A007092 -id:A007092 - cp's OEIS frontend

A033170 Positive numbers having the same set of digits in base 6 and base 7.

1, 2, 3, 4, 5, 68, 122, 246, 397, 425, 513, 625, 696, 970, 1062, 1167, 1215, 1244, 1251, 1576, 1722, 1787, 1823, 1871, 2406, 2413, 2532, 2574, 2660, 2844, 2851, 2856, 2857, 2858, 2859, 2860, 2861, 2864, 2865, 2872, 2879, 2900, 2954, 3005

Offset: 1

Clark Kimberling

John Cerkan, Table of n, a(n) for n = 1..10000

Cf. A007092, A007093.

isok(n) = Set(digits(n, 6)) == Set(digits(n, 7)); \\ Michel Marcus, Jan 21 2017

Edited by Don Reble, Apr 28 2006

Further edited by N. J. A. Sloane, Jan 17 2009 at the suggestion of R. J. Mathar

A255591 Convert n to base 6, move least significant digit to most significant digit and convert back to base 10.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 1, 7, 13, 19, 25, 31, 2, 8, 14, 20, 26, 32, 3, 9, 15, 21, 27, 33, 4, 10, 16, 22, 28, 34, 5, 11, 17, 23, 29, 35, 6, 42, 78, 114, 150, 186, 7, 43, 79, 115, 151, 187, 8, 44, 80, 116, 152, 188, 9, 45, 81, 117, 153, 189, 10, 46, 82, 118, 154, 190, 11

Offset: 0

Paolo P. Lava, Mar 02 2015

a(6*n) = n.

Fixed points of the transform are listed in A048331.

			16 in base 6 is 24: moving the least significant digit to the most significant one we have 42 that is 26 in base 10.

Indranil Ghosh, Table of n, a(n) for n = 0..46656 (terms 0..1000 from Paolo P. Lava)

Cf. A030105, A038572, A048331, A255588-A255590, A255592-A255594, A048787, A255689-A255693.

with(numtheory): P:=proc(q,h) local a,b,k,n; print(0);
for n from 1 to q do
a:=convert(n,base,h); b:=[]; for k from 2 to nops(a) do b:=[op(b),a[k]]; od; a:=[op(b),a[1]];
a:=convert(a,base,h,10); b:=0; for k from nops(a) by -1 to 1 do b:=10*b+a[k]; od;
print(b); od; end: P(10^4,6);

roll[n_, b_] := Block[{w = IntegerDigits[n, b]}, Prepend[Most@ w, Last@ w]]; b = 6; FromDigits[#, b] & /@ (roll[#, b] & /@ Range[0, 66]) (* Michael De Vlieger, Mar 04 2015 *)

def A255591(n):
    x=A007092(n)
    return int(x[-1]+x[:-1],6) # Indranil Ghosh, Feb 03 2017

A331565 The base 10 numbers with a digit product > 0 and which when written in bases 3,4,5,6,7,8,9 have two or more other base representations with the same digit product.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 91, 491, 921, 1138, 1234, 4853, 13581, 23568, 29242, 42161, 42162, 42163, 42164, 42991, 43365, 44313, 83342, 83651, 85226, 114382, 153881, 155462, 159422, 232868, 291862, 296183, 352486, 372642, 398543, 419563, 441194, 465326, 616146, 625431, 625523, 635813

Offset: 1

Scott R. Shannon, Jan 20 2020

For terms 10 < a(n) < 10^9 none have a base-3 representation whose digit product equals the base-10 product. The first such entry using the base-4 representation is 491.

			6 is a term as 6_10 = 6_7 = 6_8 = 6_9, so it has three other base representations where the digit product also equals 6.
91 is a term as 91_10 = 331_5 = 133_8, so it has two other base representations where the digit product also equals 9.
491 is a term as 491_10 = 13223_4 = 3431_5, so it has two other base representations where the digit product also equals 36.

Cf. A007954, A007089, A007090, A007091, A007092, A007093, A007094, A007095, A331561.

Subsequence of A052382 (zeroless numbers).

proDig[n_, b_] := Times @@ IntegerDigits[n, b]; seqQ[n_] := Module[{prod = proDig[n, 10], count = 0}, If[prod > 0, Do[If[proDig[n, b] == prod, count++]; If[count == 2, Break[]], {b, 3, 9}]]; count == 2]; Select[Range[650000], seqQ] (* Amiram Eldar, Jan 21 2020 *)

isok(n) = {my(p=vecprod(digits(n))); (p != 0) && (sum(k=3, 9, p==vecprod(digits(n,k))) >= 2);} \\ Michel Marcus, Jan 21 2020

A339256 Leading digit of n in base 6.

