A102489 -id:A102489 - cp's OEIS frontend

A032563 Numbers k such that A102489(k) is divisible by k.

1, 2, 3, 4, 5, 6, 7, 8, 9, 1038, 1040, 2078, 2080, 2118, 2120, 3158, 3160, 3200, 4198, 4238, 4240, 5278, 5280, 5318, 5320, 6358, 6360, 6400, 7398, 7438, 7440, 8478, 8480, 8518, 8520, 9558, 9560, 9600, 12480, 25440, 38400, 112308, 449440

Offset: 1

Patrick De Geest, Apr 15 1998

			9_10 / 9_16 = 9/9 = 1;
4152_10 / 1038_16 = 4152/1038 = 4;
4160_10 / 1040_16 = 4160/1040 = 4;
8312_10 / 2078_16 = 8312/2078 = 4.

Robert Israel, Table of n, a(n) for n = 1..71

Cf. A032564, A102489.

d:= Vector(10^7,1):
for i from 1 to 7 do
   inds:= 10^i*[$1..10^(7-i)];
   d[inds]:= (2*16^i+3)/5;
od:
b:= Vector(10^7):
b[1]:= 1:
for i from 2 to 10^7 do
  b[i]:= b[i-1]+d[i]
od:
select(t-> (b[t]/ t)::integer, [$1..10^7]); # Robert Israel, Aug 30 2015

Select[Range[16^5], IntegerQ[FromDigits[IntegerDigits[#], 16]/#] &] (* Michael De Vlieger, Aug 29 2015 *)

A237438 Double Hex-primes: let f(n) = A102489(n); then sequence lists primes p such that f(p) and f(f(p)) are also primes.

Original entry on oeis.org

2, 3, 5, 7, 11, 43, 47, 53, 59, 61, 89, 97, 101, 139, 151, 167, 199, 241, 251, 257, 269, 281, 337, 373, 443, 557, 599, 607, 647, 653, 829, 971, 1051, 1093, 1163, 1223, 1279, 1327, 1433, 1459, 1499, 1549, 1583, 1597, 1607

Offset: 1

Andreas Boe, Feb 07 2014

This sequence is a subset of A103144.

			Dec61=prime -> Hex61=Dec97=prime -> Hex97=Dec151=prime.

Andreas Boe, Table of n, a(n) for n = 1..6084

Cf. A102489.

Cf. A103144 (Hex-primes), A237439 (Triple Hex-primes), A237440 (Quadruple Hex-primes), A237441 (Quintuple Hex-primes).

A237439 Triple Hex-primes: let f(n) = A102489(n); then sequence lists primes p such that f(p), f(f(p)) and f(f(f(p))) are also primes.

Original entry on oeis.org

2, 3, 5, 7, 59, 61, 97, 101, 151, 257, 599, 647, 829, 1163, 1499, 1999, 2351, 2467, 2531, 2897, 2903, 3001, 3373, 4783, 4813, 5683, 6317, 6857, 6997, 7759, 8563, 8837, 8963, 9203, 9463, 9497, 9521, 10903, 10957

Offset: 1

Andreas Boe, Feb 07 2014

The sequence is a subset of OEIS sequences A103144 and A237438

			Dec61=prime -> Hex61=Dec97=prime -> Hex97=Dec 151=prime -> Hex151=Dec337=prime

Original research by OEIS contributor Andreas Boe, Feb 2014

Andreas Boe, Table of n, a(n) for n = 1..188

Cf. A102489

Cf. A103144(Hex-primes), A237438 (Double Hex-primes), A237440 (Quadruple Hex-primes), A237441 (Quintuple Hex-primes)

A237440 Quadruple Hex-primes: let f(n) = A102489(n); then sequence lists primes p such that f(p), f(f(p)). f(f(f(p))) and f(f(f(f(p)))) are also primes.

Original entry on oeis.org

2, 3, 5, 7, 61, 97, 101, 257, 2531, 4783, 5683, 6317, 8963, 9463, 9497, 11593, 15683, 18757, 23687, 26251, 29611, 31271, 36011, 45497, 45979, 46853, 54869, 73379, 92557, 93761, 104173, 107857, 107981, 121607, 134047, 192091, 196853, 236729, 285599, 310081

Offset: 1

Andreas Boe, Feb 07 2014

The sequence is a subset of sequences A103144, A237438, and A237439.

			Dec61=prime -> Hex61=Dec97=prime -> Hex97=Dec151=prime -> Hex151=Dec337=prime -> Hex337=Dec823=prime.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A102489.

Cf. A103144 (Hex-primes), A237438 (Double Hex-primes), A237439 (Triple Hex-primes), A237441 (Quintuple Hex-primes).

qhpQ[n_]:=AllTrue[Rest[NestList[FromDigits[IntegerDigits[#],16]&,n,4]], PrimeQ]; Select[Prime[Range[27000]],qhpQ] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Mar 13 2016 *)

isok(p)= isprime(p) && isprime(p=hd(p)) && isprime(p=hd(p)) && isprime(p=hd(p)) && isprime(p=hd(p)); \\ Michel Marcus, Feb 09 2014

More terms from Michel Marcus, Feb 09 2014

A237441 Quintuple Hex-primes: let f(n) = A102489(n); then sequence lists primes p such that f(p), f(f(p)), f(f(f(p))), f(f(f(f(p)))) and f(f(f(f(f(p))))) are also primes.

