Search: a052490 -id:a052490
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A057147
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a(n) = n times sum of digits of n.
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+10
25
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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
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OFFSET
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0,3
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COMMENTS
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LINKS
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FORMULA
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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
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MAPLE
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for n from 0 to 150 do printf(`%d, `, n*add(convert(n, base, 10)[i], i=1..nops(convert(n, base, 10)))) od:
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MATHEMATICA
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Table[n*Total[IntegerDigits[n]], {n, 0, 100}]
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PROG
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(Haskell)
(Python)
[n*sum([int(d) for d in str(n)]) for n in range(10**5)] # Chai Wah Wu, Aug 05 2014
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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More terms from James A. Sellers and Larry Reeves (larryr(AT)acm.org), Sep 13 2000
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STATUS
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approved
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A003635
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Inconsummate numbers in base 10: no number is this multiple of the sum of its digits (in base 10).
(Formerly M5325)
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+10
21
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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
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OFFSET
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1,1
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REFERENCES
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J. H. Conway, personal communication.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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MAPLE
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn,easy,nice,base
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AUTHOR
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STATUS
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approved
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A052489
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Largest number that is n times sum of its decimal digits.
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+10
11
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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
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OFFSET
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0,2
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COMMENTS
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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
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LINKS
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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A037478
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Number of positive solutions to "numbers that are n times sum of their digits".
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+10
4
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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
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listen;
history;
text;
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OFFSET
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1,1
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COMMENTS
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It appears that the largest terms occur when n=1 mod 9 and moderately large terms when n=4 or 7 mod 9.
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LINKS
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EXAMPLE
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a(13)=3 since the only three solutions are 117=9*13, 156=12*13 and 195=15*13.
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MAPLE
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read("transforms"):
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
break;
end if;
end do:
a ;
end proc:
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CROSSREFS
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KEYWORD
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nonn,base
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AUTHOR
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STATUS
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approved
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