A008591 -id:A008591 - cp's OEIS frontend

A144812 Integers having ideal digital mean up to base 7.

36990, 37230, 43350, 45390, 2149023720, 2149218300, 2149279740, 2149513020, 2149527540, 2149545960, 2151079740, 2151628020, 2151662460, 2151667320, 2152716540, 2152720860, 2152724280, 2153463540, 2154166200, 2154948600, 2155019220, 2155051980, 2155196340

Offset: 1

Reikku Kulon, Sep 21 2008

Subset of A031443, A144798, A144799, A144800 and A144801.
These numbers have digital mean dm(b, n) = (Sum_{i=1..d} 2*d_i - (b-1)) / (2*d) = 0, where d is the number of digits in the base b representation of n and d_i the individual digits, for 2 <= b <= 7.
There are no integers less than 2^32 for which this is true to base 8. It is believed there are either infinitely many starting at some larger n, or none. If they exist, it is conjectured that the set of all similar sequences continues at least to base ten, almost certainly to base 16 and likely to arbitrarily large b. Sequences for b at least ten have an intersection with A144777.

Jason Yuen, Table of n, a(n) for n = 1..10000 (first 283 terms from Pontus von Brömssen)

Cf. A007953, A008591, A031443, A144777, A144798, A144799, A144800, A144801, A364714.

A017233 a(n) = 9*n + 6.

Original entry on oeis.org

6, 15, 24, 33, 42, 51, 60, 69, 78, 87, 96, 105, 114, 123, 132, 141, 150, 159, 168, 177, 186, 195, 204, 213, 222, 231, 240, 249, 258, 267, 276, 285, 294, 303, 312, 321, 330, 339, 348, 357, 366, 375, 384, 393, 402, 411, 420, 429, 438, 447, 456, 465, 474, 483

Offset: 0

David J. Horn and Laura Krebs Gordon (lkg615(AT)verizon.net), 1985

General form: (q*n-1)*q, cf. A017233 (q=3), A098502 (q=4). - Vladimir Joseph Stephan Orlovsky, Feb 16 2009

Numbers whose digital root is 6; that is, A010888(a(n)) = 6. (Ball essentially says that Iamblichus (circa 350) announced that a number equal to the sum of three integers 3*n, 3*n - 1, and 3*n - 2 has 6 as what is now called the number's digital root.) - Rick L. Shepherd, Apr 01 2014

W. W. R. Ball, A Short Account of the History of Mathematics, Sterling Publishing Company, Inc., 2001 (Facsimile Edition) [orig. pub. 1912], pages 110-111.

Vincenzo Librandi, Table of n, a(n) for n = 0..5000
Tanya Khovanova, Recursive Sequences.
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014-2015.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A008585, A008591, A010888, A016789, A017209, A017221, A017233, A019557, A098502.

[9*n+6: n in [0..60]]; // Vincenzo Librandi, Jul 24 2011

Range[6, 1000, 9] (* Vladimir Joseph Stephan Orlovsky, May 28 2011 *)
LinearRecurrence[{2,-1},{6,15},60] (* Harvey P. Dale, Feb 01 2014 *)

a(n)=9*n+6 \\ Charles R Greathouse IV, Oct 07 2015

G.f.: 3*(2+x)/(x-1)^2. - R. J. Mathar, Mar 20 2018
Sum_{n>=0} (-1)^n/a(n) = sqrt(3)*Pi/27 - log(2)/9. - Amiram Eldar, Dec 12 2021
E.g.f.: 3*exp(x)*(2 + 3*x). - Stefano Spezia, Dec 07 2024
From Elmo R. Oliveira, Apr 10 2025: (Start)
a(n) = 3*A016789(n) = A019557(n+1)/2.
a(n) = 2*a(n-1) - a(n-2). (End)

A212704 a(n) = 9n10^(n-1).

Original entry on oeis.org

9, 180, 2700, 36000, 450000, 5400000, 63000000, 720000000, 8100000000, 90000000000, 990000000000, 10800000000000, 117000000000000, 1260000000000000, 13500000000000000, 144000000000000000, 1530000000000000000, 16200000000000000000, 171000000000000000000, 1800000000000000000000

Offset: 1

Stanislav Sykora, May 25 2012

Main transitions in systems of n particles with spin 9/2.
Please, refer to the general explanation in A212697.
This particular sequence is obtained for base b=10, corresponding to spin S = (b-1)/2 = 9/2.
Number of 0 needed to write all numbers of n+1 digits. - Bruno Berselli, Jun 30 2014
Essentially the same as A113119. - Bernard Schott, Nov 15 2022
From Bernard Schott, Nov 22 2022: (Start)
Number of nonzero digits needed to write all integers from 1 up to 10^n - 1.
a(n) is a square iff n in { A016754 union A033583\{0} } (see formulas). (End)

Stanislav Sykora, Table of n, a(n) for n = 1..100
Stanislav Sýkora, Magnetic Resonance on OEIS, Stan's NMR Blog (Dec 31, 2014), Retrieved Nov 12, 2019.
Index entries for linear recurrences with constant coefficients, signature (20,-100).

