A102487 -id:A102487 - cp's OEIS frontend

A102489 Take the decimal representation of n and read it as if it were written in hexadecimal.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 112

Offset: 0

Reinhard Zumkeller, Jan 12 2005

List of numbers in base-16 representation that can be written with decimal digits.
Early in the sequence there are blocks recurring as a(n) = a(n-10)+16, but this pattern starts to fail when we reach 160, 161, ... with hex-representations A0, A1, ... which cannot be written with decimal digits. - Rick L. Shepherd, Jun 08 2012
Binary Coded Decimal (BCD) codes, common in electronics, when interpreted as plain binary-coded integers. For example, number 39 is BCD coded in two nibbles as 0011 1001 which is the binary expansion of 57; hence, taking into account the offset, a(1+39) = 57. - Stanislav Sykora, Jun 09 2012
Integers that avoid letters in their hexadecimal expansion. - Eliora Ben-Gurion, Aug 28 2019

			10 in decimal is 16 in base 16, so a(10)=16.

Reinhard Zumkeller, Table of n, a(n) for n = 0..9999
Eric Weisstein's World of Mathematics, Hexadecimal

Complement of A102490; A102487, A102491, A102493, A122840.

Cf. A090725 (the subsequence of primes).

import Data.Maybe (fromJust, mapMaybe)
a102489 n = a102489_list !! (n-1)
a102489_list = mapMaybe dhex [0..] where
   dhex 0                         = Just 0
   dhex x | d > 9 || y == Nothing = Nothing
          | otherwise             = Just $ 16 * fromJust y + d
          where (x', d) = divMod x 16; y = dhex x'
-- Reinhard Zumkeller, Jul 06 2012

o10:= n -> min(padic:-ordp(n,2),padic:-ordp(n,5)):
d:= [0,seq((2*16^o10(n)+3)/5, n=1..1000)]:
ListTools:-PartialSums(d); # Robert Israel, Aug 30 2015

Table[FromDigits[IntegerDigits[n], 16], {n, 0, 70}] (* Ivan Neretin, Aug 12 2015 *)

a(n) - a(n-1) = (2*16^A122840(n) + 3)/5. - Robert Israel, Aug 30 2015

Edited by N. J. A. Sloane, Feb 08 2014 (changed definition, moved old definition to comment, changed offset and b-file).

A102491 Numbers whose base-20 representation can be written with decimal digits.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 120, 121, 122, 123, 124, 125, 126

Offset: 1

Reinhard Zumkeller, Jan 12 2005

a(n) = A118761(n) for n<=50. - Reinhard Zumkeller, May 01 2006

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Vigesimal
Wikipedia, Vigesimal

Complement of A102492; Cf. A102487, A102489, A102493. Cf. A037454, A037462, A007091.

import Data.List (unfoldr)
a102491 n = a102491_list !! (n-1)
a102491_list = filter (all (<= 9) . unfoldr
   (\x -> if x == 0 then Nothing else Just $ swap $ divMod x 20)) [0..]
-- Reinhard Zumkeller, Jun 27 2013

seq(n + (1/2)*add(20^k*floor(n/10^k), k = 1..floor(ln(n)/ln(10))), n = 1..100); # Peter Bala, Dec 01 2016

Select[Range@ 126, Total@ Take[Most@ DigitCount[#, 20], -10] == 0 &] (* Michael De Vlieger, Apr 09 2016 *)

isok(n) = (n==0) || ((d=digits(n, 20)) && (vecmax(d) < 10)); \\ Michel Marcus, Apr 09 2016

a(n) = fromdigits(digits(n-1),20) \\ Ruud H.G. van Tol, Dec 08 2022

A102491_list = [int(str(x), 20) for x in range(10**6)] # Chai Wah Wu, Apr 09 2016

From Peter Bala, Dec 01 2016: (Start)
If n = Sum_{i = 0..m} d(i)*10^i is the decimal expansion of n then a(n+1) = Sum_{i = 0..m} d(i)*20^i.
a(n+1) = n + 1/2*Sum_{k >= 1} 20^k*floor(n/10^k). Cf. A037454, A037462 and A007091.
a(1) = 0; a(n+1) = 20*a(n/10+1) if n == 0 (mod 10) else a(n+1) = a(n) + 1. (End)
G.f. g(x) satisfies g(x) = 20*Sum_{1<=k<=9} x^k*g(x^10)/x^9 + Sum_{1<=k<=9} k*x^(k+1)/(1-x^10). - Robert Israel, Dec 01 2016

A033048 Sums of distinct powers of 12.

