A051885 -id:A051885 - cp's OEIS frontend

A181303 Numbers of the form i*7^j-1 (i=1..6, j >= 0).

0, 1, 2, 3, 4, 5, 6, 13, 20, 27, 34, 41, 48, 97, 146, 195, 244, 293, 342, 685, 1028, 1371, 1714, 2057, 2400, 4801, 7202, 9603, 12004, 14405, 16806, 33613, 50420, 67227, 84034, 100841, 117648, 235297, 352946, 470595, 588244, 705893, 823542, 1647085, 2470628, 3294171, 4117714, 4941257

Offset: 1

N. J. A. Sloane, Jan 25 2011

Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,7,-7).

Smallest number whose base b sum of digits is n: A000225 (b=2), A062318 (b=3), A180516 (b=4), A181287 (b=5), A181288 (b=6), this sequence (b=7), A165804 (b=8), A140576 (b=9), A051885 (b=10). - Jason Kimberley, Nov 02 2011

Sort[Flatten[Table[i 7^j-1,{i,1,6},{j,0,7}]]]  (* Harvey P. Dale, Feb 03 2011 *)

G.f.: x^2*(x+1)*(x^2-x+1)*(x^2+x+1) / ((x-1)*(7*x^6-1)). - Colin Barker, Feb 01 2013

A138471 Number of numbers less than n having the same sum of digits.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 2, 2, 2, 2, 2, 2, 2, 2, 1, 0, 3, 3, 3, 3, 3, 3, 3, 2, 1, 0, 4, 4, 4, 4, 4, 4, 3, 2, 1, 0, 5, 5, 5, 5, 5, 4, 3, 2, 1, 0, 6, 6, 6, 6, 5, 4, 3, 2, 1, 0, 7, 7, 7, 6, 5, 4, 3, 2, 1, 0, 8, 8, 7, 6, 5, 4, 3, 2, 1, 0, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 2, 3, 4, 5, 6

Offset: 0

Reinhard Zumkeller, Mar 19 2008

A138470(n) + a(n) + A138472(n) = n;
a(A051885(n)) = 0.
a(A228915(n)) = a(n)+1. - Robert Israel, May 26 2017

			a(42)=#{6,15,24,33}=4.

Robert Israel, Table of n, a(n) for n = 0..10000 (terms 0..1500 from Reinhard Zumkeller)
Index entries for Colombian or self numbers and related sequences

Cf. A007953, A051885, A138470, A138472, A228915.

N:= 1000: # to get a(0) to a(N)
C:= Vector(9*(1+ilog10(N))):
A[0]:= 0:
for n from 1 to N do
  s:= convert(convert(n,base,10),`+`);
  A[n]:= C[s];
  C[s]:= C[s]+1;
od:
seq(A[i],i=0..N); # Robert Israel, May 25 2017

Module[{nn=110,sd},sd=Total[IntegerDigits[#]]&/@Range[nn];Join[{0},Table[ Count[Take[sd,i-1],sd[[i]]],{i,nn}]]] (* Harvey P. Dale, Aug 14 2013 *)

a(n) = my(sdn=sumdigits(n)); sum(k=1, n-1, sumdigits(k) == sdn); \\ Michel Marcus, May 26 2017

A061105 Smallest number whose sum of digits is n^3.

Original entry on oeis.org

0, 1, 8, 999, 19999999, 89999999999999, 999999999999999999999999, 199999999999999999999999999999999999999, 899999999999999999999999999999999999999999999999999999999

Offset: 0

Amarnath Murthy, Apr 20 2001

Except for the leading digit all the other digits of a(n), n >= 1, are 9's and the leading digit is 1 or 8. (This is because the digital sum of n^3 is congruent to 0, 1, or 8 mod 9, so the best we can do is use as many 9's as possible, prefixed if necessary by 1 or 8. - N. J. A. Sloane, Jul 19 2018)

			a(4) = 19999999, 1+9+9+9+9+9+9+9 = 64 = 4^3.

Harry J. Smith, Table of n, a(n) for n=0,...,20

Cf. A051885, A061104.

