A007095 -id:A007095 - cp's OEIS frontend

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

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

A052421 Numbers without 8 as a digit.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 16, 17, 19, 20, 21, 22, 23, 24, 25, 26, 27, 29, 30, 31, 32, 33, 34, 35, 36, 37, 39, 40, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54, 55, 56, 57, 59, 60, 61, 62, 63, 64, 65, 66, 67, 69, 70, 71, 72, 73, 74, 75, 76, 77, 79

Offset: 1

Henry Bottomley, Mar 13 2000

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
M. F. Hasler, Numbers avoiding certain digits, OEIS wiki, Jan 12 2020.
Index entries for 10-automatic sequences.

Cf. A004183, A004727, A038616 (subset of primes), A082837 (Kempner series).

Cf. A052382 (without 0), A052383 (without 1), A052404 (without 2), A052405 (without 3), A052406 (without 4), A052413 (without 5), A052414 (without 6), A052419 (without 7), A007095 (without 9).

a052421 = f . subtract 1 where
f 0 = 0
f v = 10 * f w + if r > 7 then r + 1 else r where (w, r) = divMod v 9
-- Reinhard Zumkeller, Oct 07 2014

[ n: n in [0..89] | not 8 in Intseq(n) ]; // Bruno Berselli, May 28 2011

a:= proc(n) local l, m; l, m:= 0, n-1;
      while m>0 do l:= (d->
        `if`(d<8, d, d+1))(irem(m, 9, 'm')), l
      od; parse(cat(l))/10
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Aug 01 2016

Select[Range[0,100],DigitCount[#,10,8]==0&] (* Harvey P. Dale, Oct 11 2012 *)

lista(nn)=for (n=0, nn, if (!vecsearch(vecsort(digits(n),,8), 8), print1(n, ", "));); \\ Michel Marcus, Feb 22 2015

/* See OEIS wiki page (cf. LINKS) for more programs. */
apply( {A052421(n)=fromdigits(apply(d->d+(d>7),digits(n-1,9)))}, [1..99]) \\ a(n)
select( {is_A052421(n)=!setsearch(Set(digits(n)),8)}, [0..99]) \\ used in A038616
next_A052421(n, d=digits(n+=1))={for(i=1,#d, d[i]==8&&return((1+n\d=10^(#d-i))*d)); n} \\ Least a(k) > n. Used in A038616. - M. F. Hasler, Jan 11 2020

from gmpy2 import digits
def A052421(n): return int(digits(n-1,9).replace('8','9')) # Chai Wah Wu, Jun 28 2025

seq 0 1000 | grep -v 8; # Joerg Arndt, May 29 2011

a(n) = replace digits d > 7 by d + 1 in base-9 representation of n - 1. - Reinhard Zumkeller, Oct 07 2014

Sum_{n>1} 1/a(n) = A082837 = 22.726365... (Kempner series). - Bernard Schott, Jan 12 2020, edited by M. F. Hasler, Jan 13 2020

Offset changed by Reinhard Zumkeller, Oct 07 2014

A062891 When expressed in base 3 and then interpreted in base 9, is a multiple of the original number.

Original entry on oeis.org

0, 1, 2, 3, 6, 9, 13, 18, 26, 27, 34, 39, 47, 54, 78, 81, 91, 102, 117, 121, 141, 162, 182, 234, 242, 243, 262, 273, 306, 351, 363, 423, 486, 546, 702, 726, 729, 757, 786, 819, 918, 1048, 1053, 1089, 1093, 1183, 1269, 1458, 1514, 1638, 2106, 2178, 2186, 2187

Offset: 1

Erich Friedman, Jul 21 2001

			13 in base 3 is 111, which interpreted in base 9 is 91 = 7*13.

Harvey P. Dale, Table of n, a(n) for n = 1..751 [offset shifted by _Georg Fischer_, Jun 15 2021]

Cf. A007089 (base 3), A007095 (base 9), A037314 (base 3 -> 9).

Other digit spreads: A062846 (binary), A343550 (decimal).

q:= n-> (l-> n=0 or 0=irem(add(l[i]*9^(i-1),
         i=1..nops(l)), n))(convert(n, base, 3)):
select(q, [$0..3000])[];  # Alois P. Heinz, Apr 20 2021

Join[{0},Select[Range[2200],Divisible[FromDigits[IntegerDigits[#,3],9],#]&]] (* Harvey P. Dale, Apr 11 2017 *)

Offset changed to 1 by Kevin Ryde, Apr 24 2021

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

A337239 Numbers having at least one 7 in their representation in base 8.

Original entry on oeis.org

7, 15, 23, 31, 39, 47, 55, 56, 57, 58, 59, 60, 61, 62, 63, 71, 79, 87, 95, 103, 111, 119, 120, 121, 122, 123, 124, 125, 126, 127, 135, 143, 151, 159, 167, 175, 183, 184, 185, 186, 187, 188, 189, 190, 191, 199, 207, 215, 223, 231, 239, 247, 248, 249, 250, 251, 252, 253, 254, 255

Offset: 1

François Marques, Sep 20 2020

Complementary sequence to A037474.

			54 is not in the sequence since it is 66_8 in base 8, but 55 is in the sequence since it is 67_8 in base 8.