Original entry on oeis.org

0, 1, 12, 13, 144, 145, 156, 157, 1728, 1729, 1740, 1741, 1872, 1873, 1884, 1885, 20736, 20737, 20748, 20749, 20880, 20881, 20892, 20893, 22464, 22465, 22476, 22477, 22608, 22609, 22620, 22621, 248832, 248833, 248844, 248845, 248976

Offset: 0

Clark Kimberling

Numbers without any base-12 digits greater than 1.

T. D. Noe, Table of n, a(n) for n = 0..1023
Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, Identities and periodic oscillations of divide-and-conquer recurrences splitting at half, arXiv:2210.10968 [cs.DS], 2022, p. 45.
Eric Weisstein's World of Mathematics, Duodecimal
Wikipedia, Duodecimal

Subsequence of A102487.
Cf. A000695, A005836, A033042-A033052.
Row 11 of array A104257.

import Data.List (unfoldr)
a033048 n = a033048_list !! (n-1)
a033048_list = filter (all (< 2) . unfoldr (\x ->
   if x == 0 then Nothing else Just $ swap $ divMod x 12)) [1..]
-- Reinhard Zumkeller, Apr 17 2011

With[{k = 12}, Map[FromDigits[#, k] &, Tuples[{0, 1}, 6]]] (* Michael De Vlieger, Oct 28 2022 *)

{maxn=37;
for(vv=0,maxn,
bvv=binary(vv);
ll=length(bvv);texp=0;btod=0;
forstep(i=ll,1,-1,btod=btod+bvv[i]*12^texp;texp++);
print1(btod,", "))}
\\ Douglas Latimer, Apr 16 2012

a(n)=fromdigits(binary(n),12) \\ Charles R Greathouse IV, Jan 11 2017

a(n) = Sum_{i=0..m} d(i)*12^i, where Sum_{i=0..m} d(i)*2^i is the base-2 representation of n.
a(n) = A097258(n)/11.
a(2n) = 12*a(n), a(2n+1) = a(2n)+1.
a(n) = Sum_{k>=0} A030308(n,k)*b(k) with b(k) = 12^k = A001021(k). - Philippe Deléham, Oct 19 2011
G.f.: (1/(1 - x))*Sum_{k>=0} 12^k*x^(2^k)/(1 + x^(2^k)). - Ilya Gutkovskiy, Jun 04 2017

Extended by Ray Chandler, Aug 03 2004

A102488 Numbers in base-12 representation that cannot be written with decimal digits.

Original entry on oeis.org

10, 11, 22, 23, 34, 35, 46, 47, 58, 59, 70, 71, 82, 83, 94, 95, 106, 107, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 154, 155, 166, 167, 178, 179, 190, 191, 202, 203, 214

Offset: 1

Reinhard Zumkeller, Jan 12 2005

			143 = 11*12^1 + 11*12^0 = 'BB', therefore 143 is a term.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Duodecimal
Wikipedia, Duodecimal

Complement of A102487; A102490, A102492, A102494.

import Data.List (unfoldr)
a102488 n = a102488_list !! (n-1)
a102488_list = filter (any (> 9) . unfoldr (\x ->
   if x == 0 then Nothing else Just $ swap $ divMod x 12)) [1..]
-- Reinhard Zumkeller, Apr 18 2011

Select[Range[250],Max[IntegerDigits[#,12]]>9&] (* Harvey P. Dale, Oct 20 2020 *)

A102493 Numbers in base-60 representation that can be written with decimal digits.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 240, 241, 242, 243, 244, 245, 246, 247, 248, 249, 300, 301, 302, 303, 304, 305, 306, 307, 308

Offset: 1

Reinhard Zumkeller, Jan 12 2005

Mohammad K. Azarian, Meftah al-hesab: A Summary, MJMS, Vol. 12, No. 2, Spring 2000, pp. 75-95. Mathematical Reviews, MR 1 764 526. Zentralblatt MATH, Zbl 1036.01002.
Mohammad K. Azarian, A Summary of Mathematical Works of Ghiyath ud-din Jamshid Kashani, Journal of Recreational Mathematics, Vol. 29(1), pp. 32-42, 1998.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Sexagesimal
Wikipedia, Sexagesimal

Complement of A102494; A102487, A102489, A102491.

import Data.List (unfoldr)
a102493 n = a102493_list !! (n-1)
a102493_list = filter (all (<= 9) . unfoldr
   (\x -> if x == 0 then Nothing else Just $ swap $ divMod x 60)) [0..]
-- Reinhard Zumkeller, Jun 27 2013

Table[Range[60n,60n+9],{n,0,6}]//Flatten (* Harvey P. Dale, Jul 06 2024 *)

A055983 a(n+1) = a(n) converted to base 10 from base 12.

