A095778 -id:A095778 - cp's OEIS frontend

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

N. J. A. Sloane

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

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

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.

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.

Filtered([1..300],n->9 in ListOfDigits(n)); # Muniru A Asiru, Feb 25 2019

a011539 n = a011539_list !! (n-1)
a011539_list = filter ((> 0) . a102683) [1..]  -- Reinhard Zumkeller, Dec 29 2011

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

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 *)

is(n)=n=vecsort(digits(n));n[#n]==9 \\ Charles R Greathouse IV, May 15 2013

select( is_A011539(n)=vecmax(digits(n))==9, [1..300]) \\ M. F. Hasler, Nov 16 2018

def ok(n): return '9' in str(n)
print(list(filter(ok, range(299)))) # Michael S. Branicky, Sep 19 2021

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

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

A171397 Write n in base 10, but then read it as if it were written in base 11: if n = Sum_{i >= 0} d_i10^i, with 0 <= d_i <= 9, then a(n) = Sum_{i >= 0} d_i11^i.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 66, 67, 68, 69, 70, 71, 72

Offset: 0

Paul Weisenhorn, Jul 11 2011

This is the sequence of all decimal integers that are created when base 10 numbers are interpreted as base 11 numbers.
Numbers without digit A (=10) in their representation in base 11. Complement of A095778. - François Marques, Oct 20 2020
Original definition: Earliest sequence containing no 11-term arithmetic progression.
In general, if p is prime, the earliest sequence containing no p-term arithmetic progression is created when base (p-1) numbers are interpreted as base p numbers.

			a(53)=58 because 53_11 in base 11 equals 58. - _François Marques_, Oct 20 2020

D. E. Arganbright, Mathematical Modeling with Spreadsheets, ABACUS, Vol. 3, #4(1986), 19-31.

François Marques, Table of n, a(n) for n = 0..10000

Different from A065039. - Alois P. Heinz, Sep 07 2011
CNumbers 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), A011539 (b=10), A095778 (b=11).
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), this sequence (b=11).

seq(`if`(numboccur (10, convert (n, base, 11))=0, n, NULL), n=0..122);
# second Maple program:
a:= n-> (l-> add(l[i]*11^(i-1), i=1..nops(l)))(convert(n, base, 10)):
seq(a(n), n=0..66);  # Alois P. Heinz, Aug 30 2024

Table[FromDigits[RealDigits[n, 10], 11], {n, 0, 100}] (* François Marques, Oct 20 2020 *)

a(n) = fromdigits(digits(n), 11); \\ Michel Marcus, Oct 09 2020

def A171397(n): return int(str(n),11) # Chai Wah Wu, Aug 30 2024

Edited by N. J. A. Sloane, Aug 31 2024

A037465 a(n) = Sum_{i=0..m} d(i)6^i, where Sum_{i=0..m} d(i)5^i is the base 5 representation of n.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 18, 19, 20, 21, 22, 24, 25, 26, 27, 28, 36, 37, 38, 39, 40, 42, 43, 44, 45, 46, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 60, 61, 62, 63, 64, 72, 73, 74, 75, 76, 78, 79, 80, 81, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93

Offset: 0

Clark Kimberling

Numbers without digit 5 in base 6. Complement of A333656. - François Marques, Oct 13 2020

			a(34)=46 because 34 is 114_5 in base 5 and 114_6=46. - _François Marques_, Oct 13 2020

François Marques, Table of n, a(n) for n = 0..10000 (first 1000 terms from Clark Kimberling)

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), A011539 (b=10), A095778 (b=11).

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

Table[FromDigits[RealDigits[n, 5], 6], {n, 0, 100}] (* Clark Kimberling, Aug 14 2012 *)

a(n) = fromdigits(digits(n, 5), 6); \\ François Marques, Oct 13 2020

from gmpy2 import digits
def A037465(n): return int(digits(n,5),6) # Chai Wah Wu, May 06 2025

Offset changed to 0 by Clark Kimberling, Aug 14 2012

A338090 Numbers having at least one 8 in their representation in base 9.