Cf. A008591, A011557, A016754, A033583, A033713, A033714, A053541, A081045, A113119.

Cf. A001787, A212697, A212698, A212699, A212700, A212701, A212702, A212703 (for b=2..9).

Rest@ CoefficientList[Series[9 x/(10 x - 1)^2, {x, 0, 18}], x] (* or *)
Array[9 # 10^(# - 1) &, 18] (* Michael De Vlieger, Nov 18 2019 *)

mtrans(n, b) = n*(b-1)*b^(n-1);
a(n) = mtrans(n, 10);

def a(n): return 9*n*10**(n-1)
print([a(n) for n in range(1, 21)]) # Michael S. Branicky, Nov 14 2022

a(n) = n*(b-1)*b^(n-1) with b=10.
From R. J. Mathar, Oct 15 2013: (Start)
G.f.: 9*x/(10*x-1)^2.
a(n) = 9*A053541(n). (End)
From Bernard Schott, Nov 14 2022: (Start)
a(n+1) - a(n) = 9*A081045(n).
a(n) = A113119(n) for n > 1.
a(n) = A033713(n+1) - A033713(n) = A033714(n+1) - A033714(n).
a(A016754(n)) = (3 * (2n+1) * 10^(2*n*(n+1)))^2.
a(A033583(n)) = (3 * n * 10^(5*n^2))^2. (End)
From Elmo R. Oliveira, May 13 2025: (Start)
E.g.f.: 9*x*exp(10*x).
a(n) = A008591(n)*A011557(n-1).
a(n) = 20*a(n-1) - 100*a(n-2) for n > 2. (End)

A283123 a(n) = sigma(9*n).

Original entry on oeis.org

13, 39, 40, 91, 78, 120, 104, 195, 121, 234, 156, 280, 182, 312, 240, 403, 234, 363, 260, 546, 320, 468, 312, 600, 403, 546, 364, 728, 390, 720, 416, 819, 480, 702, 624, 847, 494, 780, 560, 1170, 546, 960, 572, 1092, 726, 936, 624, 1240, 741, 1209

Offset: 1

Seiichi Manyama, Mar 01 2017

nonn

In general, for k>=1, Sum_{j=1..n} sigma(j*k) ~ A069097(k) * Pi^2 * n^2 / (12*k). - Vaclav Kotesovec, May 11 2024

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Sigma(k*n): A000203 (k=1), A062731 (k=2), A144613 (k=3), A193553 (k=4), A283118 (k=5), A224613 (k=6), A283078 (k=7), A283122 (k=8), this sequence (k=9).

Cf. A008591.

Table[DivisorSigma[1, 9 n], {n, 50}] (* Michael De Vlieger, Mar 01 2017 *)

a(n) = sigma(9*n) \\ Amiram Eldar, Dec 16 2022

a(n) = A000203(9*n).

Sum_{k=1..n} a(k) = (35*Pi^2/36) * n^2 + O(n*log(n)). - Amiram Eldar, Dec 16 2022

A279769 Numbers n such that the sum of digits of 9n is 18.

Original entry on oeis.org

11, 21, 22, 31, 32, 33, 41, 42, 43, 44, 51, 52, 53, 54, 55, 61, 62, 63, 64, 65, 66, 71, 72, 73, 74, 75, 76, 77, 81, 82, 83, 84, 85, 86, 87, 88, 91, 92, 93, 94, 95, 96, 97, 98, 99, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 121, 122, 131, 132, 133, 141

