A007093 -id:A007093 - cp's OEIS frontend

A037403 Numbers k such that every base-7 digit of k is a base-9 digit of k.

1, 2, 3, 4, 5, 6, 31, 99, 106, 107, 195, 198, 248, 257, 284, 297, 321, 498, 514, 749, 750, 751, 758, 767, 785, 936, 939, 940, 943, 950, 968, 996, 1028, 1086, 1088, 1110, 1163, 1200, 1218, 1254, 1453, 1471, 1498, 1500, 1502, 1507

Offset: 1

Clark Kimberling

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

Cf. A007093, A007095.

Cf. A037402, A037404.

import Data.List ((\\), nub)
a037403 n = a037403_list !! (n-1)
a037403_list = filter f [1..] where
   f x = null $ nub (ds 7 x) \\ nub (ds 9 x)
   ds b x = if x > 0 then d : ds b x' else []  where (x', d) = divMod x b
-- Reinhard Zumkeller, May 30 2013

A083901 Number of divisors of n with largest digit <= 6 (base 10).

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 1, 3, 2, 4, 2, 6, 2, 3, 4, 4, 1, 4, 1, 6, 3, 4, 2, 7, 3, 4, 2, 4, 1, 8, 2, 5, 4, 3, 3, 7, 1, 2, 3, 7, 2, 7, 2, 6, 5, 4, 1, 8, 1, 6, 3, 6, 2, 5, 4, 5, 2, 2, 1, 12, 2, 4, 4, 6, 4, 8, 1, 4, 3, 6, 1, 8, 1, 2, 5, 3, 2, 6, 1, 8, 2, 3, 1, 9, 2, 3, 2, 6, 1, 9, 2, 5, 3, 2, 2, 9, 1, 3, 4, 9, 2, 7, 2, 7, 7

Offset: 1

Reinhard Zumkeller, May 08 2003

Cf. A054055, A000005, A007093, A083893, A083894, A083900, A083902.

a[n_] := DivisorSum[n, 1 &, Max[IntegerDigits[#]] <= 6 &]; Array[a, 100] (* Amiram Eldar, Jan 04 2024 *)

a(n) = A083900(n) + A083893(n) = A083902(n) - A083894(n).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{k>=1} 1/A007093(k) = 7.32338572355014919630... . - Amiram Eldar, Jan 04 2024

A309958 Product of digits of (n written in base 7).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 0, 1, 2, 3, 4, 5, 6, 0, 2, 4, 6, 8, 10, 12, 0, 3, 6, 9, 12, 15, 18, 0, 4, 8, 12, 16, 20, 24, 0, 5, 10, 15, 20, 25, 30, 0, 6, 12, 18, 24, 30, 36, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 3, 4, 5, 6, 0, 2, 4, 6, 8, 10, 12, 0, 3, 6, 9, 12, 15, 18, 0, 4, 8, 12, 16, 20, 24, 0, 5, 10, 15, 20, 25, 30, 0, 6, 12, 18, 24, 30, 36, 0, 0, 0

Offset: 0

Ilya Gutkovskiy, Aug 24 2019

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Product of digits of (n written in base k): A309953 (k = 3), A309954 (k = 4), A309956 (k = 5), A309957 (k = 6), this sequence (k = 7), A309959 (k = 8), A309788 (k = 9), A007954 (k = 10).

Cf. A007093, A053828.

[0] cat [&*Intseq(n,7):n in [1..100]]; // Marius A. Burtea, Aug 25 2019

Table[Times @@ IntegerDigits[n, 7], {n, 0, 100}]

a(n) = my(v=vecprod(digits(n, 7))); n>0 && return(v) \\ Felix Fröhlich, Sep 09 2019

G.f. A(x) satisfies: A(x) = x * (1 + 2*x + 3*x^2 + 4*x^3 + 5*x^4 + 6*x^5) * (1 + A(x^7)).

A004690 Fibonacci numbers written in base 7.

Original entry on oeis.org

0, 1, 1, 2, 3, 5, 11, 16, 30, 46, 106, 155, 264, 452, 1046, 1531, 2610, 4441, 10351, 15122, 25503, 43625, 102431, 146356, 252120, 431506, 1013626, 1445435, 2462364, 4241132, 10033526, 14304661, 24341520

Offset: 0

N. J. A. Sloane

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

Cf. A000045 (Fibonacci), A007093 (numbers in base 7).

[Seqint(Intseq(Fibonacci(n),7)): n in [0..50]]; // G. C. Greubel, Oct 09 2018

Table[BaseForm[Fibonacci[n],7],{n,1,20,1}] (* Vladimir Joseph Stephan Orlovsky, Jul 23 2008 *)
FromDigits[IntegerDigits[#, 7]] & / @ Fibonacci[Range[0, 40]] (* Vincenzo Librandi, Jun 08 2013 *)

a(n)=fromdigits(digits(fibonacci(n),7)) \\ Charles R Greathouse IV, Oct 28 2014

A024639 n written in fractional base 7/2.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 20, 21, 22, 23, 24, 25, 26, 40, 41, 42, 43, 44, 45, 46, 60, 61, 62, 63, 64, 65, 66, 210, 211, 212, 213, 214, 215, 216, 230, 231, 232, 233, 234, 235, 236, 250, 251, 252, 253, 254, 255, 256, 400, 401, 402, 403, 404, 405, 406, 420, 421, 422, 423, 424, 425

Offset: 0

David W. Wilson

To represent a number in base 7, if a digit exceeds 6, subtract 7 and carry 1. In the fractional base 7/2, subtract 7 and carry 2.

