A007094 -id:A007094 - cp's OEIS frontend

A029805 Numbers k such that k^2 is palindromic in base 8.

0, 1, 2, 3, 6, 9, 11, 27, 65, 73, 79, 81, 83, 195, 219, 237, 366, 513, 543, 585, 697, 1094, 1539, 1755, 1875, 2910, 4097, 4161, 4225, 4477, 4617, 4681, 4727, 4891, 5267, 8698, 8730, 11841, 12291, 12483, 12675, 13065, 13851, 14673, 15021

Offset: 1

Patrick De Geest

The only powers of 2 in this sequence are 1 and 2. - Alonso del Arte, Feb 25 2017

			3 is in the sequence because 3^2 = 9 = 11 in base 8, which is a palindrome.
4 is not in the sequence because 4^2 = 16 = 20 in base 8, which is not a palindrome.

Seiichi Manyama, Table of n, a(n) for n = 1..100
Patrick De Geest, Palindromic Squares in bases 2 to 17

Cf. A000290, A002441, A007094.

Numbers k such that k^2 is palindromic in base b: A003166 (b=2), A029984 (b=3), A029986 (b=4), A029988 (b=5), A029990 (b=6), A029992 (b=7), this sequence (b=8), A029994 (b=9), A002778 (b=10), A029996 (b=11), A029737 (b=12), A029998 (b=13), A030072 (b=14), A030073 (b=15), A029733 (b=16), A118651 (b=17).

palQ[n_, b_:10] := Module[{idn = IntegerDigits[n, b]}, idn == Reverse[idn]]; Select[Range[0, 16000], palQ[#^2, 8] &] (* Harvey P. Dale, May 19 2012 *)

from itertools import count, islice
def A029805_gen(): # generator of terms
    return filter(lambda k: (s:=oct(k**2)[2:])[:(t:=(len(s)+1)//2)]==s[:-t-1:-1],count(0))
A029805_list = list(islice(A029805_gen(),20)) # Chai Wah Wu, Jun 23 2022

A073789 Numbers in base -8.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 170, 171, 172, 173, 174, 175, 176, 177, 160, 161, 162, 163, 164, 165, 166, 167, 150, 151, 152, 153, 154, 155, 156, 157, 140, 141, 142, 143, 144, 145, 146, 147, 130, 131, 132, 133, 134, 135, 136, 137, 120, 121, 122, 123, 124, 125, 126

Offset: 0

Robert G. Wilson v, Aug 11 2002

D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, 1969, Vol. 2, p. 189.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Prepared and presented by Matthew Szudzik of Wolfram Research, A Mathematica programming contest
Eric Weisstein's World of Mathematics, Negabinary
Wikipedia, Negative base

Cf. A007094, A039724, A073785, A007608, A073786, A073787, A073788, A073790 & A039723.

a073789 0 = 0
a073789 n = a073789 n' * 10 + m where
   (n', m) = if r < 0 then (q + 1, r + 8) else (q, r)
             where (q, r) = quotRem n (negate 8)
-- Reinhard Zumkeller, Jul 07 2012

ToNegaBases[i_Integer, b_Integer] := FromDigits[ Rest[ Reverse[ Mod[ NestWhileList[(#1 - Mod[ #1, b])/-b &, i, #1 != 0 &], b]]]]; Table[ ToNegaBases[n, 8], {n, 0, 60}]

def A073789(n):
    s, q = '', n
    while q >= 8 or q < 0:
        q, r = divmod(q, -8)
        if r < 0:
            q += 1
            r += 8
        s += str(r)
    return int(str(q)+s[::-1]) # Chai Wah Wu, Apr 09 2016

A255805 Numbers with no zeros in base-8 representation.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54, 55, 57, 58, 59, 60, 61, 62, 63, 73, 74, 75, 76, 77, 78, 79, 81, 82, 83, 84

Offset: 1

Reinhard Zumkeller, Mar 08 2015

Different from A047592, A207481.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Wikipedia, Octal
Index entries for 2-automatic sequences.

Cf. A007094, A100970 (subsequence).

