A074940 -id:A074940 - cp's OEIS frontend

A095778 Values of n for which A095777(n) is 9 (those terms which are expressible in decimal digits for bases 2 through 10, but not for base 11).

10, 21, 32, 43, 54, 65, 76, 87, 98, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 131, 142, 153, 164, 175, 186, 197, 208, 219, 230, 231, 232, 233, 234, 235, 236, 237, 238, 239, 240, 241, 252, 263, 274, 285, 296, 307, 318, 329, 340, 351, 352, 353

Offset: 1

Chuck Seggelin (seqfan(AT)plastereddragon.com), Jun 05 2004

Numbers with at least one digit A (=10) in their representation in base 11. Complementary sequence to A171397. - François Marques, Oct 11 2020

			a(5)=54 because 54 when expressed in successive bases starting at 2 will produce its first non-decimal digit at base 11. Like so: 110110, 2000, 312, 204, 130, 105, 66, 60, 54. In base 11, 54 is 4A.

François Marques, Table of n, a(n) for n = 1..10000 (first 1000 terms from Harvey P. Dale)

Cf. A095777.
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), this sequence (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).

S := []; for n from 1 to 1000 do; if 1>0 then; ct := 0; ok := true; b := 2; if (n>9) then; while ok=true do; L := convert(n, base, b); for e in L while ok=true do; if (e > 9) then ok:=false; fi; od; if ok=true then; ct := ct + 1; b := b + 1; fi; od; fi; if ct=9 then S := [op(S), n]; fi; fi; od; S;
# or
seq(`if`(numboccur(10, convert(n, base, 11))>0, n, NULL), n=0..1000); # François Marques, Oct 11 2020

Select[Range[400],Max[IntegerDigits[#,11]]>9&] (* Harvey P. Dale, Sep 30 2018 *)

isok(m) = #select(x->(x==10), digits(m, 11)) >= 1; \\ François Marques, Oct 11 2020

from gmpy2 import digits
def A095778(n):
    def f(x):
        l = (s:=digits(x,11)).find('a')
        if l >= 0: s = s[:l]+'9'*(len(s)-l)
        return n+int(s)
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m # Chai Wah Wu, Dec 04 2024

A216194 a(n) = Smallest b for which the base b representation of n contains at least one 2 (or 0 if no such base exists).

Original entry on oeis.org

0, 3, 0, 0, 3, 3, 3, 3, 4, 4, 3, 5, 5, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 10, 3, 4, 11, 3, 3, 3, 3, 4, 4, 3, 4, 4, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 6, 4, 3, 6, 5, 3, 3, 3, 3, 4, 4, 3, 6, 4, 3, 3, 3, 3, 3, 3

Offset: 1

Nathan Fox, Mar 12 2013

a(n)=3 if and only if n is in A074940.
a(n) > 0 for n >= 5 since 12 is n written in base n-2.
The only perfect k-th powers (k>=2) that can appear in this sequence are 2^k with k a prime number.
The first n for which a(n)=7 is 849.
The first n for which a(n)=8 is 1084.
The first n for which a(n)=10 is 28.  The second is 243.
The first n for which a(n)=11 is 31.  The second is 58130496.
a(n)<=11 for all n with fewer than 3000 base 10 digits.  No n for which a(n)>11 has been found.

Nathan Fox, Table of n, a(n) for n = 1..10000

Cf. A216192, A270027, A270028, A270029, A270030, A270031, A270032, A270033, A270034, A270035.

firstNTerms:=proc(n) local b,i,rep,L:
L:=[]:
for i from 5 to n do
    b:=3:
    while true do
        rep:=convert(i, base, b):
        if evalb(2 in rep) then
            L:=[op(L), b]:
            break:
        fi:
        b:=b+1:
    od:
od:
L:
end:

sb2[n_]:=Module[{b=3},While[DigitCount[n,b,2]<1,b++];b]; Array[sb2,110,5] (* Harvey P. Dale, Jan 16 2016 *)
Table[SelectFirst[Range[3, 1200], DigitCount[n, #, 2] > 0 &], {n, 5, 120}] (* Michael De Vlieger, Mar 09 2016, Version 10 *)

a(n) = if ((n<5) && (n!=2), 0, my(b=3); while (! vecsearch(vecsort(digits(n, b)), 2), b++); b);  \\ Michel Marcus, Aug 06 2014, Mar 11 2016

Modified the definition to make the offset 1 by Nathan Fox, Mar 10 2016

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

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

A081606 Numbers having at least one 1 in their ternary representation.

Original entry on oeis.org

1, 3, 4, 5, 7, 9, 10, 11, 12, 13, 14, 15, 16, 17, 19, 21, 22, 23, 25, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 55, 57, 58, 59, 61, 63, 64, 65, 66, 67, 68, 69, 70, 71, 73, 75, 76, 77, 79, 81, 82, 83, 84, 85, 86

Offset: 1

Reinhard Zumkeller, Mar 23 2003

Complement of A005823.
Integers m such that central Delannoy number A001850(m) == 0 (mod 3). - Emeric Deutsch and Bruce E. Sagan, Dec 04 2003
Integers m such that A026375(m) == 0 (mod 3). - Fabio Visonà, Aug 03 2023

Johann Cigler, Some elementary observations on Narayana polynomials and related topics, arXiv:1611.05252 [math.CO], 2016. See p.25.
Mathematics Stack Exchange, Proof that integers m in this sequence are such that A026375(m) == 0 (mod 3).

