A168183 -id:A168183 - cp's OEIS frontend

A008591 Multiples of 9: a(n) = 9*n.

0, 9, 18, 27, 36, 45, 54, 63, 72, 81, 90, 99, 108, 117, 126, 135, 144, 153, 162, 171, 180, 189, 198, 207, 216, 225, 234, 243, 252, 261, 270, 279, 288, 297, 306, 315, 324, 333, 342, 351, 360, 369, 378, 387, 396, 405, 414, 423, 432, 441, 450, 459, 468, 477

Offset: 0

N. J. A. Sloane

An Iraqi tablet dating from the Middle Babylonian period (1400-1100 BC) gives a(1)-a(20), a(30), a(40), and a(50). See CDLI link for images and more information. - Charles R Greathouse IV, Jan 21 2017
Apart from 0, numbers whose digital root is 9. - Halfdan Skjerning, Mar 15 2018
Also numbers such that when the leftmost digit is moved to the unit's place the result is divisible by 9. - Stefano Spezia, Jul 08 2025

Ivan Panchenko, Table of n, a(n) for n = 0..200
CDLI, MKT 1, 039, HS 0217a
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 321
Tanya Khovanova, Recursive Sequences
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014-2015.
Leo Tavares, Illustration: Triangular Frames
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A008588, A008589, A008590.
Cf. A168182, A168183.
Cf. A002283, A007953.
Cf. A060544.

Range[0, 1000, 9] (* Vladimir Joseph Stephan Orlovsky, May 27 2011 *)

makelist(9*n,n,0,20); /* Martin Ettl, Dec 17 2012 */

a(n)=9*n \\ Charles R Greathouse IV, Jan 30 2012

Complement of A168183; A168182(a(n)) = 0. - Reinhard Zumkeller, Nov 30 2009
a(n) = A007953(A002283(n)). - Reinhard Zumkeller, Aug 06 2010
From Vincenzo Librandi, Dec 24 2010: (Start)
a(n) = 9*n = 2*a(n-1) - a(n-2).
G.f.: 9x/(x-1)^2. (End)
a(n) = A060544(n+1) - A060544(n). - Leo Tavares, Jul 17 2022
E.g.f.: 9*x*exp(x). - Stefano Spezia, Oct 08 2022

A168182 Characteristic function of numbers that are not multiples of 9.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1

Offset: 0

Reinhard Zumkeller, Nov 30 2009

			G.f. = x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + x^8 + x^10 + x^11 + x^12 + x^13 + ...

Antti Karttunen, Table of n, a(n) for n = 0..999
Michael Somos, Rational Function Multiplicative Coefficients
Index entries for characteristic functions
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,1).

Cf. A168185, A145568, A168184, A168181, A109720, A097325, A011558, A166486, A011655, A000035, A267142, A033441.

A168182[n_]:=Boole[!Divisible[n,9]]; Array[A168182, 10, 0]
Table[If[Mod[n,9]==0,0,1],{n,0,120}] (* Harvey P. Dale, Mar 05 2025 *)

{a(n) = sign(n%9)}; /* Michael Somos, Mar 22 2011 */

Euler transform of length 9 sequence [1, 0, 0, 0, 0, 0, 0, -1, 1]. - Michael Somos, Mar 22 2011
Moebius transform is length 9 sequence [1, 0, 0, 0, 0, 0, 0, 0, -1]. - Michael Somos, Mar 22 2011
Expansion of x * (1 - x^8) / ((1 - x) * (1 - x^9)) in powers of x. - Michael Somos, Mar 22 2011
Multiplicative with a(p^e) = (if p=3 then 0^(e-1) else 1), p prime and e>0.
a(n) = a(n+9) = a(-n) for all n in Z.
a(n) = A000007(A010878(n)).
a(A168183(n)) = 1. a(A008591(n)) = 0.
A033441(n) = Sum_{k=0..n} a(k)*(n-k).
G.f.: -x*(1+x)*(1+x^2)*(1+x^4) / ( (x-1)*(1+x+x^2)*(x^6+x^3+1) ). - R. J. Mathar, Jan 07 2011
Dirichlet g.f. (1-3^(-2s))*zeta(s). - R. J. Mathar, Mar 06 2011
For the general case: the characteristic function of numbers that are not multiples of m is a(n)=floor((n-1)/m)-floor(n/m)+1, m,n > 0. - Boris Putievskiy, May 08 2013
a(n) = 1 - A267142(n). - Antti Karttunen, Oct 07 2017

A255808 Numbers with no zeros in base-9 representation.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 16, 17, 19, 20, 21, 22, 23, 24, 25, 26, 28, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 46, 47, 48, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 60, 61, 62, 64, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75

Offset: 1

Reinhard Zumkeller, Mar 08 2015

a(n) = A168183(n) for n <= 72.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for 3-automatic sequences.