Zeroless numbers in some other bases <= 10: A000042 (base 2), A032924 (base 3), A023705 (base 4), A248910 (base 6), A255808 (base 9), A052382 (base 10).

a255805 n = a255805_list !! (n-1)
a255805_list = iterate f 1 where
   f x = 1 + if r < 7 then x else 8 * f x'  where (x', r) = divMod x 8

Select[Range[100],DigitCount[#,8,0]==0&] (* Harvey P. Dale, Jun 08 2015 *)

isok(m) = vecmin(digits(m,8)) > 0; \\ Michel Marcus, Jan 23 2022

def ok(n): return '0' not in oct(n)[2:]
print([k for k in range(85) if ok(k)]) # Michael S. Branicky, Jan 23 2022

from sympy import integer_log
def A255805(n):
    m = integer_log(k:=6*n+1,7)[0]
    return sum(1+(k-7**m)//(6*7**j)%7<<3*j for j in range(m)) # Chai Wah Wu, Jun 28 2025

A097583 Octal representation of the concatenation of the first n decimal numbers with the most significant digits first.

Original entry on oeis.org

1, 14, 173, 2322, 30071, 361100, 4553207, 57060516, 726746425, 133767016076, 21756176604103, 3404420603635070, 536705213574536755, 104420417226264430242, 15305164771273206577527

Offset: 1

Cino Hilliard, Aug 29 2004

			1234 decimal is 2322 octal the 4th entry in the table.

Seiichi Manyama, Table of n, a(n) for n = 1..337

Cf. A007908, A050926, A097580, A097582.

Cf. A007094.

Table[FromDigits[IntegerDigits[FromDigits[Flatten[IntegerDigits/@ Range[ n]]],8]],{n,20}] (* Harvey P. Dale, Aug 11 2021 *)

a(n) = A007094(A007908(n)). - Seiichi Manyama, Apr 23 2022

A073795 Replace 8^k with (-8)^k in base 8 expansion of n.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, -8, -7, -6, -5, -4, -3, -2, -1, -16, -15, -14, -13, -12, -11, -10, -9, -24, -23, -22, -21, -20, -19, -18, -17, -32, -31, -30, -29, -28, -27, -26, -25, -40, -39, -38, -37, -36, -35, -34, -33, -48, -47, -46, -45, -44, -43, -42, -41, -56, -55, -54, -53, -52, -51, -50, -49

Offset: 0

Robert G. Wilson v, Aug 12 2002

Base 8 representation for n (in lexicographic order) converted from base -8 to base 10.

Dana G. Korssjoen, Biyao Li, Stefan Steinerberger, Raghavendra Tripathi, and Ruimin Zhang, Finding structure in sequences of real numbers via graph theory: a problem list, arXiv:2012.04625, Dec 08, 2020

Cf. A007094, A073789, A053985, A065369, A073791, A073792, A073793, A073794, A073796 & A073835.

f[n_Integer, b_Integer] := Block[{l = IntegerDigits[n]}, Sum[l[[ -i]]*(-b)^(i - 1), {i, 1, Length[l]}]]; a = Table[ FromDigits[ IntegerDigits[n, 8]], {n, 0, 80}]; b = {}; Do[b = Append[b, f[a[[n]], 8]], {n, 1, 80}]; b

a(8*k+m) = -8*a(k)+m for 0 <= m < 8. - Chai Wah Wu, Jan 16 2020

A088152 Value of n-th digit in octal representation of n^n.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 1, 2, 1, 1, 1, 6, 6, 5, 0, 0, 4, 4, 6, 1, 3, 3, 1, 4, 5, 4, 0, 5, 0, 3, 0, 3, 4, 1, 3, 5, 6, 2, 1, 6, 6, 5, 5, 0, 1, 0, 0, 5, 6, 3, 7, 6, 4, 1, 1, 3, 3, 6, 4, 3, 1, 0, 0, 0, 4, 4, 0, 3, 6, 1, 1, 2, 5, 0, 0, 5, 2, 6, 0, 2, 4, 7, 5, 6, 4, 2, 1, 6, 4, 3, 6, 7, 4, 6, 0, 5, 7, 5, 3, 6

Offset: 0

Reinhard Zumkeller, Sep 20 2003

a(n)=d(n) with n^n = Sum(d(k)*8^k: 0<=d(k)<8, k>=0).

			n=9, 9^9=387420489 -> '2705710511', '2---------': a(9)=2;
a(0)=1, a(k)=0 for 0<k<8 and a(8)=1.