Offset: 1

M. F. Hasler, Dec 18 2016

Differs from A084854 from a(55) = 110 on.
Numbers n such that A008591(n) is a term of A235228. - Felix Fröhlich, Dec 18 2016
The digital sum of 9n is always a multiple of 9, and never zero. For most numbers < 100, the digital sum is equal to 9, but for example in the range [91..110] all numbers except 100 have their digital sum equal to 18. The b-file / graph gives a hint on the "asymptotic" distribution / density of this set. After a "flat" range like that at [91..110] there comes a record gap. Sizes [and upper ends] of record gaps are: 10 [a(2) = 21], 11 [a(56) = 121, a(119) = 231, a(188) = 341, ..., a(553) = 891, a(616) = 1001], 21 [a(671) = 1121], 31 [a(1331) = 2231], ..., 91 [a(4339) = 8891], 101 [a(4621) = 10001], 121 [a(4841) = 11121], 231 [a(9176) = 22231], ..., 891 [a(24217) = 88891], 1001 [a(25213) = 100001], 1121 [a(25928) = 111121], 2231 [a(47510) = 222231], ..., 8891 [a(108577) = 888891], 10001 [a(111574) = 1000001], 11121 [a(113576) = 1111121], 22231 [a(202511) = 2222231], ..., 88891 [a(416215) = 8888891], ... - M. F. Hasler, Dec 22 2016

M. F. Hasler, Table of n, a(n) for n = 1..25212

Cf. A007953 (digital sum), A008591, A084854.
Cf. A279772 (sumdigits(2n) = 4), A279773 (sumdigits(3n) = 6), A279774 (sumdigits(4n) = 8), A279775 (sumdigits(5n) = 10), A279776 (sumdigits(6n) = 12), A279770 (sumdigits(7n) = 14), A279768 (sumdigits(8n) = 16), A279769 (sumdigits(9n) = 18), A279777 (sumdigits(9n) = 27).
Digital sum of m*n equals m: A088404 = A069537/2, A088405 = A052217/3, A088406 = A063997/4, A088407 = A069540/5, A088408 = A062768/6, A088409 = A063416/7, A088410 = A069543/8.
Cf. A005349 (Niven or Harshad numbers), A245062 (arranged in rows by digit sums).
Numbers with given digital sum: A011557 (1), A052216 (2), A052217 (3), A052218 (4), A052219 (5), A052220 (6), A052221 (7), A052222 (8), A052223 (9), A052224 (10), A166311 (11), A235151 (12), A143164 (13), A235225 (14), A235226 (15), A235227 (16), A166370 (17), A235228 (18), A166459 (19), A235229 (20).

Select[Range@ 141, Total@ IntegerDigits[9 #] == 18 &]

is(n) = sumdigits(9*n)==18 \\ Felix Fröhlich, Dec 18 2016

a(n) = A235228(n)/9.

A325820 Multiplication table for carryless product i X j in base 3 for i >= 0 and j >= 0, read by antidiagonals.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 2, 2, 0, 0, 3, 1, 3, 0, 0, 4, 6, 6, 4, 0, 0, 5, 8, 9, 8, 5, 0, 0, 6, 7, 12, 12, 7, 6, 0, 0, 7, 3, 15, 16, 15, 3, 7, 0, 0, 8, 5, 18, 11, 11, 18, 5, 8, 0, 0, 9, 4, 21, 24, 13, 24, 21, 4, 9, 0, 0, 10, 18, 24, 19, 21, 21, 19, 24, 18, 10, 0, 0, 11, 20, 27, 23, 26, 9, 26, 23, 27, 20, 11, 0, 0, 12, 19, 30, 36, 19, 15, 15, 19, 36, 30, 19, 12, 0

Offset: 0

Antti Karttunen, May 22 2019

			The array begins as:
  0,  0,  0,  0,  0,  0,  0,  0,  0,   0,   0,   0,   0, ...
  0,  1,  2,  3,  4,  5,  6,  7,  8,   9,  10,  11,  12, ...
  0,  2,  1,  6,  8,  7,  3,  5,  4,  18,  20,  19,  24, ...
  0,  3,  6,  9, 12, 15, 18, 21, 24,  27,  30,  33,  36, ...
  0,  4,  8, 12, 16, 11, 24, 19, 23,  36,  40,  44,  48, ...
  0,  5,  7, 15, 11, 13, 21, 26, 19,  45,  50,  52,  33, ...
  0,  6,  3, 18, 24, 21,  9, 15, 12,  54,  60,  57,  72, ...
  0,  7,  5, 21, 19, 26, 15, 13, 11,  63,  70,  68,  57, ...
  0,  8,  4, 24, 23, 19, 12, 11, 16,  72,  80,  76,  69, ...
  0,  9, 18, 27, 36, 45, 54, 63, 72,  81,  90,  99, 108, ...
  0, 10, 20, 30, 40, 50, 60, 70, 80,  90, 100,  83, 120, ...
  0, 11, 19, 33, 44, 52, 57, 68, 76,  99,  83,  91, 132, ...
  0, 12, 24, 36, 48, 33, 72, 57, 69, 108, 120, 132, 144, ...
  etc.
A(2,2) = 2*2 mod 3 = 1.