Cf. A007095, A100973 (subsequence).

Zeroless numbers in some other bases <= 10: A000042 (base 2), A032924 (base 3), A023705 (base 4), A248910 (base 6), A255805 (base 8), A052382 (base 10).

a255808 n = a255808_list !! (n-1)
a255808_list = iterate f 1 where
   f x = 1 + if r < 8 then x else 9 * f x'  where (x', r) = divMod x 9

Select[Range[100],DigitCount[#,9,0]==0&] (* or *) With[{upto=100}, Complement[ Range[upto],9*Range[Floor[upto/9]]]] (* Harvey P. Dale, May 29 2019 *)

isok(n) = vecmin(digits(n, 9)) != 0; \\ Michel Marcus, Jun 29 2019

def A255808(n):
    m = ((k:=7*n+1).bit_length()-1)//3
    return sum((1+((k-(1<<3*m))//(7<<3*j)&7))*9**j for j in range(m)) # Chai Wah Wu, Jun 28 2025

A248375 a(n) = floor(9*n/8).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 16, 18, 19, 20, 21, 22, 23, 24, 25, 27, 28, 29, 30, 31, 32, 33, 34, 36, 37, 38, 39, 40, 41, 42, 43, 45, 46, 47, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 59, 60, 61, 63, 64, 65, 66, 67, 68, 69, 70, 72, 73, 74, 75, 76, 77, 78, 79, 81, 82, 83, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95

Offset: 0

M. F. Hasler, Oct 05 2014

Also: numbers not congruent to 8 (mod 9), or numbers whose base-9 expansion does not end in the digit "8".

Paz proves that for all n>0 there is a prime in Breusch's interval [n; a(n+3)], cf A248371.

G. A. Paz, On the Interval [n; 2n]: Primes, Composites and Perfect Powers, Gen. Math. Notes 15 no. 1 (2013), 1-15.
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,1,-1).

Cf. A032766, A004773, A001068, A047226, A047368, A004777, A168183, A281899 (subsequence).

[Floor(9*n/8): n in [0..90]]; // Bruno Berselli, Oct 06 2014

Table[Floor[9 n/8], {n, 0, 90}] (* Bruno Berselli, Oct 06 2014 *)

PARI
```
a(n)=9*n\8
```

G.f.: x*(1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + 2*x^7) / ((1 + x)*(1 - x)^2*(1 + x^2)*(1 + x^4)). [Bruno Berselli, Oct 06 2014]
a(n) = n + floor(n/8) = a(n-1) + a(n-8) - a(n-9). [Bruno Berselli, Oct 06 2014]
a(n) = A168183(n+1) - 1. - Philippe Deléham, Dec 05 2013

A376418 a(n) = n - A276086(A276085(n)), where A276085 and A276086 are primorial base log and exp-functions.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 3, 0, 0, 0, 7, 0, 0, 0, 5, 0, 0, 0, 6, 0, 0, 22, 7, 0, 0, 0, 14, 0, 0, 0, 31, 0, 0, 0, 10, 0, 0, 0, 11, 0, 0, 0, 43, 0, 0, 0, 13, 0, 44, 0, 14, 0, 0, 0, 15, 0, 0, 0, 59, 0, 0, 0, 17, 0, 0, 0, 62, 0, 0, 0, 19, 0, 0, 0, 35, 66, 0, 0, 21, 0, 0, 0, 22, 0, 0, 0, 23, 0, 0, 0, 86, 0, 0, 0, 25

