A011539 "9ish numbers": decimal representation contains at least one nine.

9, 19, 29, 39, 49, 59, 69, 79, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 109, 119, 129, 139, 149, 159, 169, 179, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 209, 219, 229, 239, 249, 259, 269, 279, 289, 290, 291, 292, 293, 294, 295, 296, 297, 298

Offset

1,1

Comments

The 9ish numbers are closed under lunar multiplication. The lunar primes (A087097) are a subset.

Almost all numbers are 9ish, in the sense that the asymptotic density of this set is 1: Among the 9*10^(n-1) n-digit numbers, only a fraction of 0.8*0.9^(n-1) doesn't have a digit 9, and this fraction tends to zero (< 1/10^k for n > 22k-3). This explains the formula a(n) ~ n. - M. F. Hasler, Nov 19 2018

A 9ish number is a number whose largest decimal digit is 9. - Stefano Spezia, Nov 16 2023

Links

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

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, 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]

Index entries for sequences related to dismal (or lunar) arithmetic

Index entries for 10-automatic sequences.

Formula

Complement of A007095. A102683(a(n)) > 0 (defines this sequence). A068505(a(n)) = a(n): fixed points of A068505 are the terms of this sequence and the numbers < 9. - Reinhard Zumkeller, Dec 29 2011, edited by M. F. Hasler, Nov 16 2018

a(n) ~ n. - Charles R Greathouse IV, May 15 2013

Example

E.g. 9, 19, 69, 90, 96, 99 and 1234567890 are all 9ish.

Maple

seq(`if`(numboccur(9, convert(n, base, 10))>0, n, NULL), n=0..100); # François Marques, Oct 12 2020

Mathematica

Select[ Range[ 0, 100 ], (Count[ IntegerDigits[ #, 10 ], 9 ]>0)& ] (* François Marques, Oct 12 2020 *)
Select[Range[300], DigitCount[#, 10, 9]>0&] (* Harvey P. Dale, Mar 04 2023 *)

Prog

(Haskell)
a011539 n = a011539_list !! (n-1)
a011539_list = filter ((> 0) . a102683) [1..]  -- Reinhard Zumkeller, Dec 29 2011
(PARI) is(n)=n=vecsort(digits(n)); n[#n]==9 \\ Charles R Greathouse IV, May 15 2013
(PARI) select( is_A011539(n)=vecmax(digits(n))==9, [1..300]) \\ M. F. Hasler, Nov 16 2018
(GAP) Filtered([1..300], n->9 in ListOfDigits(n)); # Muniru A Asiru, Feb 25 2019
(Python)
def ok(n): return '9' in str(n)
print(list(filter(ok, range(299)))) # Michael S. Branicky, Sep 19 2021
(Python)
def A011539(n):
    def f(x):
        l = (s:=str(x)).find('9')
        if l >= 0: s = s[:l]+'8'*(len(s)-l)
        return n+int(s, 9)
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m # Chai Wah Wu, Dec 04 2024

Crossrefs

Cf. A088924 (number of n-digit terms).

Cf. A087062 (lunar product), A087097 (lunar primes).

A102683 (number of digits 9 in n); fixed points > 8 of A068505.

Cf. Numbers with at least one digit b-1 in base b : A074940 (b=3), A337250 (b=4), A337572 (b=5), A333656 (b=6), A337141 (b=7), A337239 (b=8), A338090 (b=9), this sequence (b=10), A095778 (b=11).

Cf. Numbers with no digit b-1 in base b: A005836 (b=3), A023717 (b=4), A020654 (b=5), A037465 (b=6), A020657 (b=7), A037474 (b=8), A037477 (b=9), A007095 (b=10), A171397 (b=11).

Supersequence of A043525.

Sequence in context: A037085 A088478 A088923 * A088479 A043525 A277596

Adjacent sequences: A011536 A011537 A011538 * A011540 A011541 A011542

Keyword

nonn,base,easy

Author

N. J. A. Sloane

Status

approved