			From _Alonso del Arte_, Apr 21 2019: (Start)
The integers 0 through 6 are written with the digits 0 through 6.
Then, since b = 7/2 is written as 10, and 7 is twice 7/2, 7 is 20 in base 7/2, and therefore a(7) = 20.
a(28) = 210 since 2 * (7/2)^2 + 1 * (7/2) = 2 * 49/4 + 1 * 7/2 = 98/4 + 14/4 = 112/4 = 28. (End)

Amiram Eldar, Table of n, a(n) for n = 0..10000
Index entries for sequences related to fractional bases.

Cf. A000079, A000420, A007093, A245342.

Select[Table[FromDigits[IntegerDigits[n, 7]], {n, 0, 230}], IntegerQ[FromDigits[IntegerDigits[#], 7/2]] &] (* Alonso del Arte, Apr 21 2019 *)
a[n_] := a[n] = If[n == 0, 0, 10 * a[2 * Floor[n/7]] + Mod[n, 7]]; Array[a, 50, 0] (* Amiram Eldar, Jul 31 2025 *)

a(n) = if(n == 0, 0, 10 * a(n\7 * 2) + n % 7); \\ Amiram Eldar, Jul 31 2025

A032548 Integer part of decimal 'base-7 looking' numbers divided by their actual base-7 values.

Original entry on oeis.org

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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2

Offset: 1

Patrick De Geest, Apr 15 1998

Antti Karttunen, Table of n, a(n) for n = 1..35000

Cf. A007093, A032549, A032550. See also A032532 for explanation.

Array[IntegerPart[FromDigits[#]/FromDigits[#, 7]] &@ IntegerDigits[#, 7] &, 100] (* Michael De Vlieger, Jan 03 2019 *)

A007093(n) = if(n<=1,n,(10*A007093(n\7))+(n%7));
A032548(n) = (A007093(n)\n); \\ Antti Karttunen, Jan 01 2019

a(n) = floor(A007093(n)/n). - Antti Karttunen, Jan 01 2019

Offset corrected by Antti Karttunen, Jan 01 2019

A032862 Numbers whose base-7 representation Sum_{i=0..m} d(i)*7^i has d(m) > d(m-1) < d(m-2) > ...

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 14, 15, 21, 22, 23, 28, 29, 30, 31, 35, 36, 37, 38, 39, 42, 43, 44, 45, 46, 47, 50, 51, 52, 53, 54, 55, 99, 100, 101, 102, 103, 104, 107, 108, 109, 110, 111, 148, 149, 150, 151, 152, 153, 156, 157, 158, 159, 160, 164

Offset: 1

Clark Kimberling

Cf. A007093.
Cf. A032858..A032865 for bases 3..10.
Cf. A306106..A306111 and A297147 for bases 3..9 and 10.

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

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

Original entry on oeis.org

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

A037440 Positive numbers having the same set of digits in base 7 and base 10.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 23, 46, 265, 316, 1030, 1234, 1336, 1366, 1401, 1431, 1443, 1454, 1464, 2060, 2116, 3261, 3515, 3621, 4631, 5052, 10144, 10342, 10542, 11134, 11425, 11524, 11544, 12415, 12450, 12532, 12564, 12651, 13035, 13045, 13245

Offset: 1

Clark Kimberling

			1336 is in the sequence because 1336 in base 7 is 3616.

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

Subsequence of A037404.

Cf. A007093.

Select[Range@ 13300, Union@ IntegerDigits@ # == Union@ IntegerDigits[#, 7] &] (* Michael De Vlieger, Feb 18 2017 *)

isok(n) = Set(digits(n, 7)) == Set(digits(n)); \\ Michel Marcus, Feb 18 2017

from sympy.ntheory import digits
def ok(n): return set(map(int, str(n))) == set(digits(n, 7)[1:])
print([k for k in range(1, 10**6) if ok(k)]) # Michael S. Branicky, Apr 22 2023

More terms from Don Reble, Apr 28 2006

Edited by John Cerkan, Feb 17 2017

A044956 Numbers with no two equally numerous base 7 digits.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 16, 24, 32, 40, 48, 49, 50, 56, 57, 58, 59, 60, 61, 62, 64, 65, 71, 73, 78, 81, 85, 89, 92, 97, 98, 100, 106, 107, 112, 113, 114, 115, 116, 117, 118, 121, 122, 128, 130, 135, 138, 142, 146, 147, 150, 155, 157, 163, 164

Offset: 1

Clark Kimberling

			106 in base 7 is 211, which has one more 1 than it has 2's.
107 in base 7 is 212, which has one more 2 than it has 1's.
108 in base 7 is 213. Since each digit occurs as many times as the others (once each), 108 is not in the sequence.

Cf. A007093 (numbers in base 7), A023802.

enb7Q[n_] := Module[{t = Tally[IntegerDigits[n, 7]][[All, 2]]}, Length[t] == Length[Union[t]]]; Select[Range[200], enb7Q] (* Harvey P. Dale, Jul 26 2018 *)

cp's OEIS Frontend

A037403 Numbers k such that every base-7 digit of k is a base-9 digit of k.

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Haskell

A083901 Number of divisors of n with largest digit <= 6 (base 10).

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A309958 Product of digits of (n written in base 7).

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Magma

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A004690 Fibonacci numbers written in base 7.

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Magma

Mathematica

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A024639 n written in fractional base 7/2.

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Mathematica

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A032548 Integer part of decimal 'base-7 looking' numbers divided by their actual base-7 values.

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A032862 Numbers whose base-7 representation Sum_{i=0..m} d(i)*7^i has d(m) > d(m-1) < d(m-2) > ...

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

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PARI

Extensions

A037440 Positive numbers having the same set of digits in base 7 and base 10.

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