Original entry on oeis.org

8, 17, 26, 35, 44, 53, 62, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 89, 98, 107, 116, 125, 134, 143, 152, 153, 154, 155, 156, 157, 158, 159, 160, 161, 170, 179, 188, 197, 206, 215, 224, 233, 234, 235, 236, 237, 238, 239, 240, 241, 242, 251, 260, 269, 278, 287, 296, 305, 314, 315

Offset: 1

François Marques, Oct 09 2020

Blocks of consecutive terms have lengths in A002452. - Devansh Singh, Oct 21 2020

			70 is not in the sequence since it is 77_9 in base 9, but 76 is in the sequence since it is 84_9 in base 9.

François Marques, Table of n, a(n) for n = 1..10000

Cf. A007095 (base 9).
Complement of A037477.
Cf. A043485 (numbers with exactly one 8 in base 9).
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), this sequence (b=9), A011539 (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).

seq(`if`(numboccur(8, convert(n, base, 9))>0, n, NULL), n=0..100);

Select[ Range[ 0, 100 ], (Count[ IntegerDigits[ #, 9 ], 8 ]>0)& ]

isok(m) = #select(x->(x==8), digits(m, 9)) >= 1;

from gmpy2 import digits
def A338090(n):
    def f(x):
        l = (s:=digits(x,9)).find('8')
        if l >= 0: s = s[:l]+'7'*(len(s)-l)
        return n+int(s,8)
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m # Chai Wah Wu, Dec 04 2024

A037474 a(n) = Sum{d(i)8^i: i=0,1,...,m}, where Sum{d(i)7^i: i=0,1,...,m} is the base 7 representation of n.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 13, 14, 16, 17, 18, 19, 20, 21, 22, 24, 25, 26, 27, 28, 29, 30, 32, 33, 34, 35, 36, 37, 38, 40, 41, 42, 43, 44, 45, 46, 48, 49, 50, 51, 52, 53, 54, 64, 65, 66, 67, 68, 69, 70, 72, 73, 74, 75, 76, 77, 78, 80, 81, 82, 83, 84, 85

Offset: 0

Clark Kimberling

Numbers without digit 7 in base 8. Complement of A337239. - François Marques, Oct 13 2020

			a(48)=54 because 48 is 66_7 in base 7 and 66_8=54. - _François Marques_, Oct 13 2020

François Marques, Table of n, a(n) for n = 0..10000 (first 1000 terms from Clark Kimberling)

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), A011539 (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), this sequence (b=8), A037477 (b=9), A007095 (b=10), A171397 (b=11).

Table[FromDigits[RealDigits[n, 7], 8], {n, 0, 100}] (* Clark Kimberling, Aug 14 2012 *)

a(n) = fromdigits(digits(n, 7), 8); \\ François Marques, Oct 13 2020

from gmpy2 import digits
def A037474(n): return int(digits(n,7),8) # Chai Wah Wu, Dec 04 2024

Offset changed to 0 by Clark Kimberling, Aug 14 2012

A037477 a(n) = Sum{d(i)9^i: i=0,1,...,m}, where Sum{d(i)8^i: i=0,1,...,m} is the base 8 representation of n.

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, 81, 82, 83, 84, 85

Offset: 0

Clark Kimberling

Numbers that do not contain the digit 8 in their base 9 expansion. - M. F. Hasler, Oct 05 2014

			a(63) = 7*9+7 = 70 since 63 = 77[8], i.e., "77" when written in base 8;
a(64) = 1*9^2 = 81 since 64 = 100[8]. - _M. F. Hasler_, Oct 05 2014

François Marques, Table of n, a(n) for n = 0..10000 (first 1000 terms from Clark Kimberling)

Cf. A248375.
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), A011539 (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), this sequence (b=9), A007095 (b=10), A171397 (b=11).

