A061105 -id:A061105 - cp's OEIS frontend

A051885 Smallest number whose sum of digits is n.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 19, 29, 39, 49, 59, 69, 79, 89, 99, 199, 299, 399, 499, 599, 699, 799, 899, 999, 1999, 2999, 3999, 4999, 5999, 6999, 7999, 8999, 9999, 19999, 29999, 39999, 49999, 59999, 69999, 79999, 89999, 99999, 199999, 299999, 399999, 499999

Offset: 0

Felice Russo, Dec 15 1999

This is also the list of lunar triangular numbers: A087052 with duplicates removed. - N. J. A. Sloane, Jan 25 2011
Numbers n such that A061486(n) = n. - Amarnath Murthy, May 06 2001
The product of digits incremented by 1 is the same as the number incremented by 1. If a(n) = abcd...(a,b,c,d, etc. are digits of a(n)) {a(n) + 1} = (a+1)*(b+1)(c+1)*(d+1)*..., e.g., 299 + 1 = (2+1)*(9+1)*(9+1) = 300. - Amarnath Murthy, Jul 29 2003
A138471(a(n)) = 0. - Reinhard Zumkeller, Mar 19 2008
a(n+1) = A108971(A179988(n)). - Reinhard Zumkeller, Aug 09 2010, Jul 10 2011
Positions of records in A003132: A080151(n) = A003132(a(n)). - Reinhard Zumkeller, Jul 10 2011
a(n) = A242614(n,1). - Reinhard Zumkeller, Jul 16 2014
A254524(a(n)) = 1. - Reinhard Zumkeller, Oct 09 2015
The slowest strictly increasing sequence of nonnegative integers such that, for any two terms, calculating the difference of their decimal representations requires no borrowing. - Rick L. Shepherd, Aug 11 2017

Iain Fox, Table of n, a(n) for n = 0..9000 (first 101 terms from Reinhard Zumkeller)
D. Applegate, M. LeBrun and N. J. A. Sloane, Dismal Arithmetic, arXiv:1107.1130 [math.NT], 2011. [Note: we have now changed the name from "dismal arithmetic" to "lunar arithmetic" - the old name was too depressing]
A. Murthy, Exploring some new ideas on Smarandache type sets, functions and sequences, Smarandache Notions Journal Vol. 11 N. 1-2-3 Spring 2000.
Index entries for sequences related to dismal (or lunar) arithmetic
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,10,-10).

Cf. A061104, A061105, A061486, A007953, A067043, A087052.
Numbers of form i*b^j-1 (i=1..b-1, j >= 0) for bases b = 2 through 9: A000225, A062318, A180516, A181287, A181288, A181303, A165804, A140576. - N. J. A. Sloane, Jan 25 2011
Cf. A002283.
Cf. A254524.

a051885 n = (m + 1) * 10^n' - 1 where (n',m) = divMod n 9
-- Reinhard Zumkeller, Jul 10 2011

Magma
```
[i*10^j-1: i in [1..9], j in [0..5]];
```

b:=10; t1:=[]; for j from 0 to 15 do for i from 1 to b-1 do t1:=[op(t1), i*b^j-1]; od: od: t1; # N. J. A. Sloane, Jan 25 2011

a[n_] := (Mod[n, 9] + 1)*10^Floor[n/9] - 1; Table[a[n], {n, 0, 49}](* Jean-François Alcover, Dec 01 2011, after Henry Bottomley *)

A051885(n) = (n%9+1)*10^(n\9)-1  \\ M. F. Hasler, Jun 17 2012

first(n) = Vec(x*(x^2 + x + 1)*(x^6 + x^3 + 1)/((x - 1)*(10*x^9 - 1)) + O(x^n), -n) \\ Iain Fox, Dec 30 2017

def A051885(n): return ((n % 9)+1)*10**(n//9)-1 # Chai Wah Wu, Apr 04 2021

These are the numbers i*10^j-1 (i=1..9, j >= 0). - N. J. A. Sloane, Jan 25 2011
a(n) = ((n mod 9) + 1)*10^floor(n/9) - 1 = a(n-1) + 10^floor((n-1)/9). - Henry Bottomley, Apr 24 2001
a(n) = A037124(n+1) - 1. - Reinhard Zumkeller, Jan 03 2008, Jul 10 2011
G.f.: x*(x^2+x+1)*(x^6+x^3+1) / ((x-1)*(10*x^9-1)). - Colin Barker, Feb 01 2013

More terms from James Sellers, Dec 16 1999

Offset fixed by Reinhard Zumkeller, Jul 10 2011

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

A061263 a(n) = floor(n^3/9).