Robert Israel, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Octal

Cf. A007094, A000312, A088150, A088151, A088153, A088154, A088155, A088156, A088157.

[Floor(n^n/8^n) mod 8:n in [0..101]]; // Marius A. Burtea, Sep 20 2019

f:= proc(n) local x,L;
   x:= n &^ n mod 8^(n+1);
   floor(x/8^n)
end proc:
f(0):= 1:
map(f, [$0..101]); # Robert Israel, Sep 19 2019

a(n) = floor(n^n / 8^n) mod 8.

A037384 Numbers k such that every base-3 digit of k is a base-8 digit of k.

Original entry on oeis.org

1, 2, 13, 17, 26, 66, 80, 112, 120, 121, 122, 129, 136, 161, 168, 202, 242, 328, 394, 401, 458, 514, 522, 528, 529, 530, 531, 532, 533, 534, 535, 538, 546, 554, 562, 570, 578, 592, 610, 634, 640, 641, 642, 643, 644, 645, 646, 647

Offset: 1

Clark Kimberling

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

Cf. A007089, A007094.

Cf. A037380, A037381, A037382, A037383, A037385, A037386.

import Data.List ((\\), nub)
a037384 n = a037384_list !! (n-1)
a037384_list = filter f [1..] where
   f x = null $ nub (ds 3 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

b3b8Q[n_]:=Module[{b3=Union[IntegerDigits[n,3]],b8=Union[ IntegerDigits[n,8]]}, And@@Table[ MemberQ[b8,b3[[i]]],{i,Length[b3]}]]; Select[Range[700],b3b8Q] (* Harvey P. Dale, Apr 17 2013 *)

is(n)=#setminus(Set(digits(n,3)), Set(digits(n,8)))==0 \\ Charles R Greathouse IV, Feb 11 2017

A190130 Numbers 1 through 10000 sorted lexicographically in octal representation (base 8).

Original entry on oeis.org

1, 8, 64, 512, 4096, 4097, 4098, 4099, 4100, 4101, 4102, 4103, 513, 4104, 4105, 4106, 4107, 4108, 4109, 4110, 4111, 514, 4112, 4113, 4114, 4115, 4116, 4117, 4118, 4119, 515, 4120, 4121, 4122, 4123, 4124, 4125, 4126, 4127, 516, 4128, 4129, 4130, 4131, 4132

Offset: 1

Reinhard Zumkeller, May 06 2011

A190131 = inverse permutation: a(A190131(n)) = A190131(a(n)) = n;

a(n) <> n for n > 1.

			a(10) = 4101 -> 10005 [oct];
a(11) = 4102 -> 10006 [oct];
a(12) = 4103 -> 10007 [oct];
a(13) =  513 -> 1001  [oct];
a(14) = 4104 -> 10010 [oct];
a(15) = 4105 -> 10011 [oct];
a(16) = 4106 -> 10012 [oct];
largest term a(67151) = 10000 -> 23420 [oct];
last term a(10000) = 4095 -> 7777 [oct], largest term lexicographically.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000 (full sequence)
Eric Weisstein's World of Mathematics, Lexicographic Order
Eric Weisstein's World of Mathematics, Octal
Wikipedia, Lexicographical order
Wikipedia, Octal

A007094; A190126 (base 2), A190128 (base 3), A190016 (base 10), A190132 (base 12), A190134 (base 16).

import Data.Ord (comparing)
import Data.List (sortBy)
a190130 n = a190130_list !! (n-1)
a190130_list = sortBy (comparing (show . a007094)) [1..10000]

A262105 Pseudoprimes to base 8, written in base 8.

Original entry on oeis.org

11, 25, 55, 77, 101, 151, 165, 205, 231, 347, 421, 525, 741, 777, 1061, 1111, 1205, 1213, 1535, 1665, 1751, 2121, 2401, 2525, 2553, 2611, 3005, 3161, 3175, 3301, 3371, 3561, 3777, 4171, 4641, 4705, 5215, 5405, 6111, 6143

Offset: 1

Abdul Gaffar Khan, Sep 11 2015

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A007094 (Numbers in base 8), A020137 (Pseudoprimes to base 8).

