A007093 -id:A007093 - cp's OEIS frontend

A037394 Numbers k such that every base-5 digit of k is a base-7 digit of k.

1, 2, 3, 4, 17, 51, 66, 102, 109, 123, 156, 158, 162, 206, 218, 312, 317, 324, 361, 381, 416, 418, 423, 458, 462, 463, 466, 467, 468, 472, 494, 518, 545, 546, 549, 556, 557, 559, 562, 584, 606, 619, 621, 630, 640, 651, 658, 687

Offset: 1

Clark Kimberling

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

Cf. A007091, A007093.

Cf. A037393, A037395, A037396, A037397.

import Data.List ((\\), nub)
a037394 n = a037394_list !! (n-1)
a037394_list = filter f [1..] where
   f x = null $ nub (ds 5 x) \\ nub (ds 7 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[700],SubsetQ[IntegerDigits[#,7],IntegerDigits[#,5]]&] (* Harvey P. Dale, Sep 29 2017 *)

A063432 Triangle read by rows in which k-th entry in row n is representation of n in base k, for 1 <= k <= n.

Original entry on oeis.org

1, 11, 10, 111, 11, 10, 1111, 100, 11, 10, 11111, 101, 12, 11, 10, 111111, 110, 20, 12, 11, 10, 1111111, 111, 21, 13, 12, 11, 10, 11111111, 1000, 22, 20, 13, 12, 11, 10, 111111111, 1001, 100, 21, 14, 13, 12, 11, 10, 1111111111, 1010, 101, 22, 20, 14, 13

Offset: 1

Henry Bottomley, Jul 20 2001

Representation of n in base 1 is defined to be a concatenation of n 1's.
It is difficult to write twenty-one in base 11 using decimal digits.
Representation in bases greater than 10 are written in base 10. This is really nasty! - N. J. A. Sloane, Dec 06 2002

			Rows start (1), (11, 10), (111, 11, 10), (1111, 100, 11, 10), etc.

Cf. A063431.
Rows are effectively the reverse of A001731, A001732, A001733, A001734, A001735, A001736, A008707, A008708, A008709, A008710, A008711, A008712, A008713, A008714, A008715, A008716, A008717, etc.
Columns are truncated versions of A000042, A007088, A007089, A007090, A007091, A007092, A007093, A007094, A007095, A000027 and perhaps A055649, etc.
Without the 1st column becomes A004053.

f[n_] := Flatten[ Append[ {FromDigits[ Table[1, {n}]] }, Table[ FromDigits[ IntegerDigits[n, i]], {i, 2, n}]]]; Flatten[ Table[ f[n], {n, 1, 10}]] (* Robert G. Wilson v *)

A110605 Numbers n whose base 7 representations, interpreted as base 10 integers, are semiprimes.

Original entry on oeis.org

4, 6, 7, 11, 12, 15, 16, 19, 20, 24, 25, 26, 34, 36, 40, 44, 47, 55, 57, 61, 64, 65, 66, 73, 74, 78, 79, 80, 82, 83, 89, 92, 97, 99, 100, 101, 103, 104, 108, 109, 110, 113, 118, 124, 136, 137, 142, 145, 148, 149, 150, 152, 158, 162, 164, 167, 172, 173, 176, 181, 187

Offset: 1

Jonathan Vos Post, Jul 30 2005

A108873 is the equivalent using base 3. A110602 is the equivalent using base 4. A110603 is the equivalent using base 5. A110604 is the equivalent using base 6. A110606 is the equivalent using base 8. A110607 is the equivalent using base 9.

			a(1) = 4 because 4 (base 7) = 4 (base 10) = 2 * 2, a semiprime (A001358).
a(2) = 6 because 6 (base 7) = 6 (base 10) = 2 * 3
a(3) = 7 because 7 (base 7) = 10 and 10 (base 10) = 2 * 5.
a(4) = 11 because 11 (base 7) = 14 and 14 (base 10) = 2 * 7.
a(5) = 12 because 12 (base 7) = 15 and 15 (base 10) = 3 * 5.