Original entry on oeis.org

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

Offset: 1

Kevin Ryde, Nov 28 2020

Kevin Ryde, Table of n, a(n) for n = 1..10000

Cf. A007092 (base 6), A109804 (partial sums).

In other bases: A122586, A122587, A339255, A000030, A099563, A276153.

Table[IntegerDigits[n,6][[1]],{n,90}] (* Harvey P. Dale, Jul 19 2023 *)

PARI
```
a(n) = n\6^logint(n,6);
```

a(n) = floor(n / 6^floor(log_6(n))).

G.f.: (x + Sum_{k>=0} Sum_{d=2..5} (x^(d*6^k)-x^(6^(k+1))) )/(1-x).

A004689 Fibonacci numbers written in base 6.

Original entry on oeis.org

0, 1, 1, 2, 3, 5, 12, 21, 33, 54, 131, 225, 400, 1025, 1425, 2454, 4323, 11221, 15544, 31205, 51153, 122402, 213555, 340401, 554400, 1335201, 2334001, 4113202, 10451203, 15004405, 25500012, 44504421, 114404433

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A000045 (Fibonacci), A007092 (numbers in base 6).

[Seqint(Intseq(Fibonacci(n),6)): n in [0..50]]; // G. C. Greubel, Oct 09 2018

bf[n_,k_]:=Module[{s=ToString[BaseForm[n,k]]},ToExpression[StringDrop[s,{Part[StringPosition[s,"\n"],1,1],-1}]]] lst={};Do[f=bf[Fibonacci[n],6];AppendTo[lst,f],{n,0,4*4!}];lst (* Vladimir Joseph Stephan Orlovsky, Jun 17 2009 *)
FromDigits[IntegerDigits[#, 6]]& / @ Fibonacci[Range[0, 40]] (* Vincenzo Librandi, Jun 08 2013 *)

vector(50, n, n--; fromdigits(digits(fibonacci(n), 6))) \\ G. C. Greubel, Oct 09 2018

A031947 Numbers in which 0,1,2,3,4,5 all occur in base 6.

Original entry on oeis.org

8345, 8350, 8375, 8385, 8410, 8415, 8525, 8530, 8585, 8600, 8620, 8630, 8735, 8745, 8765, 8780, 8835, 8840, 8950, 8955, 8980, 8990, 9015, 9020, 10505, 10510, 10535, 10545, 10570, 10575, 11045, 11050, 11165, 11190, 11200, 11220, 11255, 11265, 11345, 11370, 11415

Offset: 1

Clark Kimberling

Contains numbers like 47265, 47290, 47295, 47405 which are absent in A049357. - R. J. Mathar, Aug 24 2023

Cf. A007092 (base 6), A023744 (each base 6 digit once).

isA031947 := proc(n)
    convert(convert(n,base,6),set) ;
    if nops(%) = 6 then
        true;
    else
        false;
    end if;
end proc:
for n from 1 to 12000 do
    if isA031947(n) then
        print(n);
    end if;
end do: # R. J. Mathar, Aug 24 2023

A031951 Numbers with exactly two distinct base-6 digits.

Original entry on oeis.org

6, 8, 9, 10, 11, 12, 13, 15, 16, 17, 18, 19, 20, 22, 23, 24, 25, 26, 27, 29, 30, 31, 32, 33, 34, 36, 37, 42, 44, 45, 46, 47, 49, 50, 55, 57, 61, 64, 67, 71, 72, 74, 79, 80, 84, 85, 87, 88, 89, 92, 93, 98, 100, 104, 107, 108, 111, 115, 117

Offset: 1

Clark Kimberling

Robert Israel, Table of n, a(n) for n = 1..10000
Index entries for 6-automatic sequences.

Cf. A007092, A031948, A031949, A031950, A031952, A031953, A031954, A031955.

