A007092 -id:A007092 - cp's OEIS frontend

A030588 Write odd numbers in base 6 and juxtapose.

1, 3, 5, 1, 1, 1, 3, 1, 5, 2, 1, 2, 3, 2, 5, 3, 1, 3, 3, 3, 5, 4, 1, 4, 3, 4, 5, 5, 1, 5, 3, 5, 5, 1, 0, 1, 1, 0, 3, 1, 0, 5, 1, 1, 1, 1, 1, 3, 1, 1, 5, 1, 2, 1, 1, 2, 3, 1, 2, 5, 1, 3, 1, 1, 3, 3, 1, 3, 5, 1, 4, 1, 1, 4, 3, 1, 4, 5, 1, 5, 1, 1, 5, 3, 1, 5, 5, 2, 0, 1

Offset: 1

Clark Kimberling

			From _Antti Karttunen_, Dec 26 2018: (Start)
The first twelve odd numbers and their base-6 representations (A007092) are:
    1    1
    3    3
    5    5
    7   11
    9   13
   11   15
   13   21
   15   23
   17   25
   19   31
   21   33
   23   35
thus the sequence begins with terms 1, 3, 5, 1, 1, 1, 3, 1, 5, 2, 1, 2, 3, 2, 5, 3, 1, 3, 3, 3, 5.
(End)

Antti Karttunen, Table of n, a(n) for n = 1..100000

Cf. A007092.
Cf. A030589, A030590, A030591, A030592, A030593, A030594.
Cf. A030595, A030601, A030602, A030603.
Cf. A030596, A030597, A030598, A030599, A030600.

Flatten[Table[IntegerDigits[2n - 1, 6], {n, 50}]] (* Harvey P. Dale, Jul 22 2012 *)

up_to = 100000;
A030588list(up_to) = { my(v=vector(up_to),ds=List([]),o=1,j=1); for(i=1,up_to,if(j>#ds,ds=digits(o,6);j=1;o+=2); v[i] = ds[j]; j++); (v); };
v030588 = A030588list(up_to);
A030588(n) = v030588[n]; \\ Antti Karttunen, Dec 24 2018

(1 to 99 by 2).map(Integer.toString(, 6).toCharArray).flatten // _Alonso del Arte, Apr 02 2020

A202267 Numbers in which all digits are noncomposites (1, 2, 3, 5, 7) or 0.

Original entry on oeis.org

0, 1, 2, 3, 5, 7, 10, 11, 12, 13, 15, 17, 20, 21, 22, 23, 25, 27, 30, 31, 32, 33, 35, 37, 50, 51, 52, 53, 55, 57, 70, 71, 72, 73, 75, 77, 100, 101, 102, 103, 105, 107, 110, 111, 112, 113, 115, 117, 120, 121, 122, 123, 125, 127, 130, 131, 132, 133, 135, 137, 150

Offset: 1

Jaroslav Krizek, Dec 25 2011

If n-1 is represented as a base-6 number (see A007092) according to n-1=d(m)d(m-1)...d(3)d(2)d(1)d(0) then a(n)= sum_{j=0..m} c(d(j))*10^j, where c(k)=0,1,2,3,5,7 for k=0..5. - Hieronymus Fischer, May 30 2012

			a(1000) = 5353.
a(10^4) = 115153
a(10^5) = 2070753.
a(10^6) = 33233353.

Hieronymus Fischer, Table of n, a(n) for n = 1..10000
Robert Baillie and Thomas Schmelzer, Summing Kempner's Curious (Slowly-Convergent) Series, Mathematica Notebook kempnerSums.nb, Wolfram Library Archive, 2008.
Index entries for 10-automatic sequences.

Supersequence of A001742 and A046034.
Cf. A046034 (numbers in which all digits are primes), A001742 (numbers in which all digits are noncomposites excluding 0), A202268 (numbers in which all digits are nonprimes excluding 0), A084984 (numbers in which all digits are nonprimes), A029581 (numbers in which all digits are composites).
Cf. A007092, A001743, A001744, A193238.