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), A337141 (b=7), this sequence (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(7, convert(n, base, 8))>0, n, NULL), n=0..100);

Select[ Range[ 0, 100 ], (Count[ IntegerDigits[ #, 8 ], 7 ]>0)& ]

isok(m) = #select(x->(x==7), digits(m, 8)) >= 1;

def A337239(n):
    def f(x):
        s = oct(x)[2:]
        l = s.find('7')
        if l >= 0:
            s = s[:l]+'6'*(len(s)-l)
        return n+int(s,7)
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m # Chai Wah Wu, Dec 04 2024

A029994 Numbers k such that k^2 is palindromic in base 9.

Original entry on oeis.org

0, 1, 2, 10, 20, 82, 91, 100, 164, 730, 820, 1460, 6562, 6643, 6724, 7300, 7381, 7462, 13124, 13642, 13660, 14281, 54050, 59050, 59860, 65620, 66430, 118100, 123010, 126286, 161410, 161896, 487750, 531442, 532171

Offset: 1

Patrick De Geest

Seiichi Manyama, Table of n, a(n) for n = 1..100
Patrick De Geest, Palindromic Squares in bases 2 to 17

Cf. A007095.

Numbers k such that k^2 is palindromic in base b: A003166 (b=2), A029984 (b=3), A029986 (b=4), A029988 (b=5), A029990 (b=6), A029992 (b=7), A029805 (b=8), this sequence (b=9), A002778 (b=10), A029996 (b=11), A029737 (b=12), A029998 (b=13), A030072 (b=14), A030073 (b=15), A029733 (b=16), A118651 (b=17).

pb9Q[n_]:=Module[{idn=IntegerDigits[n^2,9]},idn==Reverse[idn]]; Select[ Range[0,600000],pb9Q] (* Harvey P. Dale, Sep 29 2013 *)

A073790 Numbers in base -9.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 180, 181, 182, 183, 184, 185, 186, 187, 188, 170, 171, 172, 173, 174, 175, 176, 177, 178, 160, 161, 162, 163, 164, 165, 166, 167, 168, 150, 151, 152, 153, 154, 155, 156, 157, 158, 140, 141, 142, 143, 144, 145, 146, 147, 148, 130, 131

Offset: 0

Robert G. Wilson v, Aug 11 2002

D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, 1969, Vol. 2, p. 189.

Chai Wah Wu, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Negabinary
Prepared and presented by Matthew Szudzik of Wolfram Research, A Mathematica programming contest

Cf. A007095, A039724, A073785, A007608, A073786, A073787, A073788, A073789 & A039723.

ToNegaBases[i_Integer, b_Integer] := FromDigits@ Rest@ Reverse@ Mod[ NestWhileList[(# - Mod[ #, b])/-b &, i, # != 0 &], b]; Table[ ToNegaBases[n, 9], {n, 0, 60}]

def A073790(n):
    s, q = '', n
    while q >= 9 or q < 0:
        q, r = divmod(q, -9)
        if r < 0:
            q += 1
            r += 9
        s += str(r)
    return int(str(q)+s[::-1]) # Chai Wah Wu, Apr 09 2016

A306111 Numbers with digits in {0,...,8} such that every other digit is strictly less than its neighbors.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 10, 20, 21, 30, 31, 32, 40, 41, 42, 43, 50, 51, 52, 53, 54, 60, 61, 62, 63, 64, 65, 70, 71, 72, 73, 74, 75, 76, 80, 81, 82, 83, 84, 85, 86, 87, 101, 102, 103, 104, 105, 106, 107, 108, 201, 202, 203, 204, 205, 206, 207, 208, 212, 213, 214, 215, 216, 217, 218, 301, 302, 303, 304, 305, 306, 307

Offset: 1

M. F. Hasler, Oct 05 2018

Terms of A032864 written in base 9.

			There are 1+2+3+4+5+6+7+8 = 9*4 = 36 terms with 2 digits.
We obtain the 3-digit terms by appending to each of these the 1-digit terms starting with a digit larger than the last digit of the prefix: 10.{1..8}, 20.{1..8}, 21.{2..8}, 30.{1..8}, ..., 86.{7..8}, 87.{8}.
We obtain the 4-digit terms by appending to each of the 2 digit terms, the 2-digit terms starting with a digit larger than the last digit of the prefix: 10.{10,...,87}, 20.{10,...,87}, 21.{20,...,87}, 30.{10,...,87}, ..., 86.{70,...,87}, 87.{80..87}.
That way we obtain all terms with n digits by taking the 2-digit terms and appending to each of these the suitable subsequence of n-2 digit terms.

Cf. A306105 .. A306110 and A297147: analog for bases 3..8 and 10.

Cf. A032864 and A032858 .. A032865 for other bases 3..10.

A(Nmax=100,K=8,A=[0..K],i=vector(2*K,i,max(1,i-K+1)),c(T,v)=apply(t->t+T,v))={for(n=0,oo, for(k=10,K*11-1,if(k%10

a(n) = A007095(A032864(n)).

Numbers in A297147 having no digit 9: Intersection of A297147 with A007095.

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

cp's OEIS Frontend

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

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A052421 Numbers without 8 as a digit.

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A062891 When expressed in base 3 and then interpreted in base 9, is a multiple of the original number.

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

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

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A337239 Numbers having at least one 7 in their representation in base 8.

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