Original entry on oeis.org

2, 3, 5, 7, 61, 101, 196853, 516151, 548239, 568627, 595039, 603833, 648887, 1996223, 2086907, 2487227, 3322757, 3711343, 4385137, 5226049, 5288929, 5853241, 8792039, 8796187, 8982191, 10203203, 12640297, 12664129, 12845561, 13156267, 13437481, 14342431

Offset: 1

Andreas Boe, Feb 07 2014

The sequence is a subset of A103144, A237438, A237439 and A237440

			Dec61=prime -> Hex61=Dec97=prime -> Hex97=Dec151=prime -> Hex151=Dec337=prime -> Hex337=Dec823=prime -> Hex823=Dec2083=prime.

Cf. A102489

Cf. A103144(Hex-primes), A237438(Double Hex-primes), A237439(Triple Hex-primes), A237440(Quadruple Hex-primes).

hd(n) = my(d = digits(n)); sum(i=1, #d, 16^(i-1)*d[#d-i+1]);
isok(p) = isprime(p) && isprime(p=hd(p)) && isprime(p=hd(p)) && isprime(p=hd(p)) && isprime(p=hd(p)) && isprime(p=hd(p)); \\ Michel Marcus, Feb 08 2014

More terms from Michel Marcus, Feb 08 2014

A102487 Numbers in base-12 representation that can be written with decimal digits.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 84, 85, 86

Offset: 1

Reinhard Zumkeller, Jan 12 2005

Numbers that are only in this sequence or only in A039274 but not in both are n= 131, 142, 275, 286, 419, 430 etc: see A039558. [From R. J. Mathar, Aug 30 2008]

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

Complement of A102488; A102489, A102491, A102493.

Cf. A033048 (subsequence).

import Data.List (unfoldr)
a102487 n = a102487_list !! (n-1)
a102487_list = filter (all (< 10) . unfoldr (\x ->
   if x == 0 then Nothing else Just $ swap $ divMod x 12)) [0..]
-- Reinhard Zumkeller, Apr 18 2011

fQ[n_] := Last@ Union@ IntegerDigits[n, 12] < 10; Select[ Range[0, 86], fQ] (* Robert G. Wilson v, Apr 17 2012 *)

{for(testn=0,87,
lgt=1;
for(i=1,1000,if(12^i > testn,lgt=i;break()));
atst=testn;pasr=1;
for(j=1,lgt,lasd=atst%12;
if(lasd<10,atst=(atst-lasd)/12,pasr=0;break()));
if(pasr==1,print1(testn,", ")))}
\\ Douglas Latimer, Apr 17 2012

A102487_list = [int(str(x), 12) for x in range(10**6)] # Chai Wah Wu, Apr 09 2016

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

A227549 Numbers that contain their base-16 representation in their decimal representation.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 357440, 357441, 357442, 357443, 357444, 357445, 357446, 357447, 357448, 357449, 1079653, 1081713, 1122966, 1123079, 1123080, 2246166, 3369253, 3371313, 3412566, 4494393, 4494400, 4535653, 5658739, 5658740, 5660793, 5660800

Offset: 1

Roland Kneer, Aug 05 2013

			357440 = (57440)_16
1079653 = (107965)_16
23132273099720801084801040 = (1322730997208010848010)_16

Roland Kneer and Giovanni Resta, Table of n, a(n) for n = 1..139 (terms < 16^22, first 55 terms from Roland Kneer)

Subsequence of A102489.

Select[Range[0,5661000],SequenceCount[IntegerDigits[#],IntegerDigits[#,16]]>0&] (* Harvey P. Dale, Apr 21 2023 *)

A102490 Numbers in base-16 representation that cannot be written with decimal digits.

Original entry on oeis.org

10, 11, 12, 13, 14, 15, 26, 27, 28, 29, 30, 31, 42, 43, 44, 45, 46, 47, 58, 59, 60, 61, 62, 63, 74, 75, 76, 77, 78, 79, 90, 91, 92, 93, 94, 95, 106, 107, 108, 109, 110, 111, 122, 123, 124, 125, 126, 127, 138, 139, 140, 141, 142, 143, 154, 155, 156, 157, 158, 159, 160

Offset: 1

Reinhard Zumkeller, Jan 12 2005

			42 = 2*16^1 + 10*16^0 = '2A', therefore 42 is a term.

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

Complement of A102489; A102488, A102492, A102494.

import Data.List (unfoldr)
a102490 n = a102490_list !! (n-1)
a102490_list = filter (any (> 9) . unfoldr
   (\x -> if x == 0 then Nothing else Just $ swap $ divMod x 16)) [0..]
-- Reinhard Zumkeller, Jun 27 2013

Select[Range[200],IntegerLength[Max[IntegerDigits[#,16]]]>1&] (* Harvey P. Dale, Jun 12 2020 *)

def ok(n): return any(hd > '9' for hd in hex(n)[2:])
print(list(filter(ok, range(161)))) # Michael S. Branicky, Oct 11 2021

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 *)

cp's OEIS Frontend

A032563 Numbers k such that A102489(k) is divisible by k.

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A237438 Double Hex-primes: let f(n) = A102489(n); then sequence lists primes p such that f(p) and f(f(p)) are also primes.

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A237439 Triple Hex-primes: let f(n) = A102489(n); then sequence lists primes p such that f(p), f(f(p)) and f(f(f(p))) are also primes.

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A237440 Quadruple Hex-primes: let f(n) = A102489(n); then sequence lists primes p such that f(p), f(f(p)). f(f(f(p))) and f(f(f(f(p)))) are also primes.

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A237441 Quintuple Hex-primes: let f(n) = A102489(n); then sequence lists primes p such that f(p), f(f(p)), f(f(f(p))), f(f(f(f(p)))) and f(f(f(f(f(p))))) are also primes.

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A102487 Numbers in base-12 representation that can be written with decimal digits.

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

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A227549 Numbers that contain their base-16 representation in their decimal representation.

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