Cf. A258189, A262101, A262102, A262103, A262104, A262154.

base = 8; t = {}; n = 1;
  While[Length[t] < 40, n++;
   If[! PrimeQ[n] && PowerMod[base, n - 1, n] == 1, AppendTo[t, FromDigits@ IntegerDigits[n, 8]]]];  t

a(n) = A007094(A020137(n)).

A293661 Base-8 circular primes that are not base-8 repunits.

Original entry on oeis.org

13, 29, 31, 41, 43, 47, 59, 61, 607, 719, 751, 761, 971, 1021, 1657, 1759, 1787, 1913, 1993, 2011, 2687, 3019, 3659, 3673, 3677, 3803, 3919, 4073, 49103, 56299, 62207, 105341, 130681, 177007, 188249, 195277, 235513, 237151, 251501, 259019, 4127707, 6807419

Offset: 1

Felix Fröhlich, Dec 30 2017

Conjecture: The sequence is finite.
From Michael De Vlieger, Dec 30 2017: (Start)
Primes in this sequence must only have odd digits.
There are 8 terms with 2 octal digits, 20 terms with 4 octal digits, 12 terms with 6 octal digits, and 8 terms with 8 octal digits.
a(49), if it exists, must be larger than 8^12 = 68719476736. (End)

			607 written in base 8 is 1137. The base-8 numbers 1137, 1371, 3711, 7113 written in base 10 are 607, 761, 1993, 3659, respectively, and all those numbers are prime, so 607, 761, 1993 and 3659 are terms of the sequence.

Cf. A007094, A293142.

Cf. base-b nonrepunit circular primes: A293657 (b=4), A293658 (b=5), A293659 (b=6), A293660 (b=7), A293662 (b=9), A293663 (b=10).

With[{b = 8}, Select[Prime@ Range[PrimePi@ b + 1, 10^6], Function[w, And[AllTrue[Array[FromDigits[RotateRight[w, #], b] &, Length@ w - 1], PrimeQ], Union@ w != {1} ]]@ IntegerDigits[#, b] &]] (* or *)
With[{b = 8}, Select[Flatten@ Array[FromDigits[#, b] & /@ Most@ Rest@ Tuples[Range[1, 7, 2], #] &, 6, 2], Function[w, And[ AllTrue[ Array[ FromDigits[ RotateRight[w, #], b] &, Length@ w], PrimeQ], Union@ w != {1} ]]@ IntegerDigits[#, b] &]] (* Michael De Vlieger, Dec 30 2017 *)

rot(n) = if(#Str(n)==1, v=vector(1), v=vector(#n-1)); for(i=2, #n, v[i-1]=n[i]); u=vector(#n); for(i=1, #n, u[i]=n[i]); v=concat(v, u[1]); v
decimal(v, base) = my(w=[]); for(k=0, #v-1, w=concat(w, v[#v-k]*base^k)); sum(i=1, #w, w[i])
is_circularprime(p, base) = my(db=digits(p, base), r=rot(db), i=0); if(vecmin(db)==0, return(0), while(1, dec=decimal(r, base); if(!ispseudoprime(dec), return(0)); r=rot(r); if(r==db, return(1))))
forprime(p=1, , if(vecmin(digits(p, 8))!=vecmax(digits(p, 8)), if(is_circularprime(p, 8), print1(p, ", "))))

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A029805 Numbers k such that k^2 is palindromic in base 8.

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A073789 Numbers in base -8.

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A255805 Numbers with no zeros in base-8 representation.

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A097583 Octal representation of the concatenation of the first n decimal numbers with the most significant digits first.

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A073795 Replace 8^k with (-8)^k in base 8 expansion of n.

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A088152 Value of n-th digit in octal representation of n^n.

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A037384 Numbers k such that every base-3 digit of k is a base-8 digit of k.

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A190130 Numbers 1 through 10000 sorted lexicographically in octal representation (base 8).

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