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.

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

Views

Author

N. J. A. Sloane, Sep 13 2000

Keywords

Comments

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

Links

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).

Programs

  • Haskell
    a057147 n = a007953 n * n  -- Reinhard Zumkeller, Mar 19 2014
  • 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}]
  • 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

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

Extensions

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

Views

Author

N. J. A. Sloane and Mira Bernstein

Keywords

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

Crossrefs

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

Programs

  • 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 *)
  • 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

A052489 Largest number that is n times sum of its decimal digits.

Original entry on oeis.org

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
Offset: 0

Views

Author

Henry Bottomley, Mar 16 2000

Keywords

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
There are, however, positive n for which no positive number is n times the sum of its digits, in which case a(n) = 0, as for n = 62, 63, 65, 75, 84, ..., cf. A003635. - M. F. Hasler, Aug 24 2025

Links

Crossrefs

Cf. A003634, A003635, A052490.

Programs

  • 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 *)
  • 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

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

Views

Author

Henry Bottomley, Sep 12 2000

Keywords

Comments

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

Examples

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

Links

Crossrefs

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

Programs

  • 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
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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