Cf. A001358, A007093, A108873, A110602, A110603, A110604, A110606, A110607.

Select[Range[187], Plus @@ Last /@ FactorInteger[FromDigits[IntegerDigits[ #, 7]]] == 2 &] (* Ray Chandler, Aug 05 2005 *)

Extended by Ray Chandler, Aug 05 2005

A353146 Decimal repunits written in base 7.

Original entry on oeis.org

0, 1, 14, 216, 3145, 44252, 641640, 12305251, 163304624, 2516300046, 36351200665, 542225612642, 11012302601310, 143163240420331, 2224515456064634, 32255340625240206, 453016062064453015, 6552245200234552232, 125142553603416142350

Offset: 0

Seiichi Manyama, Apr 26 2022

Cf. A002275, A291962, A353142, A353143, A353144, A353145, A353147, A353148.

Cf. A007093, A023000.

a(n) = fromdigits(digits((10^n-1)/9, 7));

a(n) = A007093(A002275(n)).

A382413 Numbers with at least one zero in their base-7 representation.

Original entry on oeis.org

0, 7, 14, 21, 28, 35, 42, 49, 50, 51, 52, 53, 54, 55, 56, 63, 70, 77, 84, 91, 98, 99, 100, 101, 102, 103, 104, 105, 112, 119, 126, 133, 140, 147, 148, 149, 150, 151, 152, 153, 154, 161, 168, 175, 182, 189, 196, 197, 198, 199, 200, 201, 202, 203, 210, 217, 224, 231, 238

Offset: 1

Paolo Xausa, Mar 24 2025

Paolo Xausa, Table of n, a(n) for n = 1..10000

Cf. analogous sequences in other bases: A062289 (base 2), A081605 (base 3), A196032 (base 4), A382415 (base 5), A382416 (base 6), A382417 (base 8), A382418 (base 9), A011540 (base 10).

Cf. A007093, A043393, A382412 (complement).

Select[Range[0, 250], DigitCount[#, 7, 0] > 0 &]

A043281 Maximal run length in base-7 representation of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 2, 3, 2, 2, 2, 2, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1

Offset: 1

Clark Kimberling

Winston de Greef, Table of n, a(n) for n = 1..10000

Cf. A007093 (base 7).

Cf. A043276-A043290 for base-2 to base-16 analogs.

Max[Length/@Split[IntegerDigits[#,7]]]&/@Range[100] (* Harvey P. Dale, Mar 30 2016 *)

A043281(n, b=7)={my(m,c=1); while(n>0, n%b==(n\=b)%b&&c++&&next; m=max(m, c); c=1); m} \\ M. F. Hasler, Jul 23 2013

from itertools import groupby
from sympy.ntheory.factor_ import digits
def A043281(n): return max(len(list(g)) for k, g in groupby(digits(n,7)[1:])) # Chai Wah Wu, Mar 09 2023

A004681 Primes written in base 7.

Original entry on oeis.org

2, 3, 5, 10, 14, 16, 23, 25, 32, 41, 43, 52, 56, 61, 65, 104, 113, 115, 124, 131, 133, 142, 146, 155, 166, 203, 205, 212, 214, 221, 241, 245, 254, 256, 302, 304, 313, 322, 326, 335, 344, 346, 362, 364, 401, 403, 421, 436, 443, 445, 452, 461, 463, 506, 515, 524

Offset: 1

N. J. A. Sloane

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

Cf. similar sequences listed in A004680.

Cf. A007093.

[Seqint(Intseq(NthPrime(n),7)): n in [1..60]]; // G. C. Greubel, Oct 10 2018

FromDigits/@IntegerDigits[Prime[Range[1000]], 7] (* Vincenzo Librandi, Sep 03 2016 *)

a(n)=subst(Pol(digits(prime(n),7)),'x,10) \\ Charles R Greathouse IV, Nov 06 2013

vector(60, n, fromdigits(digits(prime(n), 7))) \\ G. C. Greubel, Oct 10 2018

a(n) = A007093(prime(n)). - Michel Marcus, Sep 03 2016

A033033 Numbers all of whose base 7 digits are odd.