Union[Flatten[FromDigits/@Tuples[{0,1,2,3,5,7},3]]] (* Harvey P. Dale, Mar 11 2015 *)

From Hieronymus Fischer, May 30 2012: (Start)
a(n) = (b_m(n)+1) mod 10 + floor((b_m(n)+2)/5) + floor((b_m(n)+1)/5) - 2*floor(b_m(n)/5))*10^m + sum_{j=0..m-1} (b_j(n) mod 6 + floor((b_j(n)+1)/6) + floor((b_j(n)+2)/6) - 2*floor(b_j(n)/6)))*10^j, where n>1, b_j(n)) = floor((n-1-6^m)/6^j), m = floor(log_6(n-1)).
a(1*6^n+1) = 1*10^n.
a(2*6^n+1) = 2*10^n.
a(3*6^n+1) = 3*10^n.
a(4*6^n+1) = 5*10^n.
a(5*6^n+1) = 7*10^n.
a(n) = 10^log_6(n-1) for n=6^k+1, k>0,
a(n) < 10^log_6(n-1) else.
a(n) = A007092(n-1) iff the digits of A007092(n-1) are <= 3, a(n)>A007092(n-1), else.
a(n) <= A084984(n), equality holds if the representation of n-1 as a base-6 number only has digits 0 or 1.
G.f.: g(x) = (x/(1-x))*sum_{j>=0} 10^j*x^6^j *(1-x^6^j)* (1 + 2x^6^j + 3(x^2)^6^j + 5(x^3)^6^j + 7(x^4)^6^j)/(1-x^6^(j+1)).
Also: g(x) = (x/(1-x))*(h_(6,1)(x) + h_(6,2)(x) + h_(6,3)(x) + 2*h_(6,4)(x) + 2*h_(6,5)(x) - 7*h_(6,6)(x)), where h_(6,k)(x) = sum_{j>=0} 10^j*x^(k*6^j)/(1-x^6^(j+1)). (End)
Sum_{n>=2} 1/a(n) = 4.945325883472729555972742252181522711968119529132581193614012706741310832798... (calculated using Baillie and Schmelzer's kempnerSums.nb, see Links). - Amiram Eldar, Feb 15 2024

Examples added by Hieronymus Fischer, May 30 2012

A029990 Numbers k such that k^2 is palindromic in base 6.

Original entry on oeis.org

0, 1, 2, 7, 37, 43, 76, 91, 217, 259, 1064, 1297, 1333, 1519, 1555, 2704, 3367, 7777, 8029, 9079, 19747, 46657, 46873, 47989, 48205, 54439, 54655, 54695, 83979, 118027, 241304, 279937, 281449, 287749, 326599, 707707, 1679617

Offset: 1

Patrick De Geest

Seiichi Manyama, Table of n, a(n) for n = 1..92
Patrick De Geest, Palindromic Squares in bases 2 to 17

Cf. A007092.

Numbers k such that k^2 is palindromic in base b: A003166 (b=2), A029984 (b=3), A029986 (b=4), A029988 (b=5), this sequence (b=6), A029992 (b=7), A029805 (b=8), A029994 (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).

palindromicQ[n_, b_:10] := TrueQ[IntegerDigits[n, b] == Reverse[IntegerDigits[n, b]]]; Select[Range[1000], palindromicQ[#^2, 6] &] (* Alonso del Arte, Mar 05 2017 *)

ispal(n,base)=my(d=digits(n,base)); d==Vecrev(d)
is(n)==ispal(n^2,6) \\ Charles R Greathouse IV, Mar 09 2017

A073787 Numbers in base -6.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 150, 151, 152, 153, 154, 155, 140, 141, 142, 143, 144, 145, 130, 131, 132, 133, 134, 135, 120, 121, 122, 123, 124, 125, 110, 111, 112, 113, 114, 115, 100, 101, 102, 103, 104, 105, 250, 251, 252, 253, 254, 255, 240, 241, 242, 243, 244, 245

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. A007092, A039724, A073785, A007608, A073786, A073788, A073789, A073790 & A039723.