Do[a = {}; While[Apply[Plus, a] + 9 < n^3, a = Append[a, 9]]; If[ Apply[ Plus, a] != n^3, a = Prepend[ a, n^3 - Apply[ Plus, a]] ]; Print[ FromDigits[ a]], {n, 1, 10} ]
dsn3[n_]:=Module[{t=(n^3-{0,1,8})/9},Which[ IntegerQ[t[[1]]],FromDigits[ PadRight[ {},t[[1]],9]],IntegerQ[t[[2]]],FromDigits[ PadRight[ {1}, t[[2]]+1,9]],IntegerQ[t[[3]]],FromDigits[PadRight[{8},t[[3]]+1,9]]]]; Array[dsn3,10,0] (* Harvey P. Dale, Jul 19 2018 *)

a(n) = { ((n%3)^3 + 1)*10^(n^3\9) - 1 } \\ Harry J. Smith, Jul 19 2009

a(n) = A051885(n^3).

a(n) =((n mod 3)^3+1)*10^floor[n^3/9]-1 =(A021559(n+1)+1)*10^A061263(n)-1. - Henry Bottomley, Apr 24 2001

More terms from Robert G. Wilson v, Apr 21 2001

A331786 a(n) is the largest m such that there exists N such that none of S(N), S(N+1), ..., S(N+m-1) is divisible by n, where S(N) is the sum of digits of N.

Original entry on oeis.org

0, 2, 2, 6, 8, 8, 12, 14, 8, 18, 38, 38, 78, 98, 98, 138, 158, 98, 198, 398, 398, 798, 998, 998, 1398, 1598, 998, 1998, 3998, 3998, 7998, 9998, 9998, 13998, 15998, 9998, 19998, 39998, 39998, 79998, 99998, 99998, 139998, 159998, 99998, 199998, 399998, 399998, 799998

Offset: 1

Jianing Song, Jan 25 2020

Write n = 9*s + t, 1 <= t <= 9. The smallest N_0 such that none of S(N_0), S(N_0+1), ..., S(N_0+m-1) is divisible by n is given by N_0 = 10^(u_0) - 10^s*(t-gcd(t,9)+1) + 1, where u_0 is the smallest nonnegative solution to 9*u == -gcd(t,9) (mod n). See A331787 for more detailed information.
From Bernard Schott, Mar 25 2022: (Start)
Equivalently, a(n) is the largest number of consecutive integers whose sum of digits (A007953) is never divisible by n (this is the answer to problem of Diophante link).
a(n) ends with 8 when n = 5, 6 and n >= 9 (see formula). (End)

			The following list gives the smallest example for each 2 <= n <= 27:
   2: 9..10 (2)
   3: 1..2 (2)
   4: 997..1002 (6)
   5: 6..13 (8)
   6: 7..14 (8)
   7: 994..1005 (12)
   8: 9999993..10000006 (14)
   9: 1..8 (8)
  10: 1..18 (18)
  11: 999981..1000018 (38)
  12: 1..38 (38)
  13: 9999999961..10000000038 (78)
  14: 951..1048 (98)
  15: 961..1058 (98)
  16: 9999931..10000068 (138)
  17: 999999999999921..1000000000000078 (158)
  18: 1..98 (98)
  19: 1..198 (198)
  20: 99999999801..100000000198 (398)
  21: 1..398 (398)
  22: 99999999999999601..100000000000000398 (798)
  23: 99501..100498 (998)
  24: 99601..100598 (998)
  25: 99999999301..100000000698 (1398)
  26: 99999999999999999999201..100000000000000000000798 (1598)
  27: 1..998 (998)

Jianing Song, Table of n, a(n) for n = 1..1000
Diophante, A389 - Les décaXphobes (in French).
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,10,-10).

Cf. A007953 (S(N)), A051885, A331788.

Row 10 of A331787.

a(n) = my(s=(n-1)\9, t=(n-1)%9+1); 10^s*(2*t-gcd(t,9)+1)-2

If n = 9*s + t, 1 <= t <= 9, then a(n) = 10^s*(2*t-gcd(t,9)+1) - 2. See A331787 for a proof of the formula in base b.
Conjectures from Colin Barker, Jan 26 2020: (Start)
G.f.: 2*x^2*(1 + 2*x^2 + x^3 + 2*x^5 + x^6 - 3*x^7 + 5*x^8) / ((1 - x)*(1 - 10*x^9)).
a(n) = a(n-1) + 10*a(n-9) - 10*a(n-10) for n>10.
(End) [This conjecture is correct.]
a(n) = O(10^(n/9)).

