, 1, 11, 1, 1, 3, 1, 1, 3, 2, 2, 12, 1, 1, 3, 1, 1, 4, 1, 2, 15, 2, 1, 4, 1, 1, 3, 1, 1, 13, 2, 2, 3, 1, 1, 4, 1, 1, 13, 1, 1, 2, 1, 1, 3, 0, 0, 7, 0, 1, 4, 1, 1, 4, 1, 1, 8, 1, 0, 3, 1, 1, 4, 1, 1, 10, 1, 0, 3, 1, 1, 3, 1, 1, 9, 1, 1, 3, 0, 1, 3, 1, 1, 9, 1

Offset: 1

Henry Bottomley, Sep 12 2000

It appears that the largest terms occur when n=1 mod 9 and moderately large terms when n=4 or 7 mod 9.

			a(13)=3 since the only three solutions are 117=9*13, 156=12*13 and 195=15*13.

Giovanni Resta, Table of n, a(n) for n = 1..10000

Cf. A003634, A003635, A005349, A052489, A052490, A057147, A058913.

read("transforms"):
A037478 := proc(n)
    local a,x,k;
    a := 0 ;
    for k from 1 do
        x := n*k ;
        if digsum(x)*n = x then
            a := a+1 ;
        end if;
        # may stop if x/digsum(x)>n, so if x/#digits(x) > 9*n
        if x/A055642(x) > 9*n then
            break;
        end if;
    end do:
    a ;
end proc:
seq(A037478(n),n=1..101) ; # R. J. Mathar, May 11 2016

A052490 Numbers n with only one nonzero solution to "numbers that are n times sum of their digits".

Original entry on oeis.org

2, 3, 5, 6, 8, 9, 11, 12, 14, 15, 17, 18, 20, 21, 23, 24, 29, 30, 32, 33, 35, 39, 41, 42, 44, 45, 50, 51, 53, 54, 56, 57, 59, 60, 66, 68, 69, 71, 72, 74, 77, 78, 80, 81, 83, 86, 87, 89, 90, 92, 93, 96, 98, 99, 101, 102, 104, 105, 108, 110, 111, 113, 114, 117, 119, 120, 122

Offset: 1

Henry Bottomley, Mar 16 2000

			a(2)=3 since there is only one positive number which is three times the sum of its digits, namely 27=3*9

Cf. A003634, A052489, A003635.

A138770 Triangle read by rows: T(n,k) is the number of permutations of {1,2,...,n} such that there are exactly k entries between the entries 1 and 2 (n>=2, 0<=k<=n-2).

Original entry on oeis.org

2, 4, 2, 12, 8, 4, 48, 36, 24, 12, 240, 192, 144, 96, 48, 1440, 1200, 960, 720, 480, 240, 10080, 8640, 7200, 5760, 4320, 2880, 1440, 80640, 70560, 60480, 50400, 40320, 30240, 20160, 10080, 725760, 645120, 564480, 483840, 403200, 322560, 241920, 161280, 80640

Offset: 2

Emeric Deutsch, Apr 06 2008

Sum of row n = n! = A000142(n).

The expected value of k is (n-2)/3. [Geoffrey Critzer, Dec 19 2009]

			T(4,2)=4 because we have 1342, 1432, 2341 and 2431.
Triangle starts:
  2;
  4,2;
  12,8,4;
  48,36,24,12;
  240,192,144,96,48;
  ...

Cf. A000142, A052489, A052582, A052609, A005990, A061206, A052849, A090672.

T:=proc(n,k) if n-2 < k then 0 else (2*n-2*k-2)*factorial(n-2) end if end proc; for n from 2 to 10 do seq(T(n, k),k=0..n-2) end do; # yields sequence in triangular form

Table[Table[2 (n - r) (n - 2)!, {r, 1, n - 1}], {n, 1, 10}] // Grid (* Geoffrey Critzer, Dec 19 2009 *)

T(n,k) = 2*(n-k-1)*(n-2)!.
T(n,0) = 2(n-1)! = A052849(n-1).
T(n,1) = A052582(n-2).
T(n,2) = A052609(n-2).
T(n,3) = 12*A005990(n-3).
T(n,4) = 48*A061206(n-5).
T(n,n-2) = 2(n-2)! (A052849).
Sum_{k=0..n-2} k*T(n,k) = n!*(n-2)/3 = A090672(n-1).

A277249 Largest number that is n times sum of its digits when written in base 3.

Original entry on oeis.org

0, 2, 8, 6, 16, 25, 24, 35, 32, 18, 80, 77, 48, 78, 56, 75, 64, 0, 72, 152, 160, 105, 132, 115, 96, 150, 156, 54, 224, 232, 240, 186, 0, 231, 204, 210, 144, 296, 228, 234, 320, 287, 168, 172, 0, 225, 368, 376, 192, 392, 400, 0, 312, 371, 216, 440, 448, 456

Offset: 0

Daniel Mondot, Oct 06 2016

Daniel Mondot, Table of n, a(n) for n = 0..10000

Cf. A058899, A065880, A277250, A277291, A052489.

a(n) = 0 for n in A058899. - Michel Marcus, Oct 09 2016

A277250 Largest number that is n times sum of its digits when written in base 4.

Original entry on oeis.org

0, 3, 6, 9, 12, 30, 18, 63, 24, 54, 60, 33, 36, 91, 126, 90, 48, 102, 108, 190, 120, 189, 154, 207, 72, 250, 234, 243, 252, 0, 180, 248, 96, 198, 204, 315, 216, 148, 228, 351, 240, 0, 378, 430, 132, 270, 506, 423, 144, 490, 300, 459, 364, 477, 486, 440, 504

Offset: 0

Daniel Mondot, Oct 06 2016

Daniel Mondot, Table of n, a(n) for n = 0..10000

Cf. A058900, A065880, A277249, A277291, A052489.

a(n) = 0 for n in A058900. - Michel Marcus, Oct 09 2016

A277291 Largest number that is n times sum of its digits when written in base 5.

Original entry on oeis.org

0, 4, 8, 24, 16, 20, 48, 42, 64, 99, 40, 88, 96, 117, 112, 120, 0, 102, 144, 76, 80, 189, 0, 184, 192, 100, 208, 324, 0, 348, 240, 372, 128, 363, 272, 210, 288, 444, 304, 624, 320, 574, 336, 258, 352, 495, 0, 564, 384, 588, 200, 612, 416, 583, 432, 440, 0, 570

Offset: 0

Daniel Mondot, Oct 09 2016

Daniel Mondot, Table of n, a(n) for n = 0..37057

Cf. A058901, A065880, A277249, A277250, A052489.

a(n) = 0 for n in A058901. - Michel Marcus, Oct 09 2016

cp's OEIS Frontend

A277223 a(n) = A052489(n)/n.

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A057147 a(n) = n times sum of digits of n.

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A003635 Inconsummate numbers in base 10: no number is this multiple of the sum of its digits (in base 10).

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A065880 Largest positive number that is n times the number of 1's in its binary expansion, or 0 if no such number exists.

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A037478 Number of positive solutions to "numbers that are n times sum of their digits".

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A052490 Numbers n with only one nonzero solution to "numbers that are n times sum of their digits".

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A138770 Triangle read by rows: T(n,k) is the number of permutations of {1,2,...,n} such that there are exactly k entries between the entries 1 and 2 (n>=2, 0<=k<=n-2).

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A277249 Largest number that is n times sum of its digits when written in base 3.

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