ToNegaBases[i_Integer, b_Integer] := FromDigits[ Rest[ Reverse[ Mod[ NestWhileList[(#1 - Mod[ #1, b])/-b &, i, #1 != 0 &], b]]]]; Table[ ToNegaBases[n, 6], {n, 0, 60}]

def A073787(n):
    s, q = '', n
    while q >= 6 or q < 0:
        q, r = divmod(q, -6)
        if r < 0:
            q += 1
            r += 6
        s += str(r)
    return int(str(q)+s[::-1]) # Chai Wah Wu, Apr 09 2016

A248910 Numbers with no zeros in base-6 representation.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 61, 62, 63, 64, 65, 67, 68, 69, 70, 71, 79, 80, 81, 82, 83, 85, 86, 87, 88, 89, 91, 92

Offset: 1

Reinhard Zumkeller, Mar 08 2015

Different from A039215, A047253, A184522, A187390, A194386.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Wikipedia, Senary
Index entries for 6-automatic sequences.

Cf. A007092, A100969 (subsequence).

Zeroless numbers in some other bases <= 10: A000042 (base 2), A032924 (base 3), A023705 (base 4), A255805 (base 8), A255808 (base 9), A052382 (base 10).

a248910 n = a248910_list !! (n-1)
a248910_list = iterate f 1 where
   f x = 1 + if r < 5 then x else 6 * f x'  where (x', r) = divMod x 6

Select[Range[100], DigitCount[#,6, 0] == 0 &] (* Paolo Xausa, Jun 29 2025 *)

isok(m) = vecmin(digits(m, 6)) > 0; \\ Michel Marcus, Jan 23 2022

from sympy import integer_log
def A248910(n):
    m = integer_log(k:=(n<<2)+1,5)[0]
    return sum((1+(k-5**m)//(5**j<<2)%5)*6**j for j in range(m)) # Chai Wah Wu, Jun 28 2025

A346690 Replace 6^k with (-1)^k in base-6 expansion of n.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, -1, 0, 1, 2, 3, 4, -2, -1, 0, 1, 2, 3, -3, -2, -1, 0, 1, 2, -4, -3, -2, -1, 0, 1, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 0, 1, 2, 3, 4, 5, -1, 0, 1, 2, 3, 4, -2, -1, 0, 1, 2, 3, -3, -2, -1, 0, 1, 2, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 1, 2, 3, 4, 5, 6, 0, 1, 2, 3, 4, 5, -1, 0, 1, 2, 3, 4, -2, -1, 0, 1, 2, 3, -3, -2, -1

Offset: 0

Ilya Gutkovskiy, Jul 29 2021

If n has base-6 expansion abc..xyz with least significant digit z, a(n) = z - y + x - w + ...

			59 = 135_6, 5 - 3 + 1 = 3, so a(59) = 3.

Robert Israel, Table of n, a(n) for n = 0..10000

Cf. A007092, A053827, A055017, A065359, A065368, A346688, A346689, A346691.

f:= proc(n) option remember; (n mod 6) - procname(floor(n/6)) end proc:
f(0):= 0:
map(f, [$1..100]); # Robert Israel, Nov 21 2022

nmax = 104; A[] = 0; Do[A[x] = x (1 + 2 x + 3 x^2 + 4 x^3 + 5 x^4)/(1 - x^6) - (1 + x + x^2 + x^3 + x^4 + x^5) A[x^6] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
Table[n + 7 Sum[(-1)^k Floor[n/6^k], {k, 1, Floor[Log[6, n]]}], {n, 0, 104}]

a(n) = subst(Pol(digits(n, 6)), 'x, -1); \\ Michel Marcus, Nov 22 2022

from sympy.ntheory.digits import digits
def a(n):
    return sum(bi*(-1)**k for k, bi in enumerate(digits(n, 6)[1:][::-1]))
print([a(n) for n in range(105)]) # Michael S. Branicky, Jul 29 2021