Original entry on oeis.org

0, 0, 0, 3, 7, 13, 24, 38, 56, 81, 111, 147, 192, 244, 304, 375, 455, 545, 648, 762, 888, 1029, 1183, 1351, 1536, 1736, 1952, 2187, 2439, 2709, 3000, 3310, 3640, 3993, 4367, 4763, 5184, 5628, 6096, 6591, 7111, 7657, 8232, 8834, 9464, 10125, 10815, 11535

Offset: 0

Henry Bottomley, Apr 24 2001

			a(4) = floor(4^3 / 9) = floor(64/9) = 7.

Harry J. Smith, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,2,-3,3,-1).

Cf. A061105.

a(n) = { n^3\9 } \\ Harry J. Smith, Jul 20 2009

G.f.: x^3*(3-2*x+x^2) / ( (1+x+x^2)*(x-1)^4 ). - R. J. Mathar, Feb 20 2011

A060711 Smallest number whose sum of digits is n^4.

Original entry on oeis.org

1, 79, 999999999, 49999999999999999999999999999, 4999999999999999999999999999999999999999999999999999999999999999999999, 999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999

Offset: 1

Robert G. Wilson v, Apr 21 2001

Harry J. Smith, Table of n, a(n) for n = 1..9

Cf. A051885, A061104, A061105.

Do[ a = {}; While[ Apply[ Plus, a ] + 9 < n^4, a = Append[ a, 9 ] ]; If[ Apply[ Plus, a ] != n^4, a = Prepend[ a, n^4 - Apply[ Plus, a ] ] ]; Print[ FromDigits[ a ] ], {n, 1, 6} ]

a(n)={ my(s=n^4, x=s\9, d=s-9*x); (d+1)*10^x - 1 } \\ Harry J. Smith, Jul 10 2009

a(n) = A051885(n^4).

a(n) >= 10^(n^4/9) - 1.

A178716 a(n) = smallest prime whose sum of digits is A001651(n).

Original entry on oeis.org

17, 19999999, 99899999999999, 299989999999999999999999999999999999999, 999999999998999999999999999999999999999999999999999999999, 9999999999999999999999999999999999999999999999999999999999999991999999999999999999999999999999999999999999999999

Offset: 1

Ulrich Krug (leuchtfeuer37(AT)gmx.de), Jun 07 2010

As k runs through the numbers that are not multiples of 3, sequence gives smallest prime p whose sum of digits is k^3.
No primes exist whose sum of digits is a multiple of 3
k=4: 4^3 = 64 = 7 * 9 + 1, 8-digit prime:10^7+(10^7-1) = 2*10^7 - 1
k=5: 5^3 = 125 = 13 * 9 + 8, 14-digit prime: (10^2-1)*10^12+8*10^11+(10^11-1) = 10^14 - 10^11 - 1
k=7: 7^3 = 343 = 38 * 9 + 1: no such prime formed with 38 digits "9" and one digit "1" 343 = 37 * 9 + 2 + 8, gives as 4th term a 39-digit prime: 2*10^38+(10^3-1)*10^35+8*10^34+(10^34-1) = 3*10^38 - 10^34 - 1
k=8: 8^3 = 512 = 56 * 9 + 8, 57-digit prime: (10^11-1)*10^46+8*10^45+(10^45-1) = 10^57 - 10^45 - 1
k=10: 10^3 = 1000 = 111 * 9 + 1, 112-digit prime, 63 "leading" "9's", "1", 48 "ending" "9's": (10^63-1)*10^49+10^48+(10^48-1) = 10^112 - 8 * 10^48 - 1 = ((2*5)^(2^2^2))^7 - (2*((2*5)^(2^2^2)))^3 - 1
k=11: 11^3 = 1331 = 147 * 9 + 8: no such prime formed with 147 digits "9" and one digit "8" 1331 = 147 * 9 + 1 + 7, 149-digit prime: 10^148+(10^143-1)*10^5+7*10^4+(10^4-1) = 2*10^148 - 2*10^4 - 1

Cf. A061105

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A051885 Smallest number whose sum of digits is n.

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

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A061263 a(n) = floor(n^3/9).

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A060711 Smallest number whose sum of digits is n^4.

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A178716 a(n) = smallest prime whose sum of digits is A001651(n).

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