Cf. A169999 (the main diagonal).
Row/Column 0: A000004, Row/Column 1: A001477, Row/Column 2: A004488, Row/Column 3: A008585, Row/Column 4: A242399, Row/Column 9: A008591.
Cf. A007089, A207669, A207670, A263273, A325825.
Cf. A325821 (same table without the zero row and column).
Cf. A048720 (binary), A059692 (decimal), A004247 (full multiply).

up_to = 105;
A325820sq(b, c) = fromdigits(Vec(Pol(digits(b,3))*Pol(digits(c,3)))%3, 3);
A325820list(up_to) = { my(v = vector(up_to), i=0); for(a=0,oo, for(col=0,a, if(i++ > up_to, return(v)); v[i] = A325820sq(a-col,col))); (v); };
v325820 = A325820list(up_to);
A325820(n) = v325820[1+n];

A279777 Numbers k such that the sum of digits of 9k is 27.

Original entry on oeis.org

111, 211, 221, 222, 311, 321, 322, 331, 332, 333, 411, 421, 422, 431, 432, 433, 441, 442, 443, 444, 511, 521, 522, 531, 532, 533, 541, 542, 543, 544, 551, 552, 553, 554, 555, 611, 621, 622, 631, 632, 633, 641, 642, 643, 644, 651, 652, 653, 654, 655, 661

Offset: 1

M. F. Hasler, Dec 23 2016

The digital sum of 9k is always a multiple of 9. For most numbers below 100 it is actually equal to 9. Numbers such that the digital sum of 9k is 18 are listed in A279769. Only every third term of the present sequence is divisible by 3.

The sequence of record gaps [and upper end of the gap] is: 100 [a(2) = 211], 101 [a(221) = 1211], 111 [a(4841) = 11211], 111 [a(10121) = 22311], 111 [a(15752) = 33411], ..., 111 [a(45133) = 88911], 111 [a(50413) = 100011], 211 [a(55253) = 111211], 311 [a(110000) = 222311], ..., 911 [a(380557) = 888911], 1011 [a(411049) = 1000011], 1211 [a(436976) = 1111211], 2311 [a(840281) = 2222311], ..., 8911 [a(2451241) = 8888911], ...

Cf. A008591, A084854, A003991, A004247, A279769 (sumdigits(9n) = 18).
Digital sum of m*n equals m: A088404 = A069537/2, A088405 = A052217/3, A088406 = A063997/4, A088407 = A069540/5, A088408 = A062768/6, A088409 = A063416/7, A088410 = A069543/8.
Numbers with given digital sum: A011557 (1), A052216 (2), A052217 (3), A052218 (4), A052219 (5), A052220 (6), A052221 (7), A052222 (8), A052223 (9), A052224 (10), A166311 (11), A235151 (12), A143164 (13), A235225 (14), A235226 (15), A235227 (16), A166370 (17), A235228 (18), A166459 (19), A235229 (20).
Cf. A007953 (digital sum), A005349 (Niven or Harshad numbers), A245062 (arranged in rows by digit sums).
Cf. A082259.

Select[Range@ 661, Total@ IntegerDigits[9 #] == 27 &] (* Michael De Vlieger, Dec 23 2016 *)

PARI
```
is(n)=sumdigits(9*n)==27
```

A373992 Numbers k such that A328768(k) is a multiple of 3, where A328768 is the first primorial based variant of arithmetic derivative.

Original entry on oeis.org

0, 1, 5, 7, 8, 9, 11, 13, 17, 18, 19, 23, 25, 27, 29, 31, 35, 36, 37, 40, 41, 43, 45, 47, 49, 53, 54, 55, 56, 59, 61, 63, 64, 65, 67, 71, 72, 73, 77, 79, 81, 83, 85, 88, 89, 90, 91, 95, 97, 99, 101, 103, 104, 107, 108, 109, 113, 115, 117, 119, 121, 125, 126, 127, 131, 133, 135, 136, 137, 139, 143, 144, 145, 149

Offset: 1

Antti Karttunen, Jun 26 2024

Term is present if and only if it is either a multiple of 9, or it is not a multiple of 3 but its 2-adic valuation is (a multiple of 3).
A multiplicative semigroup: if m and n are in the sequence, then so is m*n.
The asymptotic density of this sequence is 31/63. - Amiram Eldar, Jun 28 2024

Antti Karttunen, Table of n, a(n) for n = 1..12000
Index entries for sequences related to primorial numbers.