Cf. A007089, A062756, A081609, A081605, A074940.

Select[Range[100],DigitCount[#,3,1]>0&] (* Harvey P. Dale, Nov 26 2022 *)

from itertools import count, islice
def A081606_gen(): # generator of terms
    a = 0
    for n in count(1):
        b = int(bin(n)[2:],3)<<1
        yield from range(a+1,b)
        a = b
A081606_list = list(islice(A081606_gen(),30)) # Chai Wah Wu, Oct 13 2023

from gmpy2 import digits
def A081606(n):
    def f(x):
        s = digits(x>>1,3)
        for i in range(l:=len(s)):
            if s[i]>'1':
                break
        else:
            return n+int(s,2)
        return n-1+(int(s[:i] or '0',2)+1<Chai Wah Wu, Oct 29 2024

More terms from Emeric Deutsch and Bruce E. Sagan, Dec 04 2003

A006996 a(n) = C(2n,n) mod 3.

Original entry on oeis.org

1, 2, 0, 2, 1, 0, 0, 0, 0, 2, 1, 0, 1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 1, 0, 1, 2, 0, 0, 0, 0, 1, 2, 0, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 1, 0, 1, 2, 0, 0, 0, 0, 1, 2, 0, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

N. J. A. Sloane, James Propp

Removing 0's from the sequence gives Thue-Morse sequence A001285 : 1,2,0,2,1,0,0,0,0,2,1,0,1,2,..->1,2,2,1,2,1,1,2,... - Benoit Cloitre, Jan 04 2004
a(n) = 0 if n in A074940, a(n) = 1 if n in A074939, a(n) = 2 if n in A074938.
Central terms of the triangle in A083093. - Reinhard Zumkeller, Jul 11 2013

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Reinhard Zumkeller, Table of n, a(n) for n = 0..2187=3^7
Michael Gilleland, Some Self-Similar Integer Sequences
Index entries for sequences that are fixed points of mappings

Cf. A001285.
Cf. A074938, A074939, A074940.
Cf. A083093.
Cf. A000984, A005704.

a006996 n = a083093 (2 * n) n  -- Reinhard Zumkeller, Jul 11 2013

Table[ Mod[ Binomial[2n, n], 3], {n, 0, 104}] (* Or *)
Nest[ Function[ l, {Flatten[(l /. {0 -> {0, 0, 0}, 1 -> {1, 2, 0}, 2 -> {2, 1, 0}})]}], {1}, 7] (* Robert G. Wilson v, Mar 28 2005 *)

a(n)=if(n==0, return(1)); if(vecmax(Set(digits(n,3)))>1, 0, 1 + n%2) \\ Charles R Greathouse IV, May 09 2016

from gmpy2 import digits
def A006996(n): return 0 if '2' in digits(n,3) else 1+(n&1) # Chai Wah Wu, Jun 26 2025

a(n) = A000984(n) mod 3.
a(n) = A005704(n) mod 3. - Benoit Cloitre, Jan 04 2004
A fixed point of the morphism : 1 -> 120, 2 -> 210, 0 -> 000. - Philippe Deléham, Jan 08 2004

A370920 Positive integers in whose ternary representation 2 occurs at least once, and every 2 is followed by 0.

Original entry on oeis.org

6, 15, 18, 19, 33, 42, 45, 46, 54, 55, 57, 58, 60, 87, 96, 99, 100, 114, 123, 126, 127, 135, 136, 138, 139, 141, 162, 163, 165, 166, 168, 171, 172, 174, 175, 177, 180, 181, 249, 258, 261, 262, 276, 285, 288, 289, 297, 298, 300, 301, 303, 330, 339, 342, 343

Offset: 1

Clark Kimberling, Mar 16 2024

Cf. A007089, A074940.

Cf. A370916, A370917, A370918, A370919, A370921, A370922, A370923.

Map[#[[1]] &, Select[Map[{#, #[[1]] > 0 && #[[1]] == #[[2]] &[{Length[
StringCases[#, "2"]], Length[StringCases[#, "20"]]}] &[
IntegerString[#, 3]]} &, Range[500]], #[[2]] &]]   (* Peter J. C. Moses, Mar 05 2024 *)

The ternary representations of 6, 15, and 18 are 20, 120, and 200.

cp's OEIS Frontend

A095778 Values of n for which A095777(n) is 9 (those terms which are expressible in decimal digits for bases 2 through 10, but not for base 11).

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A216194 a(n) = Smallest b for which the base b representation of n contains at least one 2 (or 0 if no such base exists).

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

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

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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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