A062388 Smallest palindrome with digit sum = n.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 55, 191, 66, 292, 77, 393, 88, 494, 99, 595, 686, 696, 787, 797, 888, 898, 989, 999, 5995, 19991, 6996, 29992, 7997, 39993, 8998, 49994, 9999, 59995, 69896, 69996, 79897, 79997, 89898, 89998, 99899, 99999, 599995, 1999991

Offset: 0

Amarnath Murthy, Jun 27 2001

			191 is the smallest palindrome to have digits sum to 11.
a(28) = 5995 and no palindrome less than 5995 has digit sum 28.

R. J. Mathar, Table of n, a(n) for n = 0..88

Cf. A043269, A051885.

With[{p={#,Total[IntegerDigits[#]]}&/@Select[Range[0,3*10^6],PalindromeQ]},Table[SelectFirst[p,#[[2]]==n&],{n,0,50}]][[;;,1]] (* Harvey P. Dale, Aug 13 2025 *)

ispal(n) = my(d=digits(n)); Vecrev(d)==d;
a(n) = {k = 0; while ((sumdigits(k) != n) || !ispal(k), k++); k;} \\ Michel Marcus, Aug 20 2015

More terms from Erich Friedman, Jul 04 2001

More terms from David Wasserman, Jun 27 2002

A067043 Nondecreasing sums of digits: a(0) = 0 and for n>0: a(n) = Min{m>n|SumOfDigits(m)>= SumOfDigits(a(n-1))}, where SumOfDigits = A007953.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 18, 19, 28, 29, 38, 39, 48, 49, 58, 59, 68, 69, 78, 79, 88, 89, 98, 99, 189, 198, 199, 289, 298, 299, 389, 398, 399, 489, 498, 499, 589, 598, 599, 689, 698, 699, 789, 798, 799, 889, 898, 899, 989, 998

Offset: 0

Reinhard Zumkeller, Feb 17 2002

A138472(a(n)) = 0. - Reinhard Zumkeller, Mar 19 2008

Positions of records in A048377: A192686(n) = A048377(a(n)). [Reinhard Zumkeller, Jul 10 2011]

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A007953, A051885.

a067043 n = a067043_list !! n
a067043_list = 0 : f 1 1 0 1 where
   f k x y z
     | y > 0     = (x-y) : f k x (y `div` 10) z
     | k < 9     = x : f (k+1) (2*x-k*z+1) (z `div` 10) z
     | otherwise = x : f 1 (20*z-1) z (10*z)
-- Reinhard Zumkeller, Jul 10 2011

A087052 Lunar triangular numbers: 0+1+2+3+...+n, where + is lunar addition.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 39, 39, 39, 39, 39, 39, 39, 39, 39, 39, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 59, 59, 59, 59, 59, 59, 59, 59, 59, 59, 69, 69, 69, 69, 69, 69, 69, 69, 69, 69, 79, 79, 79, 79, 79, 79, 79, 79, 79, 79, 89, 89, 89, 89, 89, 89, 89, 89, 89, 89, 99, 99, 99, 99, 99, 99, 99, 99, 99, 99, 199, 199, 199, 199, 199, 199, 199, 199, 199, 199, 199, 199, 199, 199, 199, 199, 199, 199, 199, 199, 199

Offset: 0

Marc LeBrun and N. J. A. Sloane, Oct 19 2003

nonn

Differs from A087121 after 100 terms.

If duplicates are removed we get A051885. - N. J. A. Sloane, Jan 25 2011

D. Applegate, C program for lunar arithmetic and number theory [Note: we have now changed the name from "dismal arithmetic" to "lunar arithmetic" - the old name was too depressing]
D. Applegate, M. LeBrun and N. J. A. Sloane, Dismal Arithmetic [Note: we have now changed the name from "dismal arithmetic" to "lunar arithmetic" - the old name was too depressing]
D. Applegate, M. LeBrun, N. J. A. Sloane, Dismal Arithmetic, J. Int. Seq. 14 (2011) # 11.9.8.
Index entries for sequences related to dismal (or lunar) arithmetic

Cf. A051885.