G.f. A(x) satisfies: A(x) = x * (1 + 2*x + 3*x^2 + 4*x^3 + 5*x^4) / (1 - x^6) - (1 + x + x^2 + x^3 + x^4 + x^5) * A(x^6).
a(n) = n + 7 * Sum_{k>=1} (-1)^k * floor(n/6^k).
a(6*n+j) = j - a(n) for 0 <= j <= 5. - Robert Israel, Nov 21 2022

A073793 Replace 6^k with (-6)^k in base 6 expansion of n.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, -6, -5, -4, -3, -2, -1, -12, -11, -10, -9, -8, -7, -18, -17, -16, -15, -14, -13, -24, -23, -22, -21, -20, -19, -30, -29, -28, -27, -26, -25, 36, 37, 38, 39, 40, 41, 30, 31, 32, 33, 34, 35, 24, 25, 26, 27, 28, 29, 18, 19, 20, 21, 22, 23, 12, 13, 14, 15, 16, 17, 6, 7, 8, 9, 10, 11

Offset: 0

Robert G. Wilson v, Aug 12 2002

Base 6 representation for n converted from base -6 to base 10.

Dana G. Korssjoen, Biyao Li, Stefan Steinerberger, Raghavendra Tripathi, and Ruimin Zhang, Finding structure in sequences of real numbers via graph theory: a problem list, arXiv:2012.04625 [math.CO], 2020-2021.

Cf. A007092, A073787, A053985, A065369, A073791, A073792, A073794, A073795, A073796 & A073835.

f[n_Integer, b_Integer] := Block[{l = IntegerDigits[n]}, Sum[l[[ -i]]*(-b)^(i - 1), {i, 1, Length[l]}]]; a = Table[ FromDigits[ IntegerDigits[n, 6]], {n, 0, 80}]; b = {}; Do[b = Append[b, f[a[[n]], 6]], {n, 1, 80}]; b

a(6*k+m) = -6*a(k)+m for 0 <= m < 6. - Chai Wah Wu, Jan 16 2020

A037382 Numbers k such that every base-3 digit of k is a base-6 digit of k.

Original entry on oeis.org

1, 2, 8, 13, 26, 36, 37, 38, 39, 40, 44, 48, 49, 50, 52, 53, 68, 72, 73, 74, 78, 79, 80, 109, 121, 152, 157, 182, 218, 224, 228, 229, 230, 231, 232, 233, 236, 242, 243, 244, 246, 247, 248, 252, 253, 254, 255, 256, 264, 270, 282, 288

Offset: 1

Clark Kimberling

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

Cf. A007089, A007092.

Cf. A037380, A037381, A037383, A037384, A037385, A037386.

import Data.List ((\\), nub)
a037382 n = a037382_list !! (n-1)
a037382_list = filter f [1..] where
   f x = null $ nub (ds 3 x) \\ nub (ds 6 x)
   ds b x = if x > 0 then d : ds b x' else []  where (x', d) = divMod x b
-- Reinhard Zumkeller, May 30 2013

Select[Range[300],SubsetQ[IntegerDigits[#,6],IntegerDigits[#,3]]&] (* Harvey P. Dale, Jun 05 2015 *)

A262103 Pseudoprimes to base 6, written in base 6.

Original entry on oeis.org

55, 505, 1001, 1221, 2121, 5041, 5051, 5501, 10101, 12001, 15225, 20301, 21021, 23501, 24301, 24341, 30041, 31031, 32451, 42241, 50125, 50321, 101101, 102421, 105131, 111111, 113425, 121001, 121101, 123041, 123321, 132305, 150135, 152021, 201201, 204445, 212121, 221001, 222401, 232401

Offset: 1

Abdul Gaffar Khan, Sep 11 2015

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A007092 (numbers in base 6), A005937 (pseudoprimes to base 6).