Cf. A328768, A373991 (characteristic function).
Union of A008591 and A374044.
Cf. A374042 (subsequence).
Cf. also A042965 (where A328768 is a multiple of 2), A327863 (where A003415 is a multiple of 3).

Select[Range[0, 150], Divisible[#, 9] || (!Divisible[#, 3] && Divisible[IntegerExponent[#, 2], 3]) &] (* Amiram Eldar, Jun 28 2024 *)

PARI
```
isA373992 = A373991;
```

A045644 Palindromic and divisible by 9.

Original entry on oeis.org

0, 9, 99, 171, 252, 333, 414, 585, 666, 747, 828, 909, 999, 1881, 2772, 3663, 4554, 5445, 6336, 7227, 8118, 9009, 9999, 10701, 11511, 12321, 13131, 14841, 15651, 16461, 17271, 18081, 18981, 19791, 20502, 21312, 22122, 23832, 24642, 25452, 26262

Offset: 1

Jeff Burch

Harvey P. Dale, Table of n, a(n) for n = 1..1001

Intersection of A002113 and A008591.

Cf. A029951, A045638, A045639, A043040, A043041, A045642, A045643, A083852.

Select[9*Range[0,3000],PalindromeQ] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Mar 30 2020 *)

Edited to include a(1) = 0 by Paolo Xausa, Jul 07 2025

A054966 Numbers that are congruent to {0, 1, 8} mod 9.

Original entry on oeis.org

0, 1, 8, 9, 10, 17, 18, 19, 26, 27, 28, 35, 36, 37, 44, 45, 46, 53, 54, 55, 62, 63, 64, 71, 72, 73, 80, 81, 82, 89, 90, 91, 98, 99, 100, 107, 108, 109, 116, 117, 118, 125, 126, 127, 134, 135, 136, 143, 144, 145, 152, 153, 154, 161, 162, 163, 170, 171, 172, 179, 180

Offset: 1

Henry Bottomley, May 24 2000

n == n^3 mod 9, so the iterated sum of the decimal digits of n and n^3 are equal.

H. I. Okagbue, M.O.Adamu, S.A. Bishop and A.A. Opanuga, Properties of Sequences Generated by Summing the Digits of Cubed Positive Integers, Indian Journal Of Natural Sciences, Vol. 6 / Issue 32 / October 2015

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

Cf. A047523. Complement of A275910.

[n : n in [0..200] | n mod 9 in [0, 1, 8]]; // Wesley Ivan Hurt, Jun 14 2016

A054966:=n->3*n-3+2*cos(2*n*Pi/3)+2*sin(2*n*Pi/3)/sqrt(3): seq(A054966(n), n=1..100); # Wesley Ivan Hurt, Jun 14 2016

Select[Range[0, 200], MemberQ[{0, 1, 8}, Mod[#, 9]] &] (* Wesley Ivan Hurt, Jun 14 2016 *)
LinearRecurrence[{1, 0, 1, -1}, {0, 1, 8, 9}, 100] (* Vincenzo Librandi, Jun 15 2016 *)

G.f.: x^2*(1+7*x+x^2) / ((1+x+x^2)*(x-1)^2). - R. J. Mathar, Oct 08 2011
From Wesley Ivan Hurt, Jun 14 2016: (Start)
a(n) = a(n-1) + a(n-3) - a(n-4) for n>4.
a(n) = 3*n-3+2*cos(2*n*Pi/3)+2*sin(2*n*Pi/3)/sqrt(3).
a(3k) = 9k-1, a(3k-1) = 9k-8, a(3k-2) = 9k-9. (End)
A008591 UNION A056020. - R. J. Mathar, Jul 19 2024
a(n) -a(n-1) = A105395(n+1), n>1. - R. J. Mathar, Jul 19 2024

cp's OEIS Frontend

A144812 Integers having ideal digital mean up to base 7.

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A017233 a(n) = 9*n + 6.

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A212704 a(n) = 9*n*10^(n-1).

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A283123 a(n) = sigma(9*n).

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A279769 Numbers n such that the sum of digits of 9n is 18.

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A325820 Multiplication table for carryless product i X j in base 3 for i >= 0 and j >= 0, read by antidiagonals.

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A279777 Numbers k such that the sum of digits of 9k is 27.

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Mathematica

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A373992 Numbers k such that A328768(k) is a multiple of 3, where A328768 is the first primorial based variant of arithmetic derivative.

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A212704 a(n) = 9n10^(n-1).