Original entry on oeis.org

1, 3, 5, 8, 10, 12, 22, 24, 26, 36, 38, 40, 57, 59, 61, 71, 73, 75, 85, 87, 89, 155, 157, 159, 169, 171, 173, 183, 185, 187, 253, 255, 257, 267, 269, 271, 281, 283, 285, 400, 402, 404, 414, 416, 418, 428, 430, 432, 498, 500, 502, 512

Offset: 1

Clark Kimberling

			38 in base 7 is 53_7. All the digits of 38 in base 7; 5 and 3; are odd. So 38 is in the sequence. - _David A. Corneth_, Aug 24 2019

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A007093, A014261.

[m:m in [1..600]| Intseq(m,7) subset {1,3,5}]; // Marius A. Burtea, Aug 24 2019

Select[Range[600],AllTrue[IntegerDigits[#,7],OddQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Sep 28 2014 *)

is(n) = {my(d = Set(digits(n, 7))); for(i = 1, #d, if(d[i]%2 == 0, return(0))); 1} \\ David A. Corneth, Aug 24 2019

A037398 Numbers k such that every base-6 digit of k is a base-7 digit of k.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 14, 21, 28, 35, 43, 68, 79, 86, 122, 123, 129, 165, 172, 208, 215, 246, 252, 260, 361, 373, 397, 425, 427, 431, 444, 445, 446, 469, 475, 476, 479, 481, 482, 504, 513, 520, 527, 556, 562, 583, 625, 696, 738, 756

Offset: 1

Clark Kimberling

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

Cf. A007092, A007093.

Cf. A037399, A037400, A037401.

import Data.List ((\\), nub)
a037398 n = a037398_list !! (n-1)
a037398_list = filter f [1..] where
   f x = null $ nub (ds 6 x) \\ nub (ds 7 x)
   ds b x = if x > 0 then d : ds b x' else []  where (x', d) = divMod x b
-- Reinhard Zumkeller, May 30 2013

A037402 Numbers k such that every base-7 digit of k is a base-8 digit of k.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 16, 24, 32, 40, 48, 57, 85, 94, 106, 114, 133, 142, 163, 171, 196, 204, 212, 220, 224, 225, 226, 227, 228, 229, 230, 236, 244, 277, 285, 305, 334, 342, 385, 392, 401, 540, 546, 547, 550, 597, 620, 629, 646, 688

Offset: 1

Clark Kimberling

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

Cf. A007093, A007094.

Cf. A037403, A037404.

import Data.List ((\\), nub)
a037402 n = a037402_list !! (n-1)
a037402_list = filter f [1..] where
   f x = null $ nub (ds 7 x) \\ nub (ds 8 x)
   ds b x = if x > 0 then d : ds b x' else []  where (x', d) = divMod x b
-- Reinhard Zumkeller, May 30 2013

b7b8Q[n_]:=Module[{idn7=Union[IntegerDigits[n,7]]},Intersection[ idn7, Union[ IntegerDigits[n,8]]]==idn7]; Select[Range[700],b7b8Q] (* Harvey P. Dale, Dec 15 2013 *)

cp's OEIS Frontend

A037394 Numbers k such that every base-5 digit of k is a base-7 digit of k.

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Mathematica

A063432 Triangle read by rows in which k-th entry in row n is representation of n in base k, for 1 <= k <= n.

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Mathematica

A110605 Numbers n whose base 7 representations, interpreted as base 10 integers, are semiprimes.

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Mathematica

Extensions

A353146 Decimal repunits written in base 7.

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PARI

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A382413 Numbers with at least one zero in their base-7 representation.

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Mathematica

A043281 Maximal run length in base-7 representation of n.

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Mathematica

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Python

A004681 Primes written in base 7.

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Magma

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A033033 Numbers all of whose base 7 digits are odd.

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Magma

Mathematica

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A037398 Numbers k such that every base-6 digit of k is a base-7 digit of k.

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