Offset: 1

Antti Karttunen, Nov 03 2024

nonn

All terms are nonnegative because for all n, x = A276086(A276085(n)) <= n, as any factor prime(i)^k || n (with k >= prime(i)) will propagate carries (in the image of fully additive A276085) towards more significant digit positions, which A276086 will convert back to the exponents of larger primes, but for each new instance of such larger prime present in x, enough instances of smaller primes in n have been eliminated (by the carry process) so that the net change of magnitude is negative, unless there are no such factors present at all in n (i.e., when n is a term of A048103), then A276086(A276085(n)) = n, and a(n) = 0.
  This implies also that the least k for which A276085(k) = n is k = A276086(n).
There are several conspicuous patterns among the terms. For example, for n = 392, 3992, 39992, 399992, 3999992, ..., a(n) = 98, 998, 9998, 99998, 999998, ..., = n/4, but this holds only if n/8 is not in A100716, as generally, for all terms x that are in the intersection of A051062 and A168183 and x/8 is in A048103, it follows that a(x) = x/4. There are many other similar identities.
Differs from similar A376417 for the first time at n=625, 1250, 1875, 2500, 3125, 3375, 3750, 4375, 4500, 5000, 5625, ...

			a(4) = 1, as 4 = prime(1)^2, thus A276085(4) = 2 * A002110(1-1) = 2, and A276086(2) = prime(2) = 3, and 4-3 = 1.
a(625) = 0, as 625 = prime(3)^4, thus A276085(625) = 4 * A002110(3-1) = 4*6 = 24, and A276086(24) = prime(3)^4 [because A049345(24) = 400] = 625, and 625-625 = 0.
a(2500) = 625, as 2500 = 2^2 * 5^4 = prime(1)^2 * prime(3)^4, thus A276085(2500) = 2 * A002110(1-1) + 4 * A002110(3-1) = 2*1 + 4*6 = 26, but on the other hand, A276086(26) = prime(2) * prime(3)^4 [because A049345(26) = 410] = 3 * 5^4 = 1875, and 2500 - 1875 = 625.
a(3999999992) = 999999998, as 3999999992 = 2^3 * 691 * 723589 = prime(1)^3 * prime(125) * prime(58312), thus x = A276085(3999999992) = A002110(1-1) + A002110(2-1) + A002110(125-1) + A002110(58312-1), so A276086(x) = prime(1) * prime(2) * prime(125) * prime(58312), therefore a(3999999992) = (8-6)*prime(125)*prime(58312) = 3999999992/4 = 999999998. Note that A049345(8) = "110", as 8 = 6+2.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences related to primorial base

Cf. A049345, A048103 (indices of 0's), A100716 (of terms > 0), A276085, A276086, A376417.

A276085(n) = { my(f=factor(n)); sum(k=1, #f~, f[k, 2]*prod(i=1,primepi(f[k, 1]-1),prime(i))); };
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A376418(n) = (n - A276086(A276085(n)));

A382750 If k appears, 9*k does not.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 16, 17, 19, 20, 21, 22, 23, 24, 25, 26, 28, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 46, 47, 48, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 60, 61, 62, 64, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 80, 81

Offset: 1

Jan Snellman, May 09 2025

Integers k with val(k, 9) even, where val(k, 9) is the 9-adic valuation of k.
Natural density 9/10.
Differs from A168183: 81 for example is not in A168183 but in this sequence. - R. J. Mathar, May 26 2025

			18 = 9*2 is not a term because 2 is a term.
162 = 9*18 is a term since 18 is not a term.

Jan Snellman, Table of n, a(n) for n = 1..9000
Jan Snellman, Greedy Regular Convolutions, arXiv:2504.02795 [math.NT], 2025.

Cf. A003159, A007417, A382744, A382745, A382746.

Select[Range[100], Mod[IntegerExponent[#, 3], 4] < 2 &] (* Amiram Eldar, May 12 2025 *)

from sympy import integer_log
def A382750(n):
    def f(x): return n+x-sum((k:=x//9**m)-k//9 for m in range(0,integer_log(x,9)[0]+1,2))
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m # Chai Wah Wu, May 24 2025

[ for  in range(1,100+1) if (valuation(_,3) % 4) < 2 ]

cp's OEIS Frontend

A008591 Multiples of 9: a(n) = 9*n.

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A168182 Characteristic function of numbers that are not multiples of 9.

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A255808 Numbers with no zeros in base-9 representation.

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A248375 a(n) = floor(9*n/8).

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A376418 a(n) = n - A276086(A276085(n)), where A276085 and A276086 are primorial base log and exp-functions.

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A382750 If k appears, 9*k does not.

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