M:= 6: # get all terms < 6^M
sort([seq(seq(seq(seq(add(6^(m-j)*`if`(member(j,S2),d2,d1),j=1..m)  ,
S2 = combinat:-powerset({$2..m}) minus {{}}),
d2 = {$0..5} minus {d1}), d1 = 1..5), m=2..M)]);# Robert Israel, Dec 03 2015

fQ[n_] := Length@ Union@ IntegerDigits[n, 6] == 2; Select[Range@117, fQ] (* Robert G. Wilson v, Dec 03 2015 *)

A032861 Numbers whose base-6 representation Sum_{i=0..m} d(i)*6^i has d(m) > d(m-1) < d(m-2) > ...

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 12, 13, 18, 19, 20, 24, 25, 26, 27, 30, 31, 32, 33, 34, 37, 38, 39, 40, 41, 73, 74, 75, 76, 77, 80, 81, 82, 83, 109, 110, 111, 112, 113, 116, 117, 118, 119, 123, 124, 125, 145, 146, 147, 148, 149, 152, 153, 154, 155

Offset: 1

Clark Kimberling

Cf. A007092.
Cf. A032858..A032865 for bases 3..10.
Cf. A306106..A306111 and A297147 for bases 3..9 and 10.

sdQ[n_]:=Module[{s=Sign[Differences[IntegerDigits[n,6]]]},s==PadRight[{}, Length[ s],{-1,1}]]; Select[Range[0,200],sdQ] (* Harvey P. Dale, Dec 15 2017 *)

a(1)=0 inserted by Georg Fischer, Dec 18 2020

A033019 Numbers whose base-6 expansion has no run of digits with length < 2.

Original entry on oeis.org

7, 14, 21, 28, 35, 43, 86, 129, 172, 215, 252, 259, 266, 273, 280, 287, 504, 511, 518, 525, 532, 539, 756, 763, 770, 777, 784, 791, 1008, 1015, 1022, 1029, 1036, 1043, 1260, 1267, 1274, 1281, 1288, 1295, 1512, 1548, 1555, 1562

Offset: 1

Clark Kimberling

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A007092.

Select[Range[10000], Min[Length/@Split[IntegerDigits[#, 6]]]>1&] (* Vincenzo Librandi, Feb 05 2014 *)

A166479 Lesser of twin primes, written in base 6.

Original entry on oeis.org

3, 5, 15, 25, 45, 105, 135, 155, 245, 255, 345, 405, 455, 515, 525, 1015, 1035, 1125, 1145, 1235, 1335, 1535, 1555, 2045, 2225, 2345, 2435, 2505, 2545, 3015, 3425, 3445, 3455, 3545, 4025, 4415, 4435, 4505, 4525, 5015, 5155, 5405, 5525, 5545, 10005, 10035

Offset: 1

Jonathan Vos Post, Oct 14 2009

All but the first value is of the form 6n-1, hence in base 6 end with the digit 5.

Nathaniel Johnston, Table of n, a(n) for n = 1..10000

Cf. A001359, A007092.

A001359 := proc(n) option remember: local p: if(n=1)then return 3: fi: p:=procname(n-1): do p:=nextprime(p): if(isprime(p+2))then return p: fi: od: end: A166479 := proc(n) local l: l:=convert(A001359(n),base,6): return op(convert(l,base,10,10^nops(l))): end: seq(A166479(n),n=1..60); # Nathaniel Johnston, May 06 2011

FromDigits[IntegerDigits[#,6]]&/@Transpose[Select[Partition[ Prime[ Range[ 300]],2,1],#[[2]]-#[[1]]==2&]][[1]] (* Harvey P. Dale, Aug 19 2015 *)

a(n) = A007092(A001359(n)).

Extended by Nathaniel Johnston, May 06 2011

cp's OEIS Frontend

A033170 Positive numbers having the same set of digits in base 6 and base 7.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Extensions

A255591 Convert n to base 6, move least significant digit to most significant digit and convert back to base 10.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Python

A331565 The base 10 numbers with a digit product > 0 and which when written in bases 3,4,5,6,7,8,9 have two or more other base representations with the same digit product.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI

A339256 Leading digit of n in base 6.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A004689 Fibonacci numbers written in base 6.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

A031947 Numbers in which 0,1,2,3,4,5 all occur in base 6.

Views

Author

Keywords

Comments

Crossrefs

Programs

Maple

A031951 Numbers with exactly two distinct base-6 digits.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

A032861 Numbers whose base-6 representation Sum_{i=0..m} d(i)*6^i has d(m) > d(m-1) < d(m-2) > ...

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Extensions

A033019 Numbers whose base-6 expansion has no run of digits with length < 2.

Views

Author