Table[FromDigits[RealDigits[n, 8], 9], {n, 0, 100}]
Select[Range[0,100],DigitCount[#,9,8]==0&] (* Harvey P. Dale, Aug 06 2024 *)

a(n) = vector(#n=digits(n,8),i,9^(#n-i))*n~ \\ M. F. Hasler, Oct 05 2014

a(n) = fromdigits(digits(n, 8), 9); \\ François Marques, Oct 15 2020

def A037477(n): return int(oct(n)[2:],9) # Chai Wah Wu, Jan 27 2025

For n<64, a(n) = floor(9n/8) = A248375(n). - M. F. Hasler, Oct 05 2014

Offset changed to 0 by Clark Kimberling, Aug 14 2012

A337250 Numbers having at least one 3 in their representation in base 4.

Original entry on oeis.org

3, 7, 11, 12, 13, 14, 15, 19, 23, 27, 28, 29, 30, 31, 35, 39, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 67, 71, 75, 76, 77, 78, 79, 83, 87, 91, 92, 93, 94, 95, 99, 103, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119

Offset: 1

François Marques, Sep 19 2020

Complementary sequence of A023717.

			18 is not in the sequence since it is 102_4 in base 4, but 19 is in the sequence since it is 103_4 in base 4.

François Marques, Table of n, a(n) for n = 1..10000

Cf. A196032 (at least one 0 in base 4).
Cf. Numbers with at least one digit b-1 in base b : A074940 (b=3), this sequence, A337572 (b=5), A333656 (b=6), A337141 (b=7), A337239 (b=8), A338090 (b=9), A011539 (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).

seq(`if`(numboccur(3, convert(n, base, 4))>0, n, NULL), n=0..100);

Select[ Range[ 0, 100 ], (Count[ IntegerDigits[ #, 4 ], 3 ]>0)& ]

isok(m) = #select(x->(x==3), digits(m, 4)) >= 1; \\ Michel Marcus, Sep 20 2020

from gmpy2 import digits
def A337250(n):
    def f(x):
        l = (s:=digits(x,4)).find('3')
        if l >= 0: s = s[:l]+'2'*(len(s)-l)
        return n+int(s,3)
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m # Chai Wah Wu, Dec 04 2024

A337572 Numbers having at least one 4 in their representation in base 5.

Original entry on oeis.org

4, 9, 14, 19, 20, 21, 22, 23, 24, 29, 34, 39, 44, 45, 46, 47, 48, 49, 54, 59, 64, 69, 70, 71, 72, 73, 74, 79, 84, 89, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 129, 134

Offset: 1

François Marques, Sep 19 2020

Complementary sequence to A020654.

			75 is not in the sequence since it is 300_5 in base 5, but 74 is in the sequence since it is 244_5 in base 5.

François Marques, Table of n, a(n) for n = 1..10000

Cf. Numbers with at least one digit b-1 in base b : A074940 (b=3), A337250 (b=4), this sequence (b=5), A333656 (b=6), A337141 (b=7), A337239 (b=8), A338090 (b=9), A011539 (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).

seq(`if`(numboccur(4, convert(n, base, 5))>0, n, NULL), n=0..100);

Select[ Range[ 0, 100 ], (Count[ IntegerDigits[ #, 5 ], 4 ]>0)& ]

isok(m) = #select(x->(x==4), digits(m, 5)) >= 1; \\ Michel Marcus, Sep 20 2020

from gmpy2 import digits
def A337572(n):
    def f(x):
        l = (s:=digits(x,5)).find('4')
        if l >= 0: s = s[:l]+'3'*(len(s)-l)
        return n+int(s,4)
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m # Chai Wah Wu, Dec 04 2024

A333656 Numbers having at least one 5 in their representation in base 6.

Original entry on oeis.org

5, 11, 17, 23, 29, 30, 31, 32, 33, 34, 35, 41, 47, 53, 59, 65, 66, 67, 68, 69, 70, 71, 77, 83, 89, 95, 101, 102, 103, 104, 105, 106, 107, 113, 119, 125, 131, 137, 138, 139, 140, 141, 142, 143, 149, 155, 161, 167, 173, 174, 175, 176, 177, 178, 179, 180, 181, 182, 183, 184

Offset: 1

François Marques, Sep 20 2020

Complementary sequence to A037465.

			22 is not in the sequence since it is 34_6 in base 6, but 23 is in the sequence since it is 35_6 in base 6.