Original entry on oeis.org

10, 12, 14, 16, 18, 20, 24, 28, 32, 38, 44, 52, 62, 74, 88, 104, 148, 200, 288, 392, 542, 770, 1092, 1838, 2924, 4780, 8016, 13842, 27122, 53738, 109916, 265698, 631700, 1557936, 4347258, 12785828, 43721312, 154070654, 621230752, 2655100718

Offset: 0

Henry Bottomley, Jun 01 2000

Vincenzo Librandi, Table of n, a(n) for n = 0..100

Cf. A008557, A055982, A055984.

a055983 n = a055983_list !! (n-1)
a055983_list = iterate (a102487 . (+ 1)) 10  -- Reinhard Zumkeller, Aug 29 2013

NestList[FromDigits[IntegerDigits[#],12]&,10,40] (* Vincenzo Librandi, Apr 06 2012 *)

a(n+1) = A102487(a(n)+1), a(1) = 8. - Reinhard Zumkeller, Aug 29 2013

A031344 Write primes in base 10 but interpret as if in base 12.

Original entry on oeis.org

2, 3, 5, 7, 13, 15, 19, 21, 27, 33, 37, 43, 49, 51, 55, 63, 69, 73, 79, 85, 87, 93, 99, 105, 115, 145, 147, 151, 153, 159, 175, 181, 187, 189, 201, 205, 211, 219, 223, 231, 237, 241, 253, 255, 259, 261, 301, 315, 319, 321, 327, 333, 337, 349, 355

Offset: 1

Jeff Burch

A102487 := proc(n) local dgs; dgs := convert(n,base,10) ; add(op(d,dgs)*12^(d-1), d=1..nops(dgs)) ; end proc:
A031344 := proc(n) A102487(ithprime(n)) ; end proc: # R. J. Mathar, Apr 09 2011

a(n) = A102487(A000040(n)). - R. J. Mathar, Apr 09 2011

A113016 Primes that remain primes when their decimal representation is interpreted duodecimally.

Original entry on oeis.org

2, 3, 5, 7, 11, 17, 31, 37, 61, 67, 107, 131, 157, 167, 181, 241, 251, 271, 277, 307, 347, 397, 401, 421, 431, 457, 541, 557, 577, 587, 617, 647, 661, 701, 727, 751, 797, 881, 907, 971, 1021, 1051, 1061, 1087, 1151, 1201, 1231, 1297, 1301, 1367, 1471, 1601

Offset: 1

Reinhard Zumkeller, Oct 11 2005

			A000040(19) = 67 -> 6*12^1 + 7*12^0 = 72 + 7 = 79 = A000040(22).

Robert Price, Table of n, a(n) for n = 1..3642
Eric Weisstein's World of Mathematics, Duodecimal

Cf. A113017, A102487, A000040.

Select[ Prime[ Range[256]], PrimeQ[ FromDigits[ IntegerDigits[ # ], 12]] &] (* Robert G. Wilson v, Oct 12 2005 *)

A113017 Primes that become composites when their decimal representation is interpreted duodecimally.

Original entry on oeis.org

13, 19, 23, 29, 41, 43, 47, 53, 59, 71, 73, 79, 83, 89, 97, 101, 103, 109, 113, 127, 137, 139, 149, 151, 163, 173, 179, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 257, 263, 269, 281, 283, 293, 311, 313, 317, 331, 337, 349, 353, 359, 367, 373, 379, 383

Offset: 1

Reinhard Zumkeller, Oct 11 2005

			A000040(25)=97 -> 9*12^1+7*12^0=108+7=115=5*23.

Robert Price, Table of n, a(n) for n = 1..26358
Eric Weisstein's World of Mathematics, Duodecimal

Cf. A000040, A002808, A113016, A102487.

cp's OEIS Frontend

A102489 Take the decimal representation of n and read it as if it were written in hexadecimal.

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A102491 Numbers whose base-20 representation can be written with decimal digits.

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A033048 Sums of distinct powers of 12.

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A102488 Numbers in base-12 representation that cannot be written with decimal digits.

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A102493 Numbers in base-60 representation that can be written with decimal digits.

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A055983 a(n+1) = a(n) converted to base 10 from base 12.

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A031344 Write primes in base 10 but interpret as if in base 12.

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