(Continuing from A087062) dt := proc(n) local i,t1; t1 := 0; for i from 1 to n do t1 := dadd(t1,i); od: t1; end;

A331788 a(n) is the smallest m such that for any N, at least one of S(N), S(N+1), ..., S(N+m-1) is divisible by n, where S(N) is the sum of digits of N.

Original entry on oeis.org

1, 3, 3, 7, 9, 9, 13, 15, 9, 19, 39, 39, 79, 99, 99, 139, 159, 99, 199, 399, 399, 799, 999, 999, 1399, 1599, 999, 1999, 3999, 3999, 7999, 9999, 9999, 13999, 15999, 9999, 19999, 39999, 39999, 79999, 99999, 99999, 139999, 159999, 99999, 199999, 399999, 399999, 799999

Offset: 1

Jianing Song, Jan 25 2020

The main sequence is A331786; this is added because some people may search for this.

			See A331786.

Jianing Song, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,10,-10).

Cf. A007953 (S(N)), A051885, A331786.

Row 10 of A331789.

a(n) = my(s=(n-1)\9, t=(n-1)%9+1); 10^s*(2*t-gcd(t,9)+1)-1

If n = 9*s + t, 1 <= t <= 9, then a(n) = 10^s*(2*t-gcd(t,9)+1) - 1. See A331787 for a proof of the formula in base b.
a(n) = A331786(n) + 1.
Conjectures from Colin Barker, Jan 26 2020: (Start)
G.f.: x*(1 + 2*x + 4*x^3 + 2*x^4 + 4*x^6 + 2*x^7 - 6*x^8) / ((1 - x)*(1 - 10*x^9)).
a(n) = a(n-1) + 10*a(n-9) - 10*a(n-10) for n>10.
(End) [This conjecture is correct.]
a(n) = O(10^(n/9)).

A061104 Smallest number whose sum of digits is n^2.

Original entry on oeis.org

0, 1, 4, 9, 79, 799, 9999, 499999, 19999999, 999999999, 199999999999, 49999999999999, 9999999999999999, 7999999999999999999, 7999999999999999999999, 9999999999999999999999999

Offset: 0

Amarnath Murthy, Apr 20 2001

			a(5) = 799, 7 + 9 + 9 = 25 = 5^2.

Harry J. Smith, Table of n, a(n) for n = 0..94

Cf. A051885, A061105.

a(n) = { (n^2%9 + 1)*10^(n^2\9) - 1 } \\ Harry J. Smith, Jul 18 2009

a(n) = A051885(n^2).

a(n) = ((n^2 mod 9) + 1)*10^floor(n^2/9) - 1. - Henry Bottomley, Apr 24 2001

A179988 Smallest m such that n = sum of digits of A108971(m).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 26, 29, 46, 49, 66, 69, 86, 89, 176, 273, 376, 473, 576, 673, 776, 873, 976, 1773, 2776, 3773, 4776, 5773, 6776, 7773, 8776, 9773, 17776, 27773, 37776

Offset: 1

Reinhard Zumkeller, Aug 09 2010

A179987(a(n)) = n and A179987(m) <> n for m < a(n);
A108971(a(n)) = A051885(n+1);
A007953(A108971(a(n))) = n;
conjecture for n>=18: a(n) = A051885(n+1) + 7*(10^floor(n/9) - 1)/9 - 3*(n mod 2).

cp's OEIS Frontend

A181303 Numbers of the form i*7^j-1 (i=1..6, j >= 0).

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A138471 Number of numbers less than n having the same sum of digits.

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Maple

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A061105 Smallest number whose sum of digits is n^3.

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A331786 a(n) is the largest m such that there exists N such that none of S(N), S(N+1), ..., S(N+m-1) is divisible by n, where S(N) is the sum of digits of N.

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A062388 Smallest palindrome with digit sum = n.

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A067043 Nondecreasing sums of digits: a(0) = 0 and for n>0: a(n) = Min{m>n|SumOfDigits(m)>= SumOfDigits(a(n-1))}, where SumOfDigits = A007953.

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A087052 Lunar triangular numbers: 0+1+2+3+...+n, where + is lunar addition.

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A331788 a(n) is the smallest m such that for any N, at least one of S(N), S(N+1), ..., S(N+m-1) is divisible by n, where S(N) is the sum of digits of N.

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