François Marques, Table of n, a(n) for n = 1..10000

Cf. Numbers with at least one digit b-1 in base b : A074940 (b=3), A337250 (b=4), A337572 (b=5), this sequence (b=6), A337141 (b=7), A337239 (b=8), A338090 (b=9), A011539 (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).

seq(`if`(numboccur(5, convert(n, base, 6))>0, n, NULL), n=0..100);

Select[ Range[ 0, 100 ], (Count[ IntegerDigits[ #, 6 ], 5 ]>0)& ]

isok(m) = #select(x->(x==5), digits(m, 6)) >= 1;

from gmpy2 import digits
def A333656(n):
    def f(x):
        l = (s:=digits(x,6)).find('5')
        if l >= 0: s = s[:l]+'4'*(len(s)-l)
        return n+int(s,5)
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m # Chai Wah Wu, Dec 04 2024

A337141 Numbers having at least one 6 in their representation in base 7.

Original entry on oeis.org

6, 13, 20, 27, 34, 41, 42, 43, 44, 45, 46, 47, 48, 55, 62, 69, 76, 83, 90, 91, 92, 93, 94, 95, 96, 97, 104, 111, 118, 125, 132, 139, 140, 141, 142, 143, 144, 145, 146, 153, 160, 167, 174, 181, 188, 189, 190, 191, 192, 193, 194, 195, 202, 209, 216, 223, 230, 237, 238, 239, 240

Offset: 1

François Marques, Sep 20 2020

Complementary sequence to A020657.

			33 is not in the sequence since it is 45_7 in base 7, but 34 is in the sequence since it is 46_7 in base 7.

François Marques, Table of n, a(n) for n = 1..10000

Cf. Numbers with at least one digit b-1 in base b: A074940 (b=3), A337250 (b=4), A337572 (b=5), A333656 (b=6), this sequence (b=7), A337239 (b=8), A338090 (b=9), A011539 (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).

seq(`if`(numboccur(6, convert(n, base, 7))>0, n, NULL), n=0..100);

Select[ Range[ 0, 100 ], (Count[ IntegerDigits[ #, 7 ], 6 ]>0)& ]
Select[Range[300],DigitCount[#,7,6]>0&] (* Harvey P. Dale, Dec 23 2020 *)

isok(m) = #select(x->(x==6), digits(m, 7)) >= 1;

from gmpy2 import digits
def A337141(n):
    def f(x):
        l = (s:=digits(x,7)).find('6')
        if l >= 0: s = s[:l]+'5'*(len(s)-l)
        return n+int(s,6)
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m # Chai Wah Wu, Dec 04 2024

cp's OEIS Frontend

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

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A171397 Write n in base 10, but then read it as if it were written in base 11: if n = Sum_{i >= 0} d_i*10^i, with 0 <= d_i <= 9, then a(n) = Sum_{i >= 0} d_i*11^i.

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A037465 a(n) = Sum_{i=0..m} d(i)*6^i, where Sum_{i=0..m} d(i)*5^i is the base 5 representation of n.

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A338090 Numbers having at least one 8 in their representation in base 9.

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A037474 a(n) = Sum{d(i)*8^i: i=0,1,...,m}, where Sum{d(i)*7^i: i=0,1,...,m} is the base 7 representation of n.

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A037477 a(n) = Sum{d(i)*9^i: i=0,1,...,m}, where Sum{d(i)*8^i: i=0,1,...,m} is the base 8 representation of n.

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A171397 Write n in base 10, but then read it as if it were written in base 11: if n = Sum_{i >= 0} d_i10^i, with 0 <= d_i <= 9, then a(n) = Sum_{i >= 0} d_i11^i.

A037465 a(n) = Sum_{i=0..m} d(i)6^i, where Sum_{i=0..m} d(i)5^i is the base 5 representation of n.

A037474 a(n) = Sum{d(i)8^i: i=0,1,...,m}, where Sum{d(i)7^i: i=0,1,...,m} is the base 7 representation of n.

A037477 a(n) = Sum{d(i)9^i: i=0,1,...,m}, where Sum{d(i)8^i: i=0,1,...,m} is the base 8 representation of n.