Cf. A258189, A262101, A262102, A262104, A262105, A262154.

base = 6; t = {}; n = 1;
While[Length[t] < 40, n++;
If[! PrimeQ[n] && PowerMod[base, n - 1, n] == 1,
  AppendTo[t, FromDigits@IntegerDigits[n, 6]]]]; t

lista(nn, b=6) = {for (n=1, nn, if (Mod(b, n)^(n-1)==1 && !ispseudoprime(n) && n>1, print1(subst(Pol(digits(n,b), x), x, 10), ", ");););} \\ Michel Marcus, Sep 30 2015

a(n) = A007092(A005937(n)).

A293659 Base-6 circular primes that are not base-6 repunits.

Original entry on oeis.org

11, 31, 71, 191, 211

Offset: 1

Felix Fröhlich, Oct 28 2017

Conjecture: The sequence is finite, with 211 being the last term (see A293142).
Written in base 6 (A007092), the terms are 15, 51, 155, 515, 551. - Antti Karttunen, Nov 26 2017
From Michael De Vlieger, Dec 30 2017: (Start)
This sequence may be particularly constrained to few terms since only {1, 5} are coprime to 6, and any senary circular prime involves just these 2 senary digits. This is because all primes aside from {2, 3} are congruent to {1, 5} (mod 6). Since a senary number consisting of all 5's is divisible by 5 and since we have disqualified prime repunits, the sequence is probably finite.
a(6), if it exists, must be larger than 6^21 = 21936950640377856. (End)

			71 written in base 6 is 155. The base-6 numbers 155, 515, 551 written in base 10 are 71, 191, 211, respectively and all those numbers are prime, so 71, 191 and 211 are terms of the sequence.

Cf. A007092, A293142.

Cf. base-b nonrepunit circular primes: A293657 (b=4), A293658 (b=5), A293660 (b=7), A293661 (b=8), A293662 (b=9), A293663 (b=10).

With[{b = 6}, Select[Prime@ Range[PrimePi@ b + 1, 10^6], Function[w, And[AllTrue[Array[FromDigits[RotateRight[w, #], b] &, Length@ w - 1], PrimeQ], Union@ w != {1} ]]@ IntegerDigits[#, b] &]] (* or *)
With[{b = 6}, Select[Flatten@ Array[FromDigits[#, 6] & /@ Most@ Rest@ Tuples[{1, 5}, #] &, 18, 2], Function[w, And[ AllTrue[ Array[ FromDigits[ RotateRight[w, #], b] &, Length@ w], PrimeQ], Union@ w != {1} ]]@ IntegerDigits[#, b] &]] (* Michael De Vlieger, Dec 30 2017 *)

rot(n) = if(#Str(n)==1, v=vector(1), v=vector(#n-1)); for(i=2, #n, v[i-1]=n[i]); u=vector(#n); for(i=1, #n, u[i]=n[i]); v=concat(v, u[1]); v
decimal(v, base) = my(w=[]); for(k=0, #v-1, w=concat(w, v[#v-k]*base^k)); sum(i=1, #w, w[i])
is_circularprime(p, base) = my(db=digits(p, base), r=rot(db), i=0); if(vecmin(db)==0, return(0), while(1, dec=decimal(r, base); if(!ispseudoprime(dec), return(0)); r=rot(r); if(r==db, return(1))))
forprime(p=1, , if(vecmin(digits(p, 6))!=vecmax(digits(p, 6)), if(is_circularprime(p, 6), print1(p, ", "))))

cp's OEIS Frontend

A030588 Write odd numbers in base 6 and juxtapose.

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A202267 Numbers in which all digits are noncomposites (1, 2, 3, 5, 7) or 0.

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A029990 Numbers k such that k^2 is palindromic in base 6.

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A073787 Numbers in base -6.

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A248910 Numbers with no zeros in base-6 representation.

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Haskell

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A346690 Replace 6^k with (-1)^k in base-6 expansion of n.

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A073793 Replace 6^k with (-6)^k in base 6 expansion of n.

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A037382 Numbers k such that every base-3 digit of k is a base-6 digit of k.