A052490 -id:A052490

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A003635

Inconsummate numbers in base 10: no number is this multiple of the sum of its digits (in base 10).
(Formerly M5325)

+0
21

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

REFERENCES

J. H. Conway, personal communication.

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

LINKS

Daniel Mondot, Table of n, a(n) for n = 1..10867

MAPLE

For Maple code see A058906.

MATHEMATICA

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 *)

PROG

(Python)

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

CROSSREFS

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

KEYWORD

nonn,easy,nice,base

AUTHOR

N. J. A. Sloane and Mira Bernstein

STATUS

approved

A037478

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

+0
4

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

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

LINKS

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

EXAMPLE

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

MAPLE

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

CROSSREFS

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

KEYWORD

nonn,base

AUTHOR

Henry Bottomley, Sep 12 2000

STATUS

approved

A052489

Largest number that is n times sum of its decimal digits.

+0
11

0, 9, 18, 27, 48, 45, 54, 84, 72, 81, 90, 198, 108, 195, 126, 135, 288, 153, 162, 399, 180, 378, 396, 207, 216, 375, 468, 486, 588, 261, 270, 558, 576, 594, 408, 315, 648, 999, 684, 351, 480, 738, 756, 774, 792, 405, 966, 846, 864, 882, 450, 918, 936, 954

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

It is infinite, as pointed out by Dr. Geoffrey Landis: Clearly if you have one integer that is N times the sum of its decimal digits, then when you add a 0 to the end, you have an integer that is 10N times the sum of its decimal digits. - Jonathan Vos Post, Feb 06 2011

LINKS

Daniel Mondot, Table of n, a(n) for n = 0..9999

MATHEMATICA

p[n_] := 10(Length[IntegerDigits[n]]+1); a[0]=0; a[n_] := Catch[For[k = p[n]*n, k >= 0, k--, If[k == n*Total[IntegerDigits[k]], If[k == 0, Print["a(", n, ") not found"]]; Throw[k]]]]; Table[a[n], {n, 0, 1000}] (* Jean-François Alcover, Jul 19 2012 updated Oct 06 2016 after Daniel Mondot's observations *)

PROG

(PARI) 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)); ); 0; } \\ Michel Marcus, Oct 05 2016

CROSSREFS

Cf. A003634, A003635, A052490.

KEYWORD

base,nonn,nice,look

AUTHOR

Henry Bottomley, Mar 16 2000

STATUS

approved

A057147

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

+0
25

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

COMMENTS

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

LINKS

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.

FORMULA

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

MAPLE

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

MATHEMATICA

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

PROG

(Haskell)

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

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

(Python)

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

CROSSREFS

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).

KEYWORD

nonn,base,easy,look

AUTHOR

N. J. A. Sloane, Sep 13 2000

EXTENSIONS

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

STATUS

approved

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