A071590 -id:A071590 - cp's OEIS frontend

A103168 a(n) is the remainder when (n written backwards) is divided by n.

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 9, 5, 13, 6, 13, 3, 9, 15, 2, 12, 0, 9, 18, 2, 10, 18, 26, 5, 3, 13, 23, 0, 9, 18, 27, 36, 7, 15, 4, 14, 24, 34, 0, 9, 18, 27, 36, 45, 5, 15, 25, 35, 45, 0, 9, 18, 27, 36, 6, 16, 26, 36, 46, 56, 0, 9, 18, 27, 7, 17, 27, 37, 47, 57, 67, 0, 9, 18, 8, 18, 28, 38

Offset: 1

Labos Elemer, Jan 28 2005

			a(n) = 0 for palindromic numbers.

Rémy Sigrist, Table of n, a(n) for n = 1..10000

Cf. A071955, A103166-A103169, A002113, A071590.

rd[x_]:=FromDigits[Reverse[IntegerDigits[x]]] Table[Mod[rd[n], n], {n, 1, 256}]
Table[Mod[IntegerReverse[n],n],{n,90}] (* Harvey P. Dale, Jun 20 2025 *)

a(n, base=10) = my (r=fromdigits(Vecrev(digits(n, base)), base)); r%n \\ Rémy Sigrist, Apr 05 2020

def a(n): return int(str(n)[::-1])%n
print([a(n) for n in range(1, 84)]) # Michael S. Branicky, Dec 12 2021

Definition corrected by N. J. A. Sloane, Jul 14 2007

A161602 Positive integers k that are greater than the value of the reversal of k's binary representation.

Original entry on oeis.org

2, 4, 6, 8, 10, 12, 13, 14, 16, 18, 20, 22, 24, 25, 26, 28, 29, 30, 32, 34, 36, 38, 40, 41, 42, 44, 46, 48, 49, 50, 52, 53, 54, 56, 57, 58, 59, 60, 61, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 81, 82, 84, 86, 88, 89, 90, 92, 94, 96, 97, 98, 100, 101, 102, 104, 105, 106, 108, 109

Offset: 1

Leroy Quet, Jun 14 2009

By "reversal" of k's binary representation, it is meant: write k in binary, reverse the order of its digits, and read the result as a binary value.

This sequence contains all the positive even integers.

			29 in binary is 11101. Its digital reversal is 10111, which is 23 in decimal. Since 29 > 23, 29 is in this sequence.

Jonathan Frech, Table of n, a(n) for n = 1..10000

Cf. A030101, A006995, A161601, A161603 (odd terms).

Cf. A071590 (using decimal reversal).

Select[Range[109], # > IntegerReverse[#, 2] &] (* Michael De Vlieger, Apr 07 2021 *)

isok(k) = k > fromdigits(Vecrev(binary(k)), 2); \\ Michel Marcus, Apr 06 2021

from itertools import count, islice
def A161602_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n:n>int(bin(n)[-1:1:-1],2),count(max(startvalue,1)))
A161602_list = list(islice(A161602_gen(),20)) # Chai Wah Wu, Jan 19 2023

More terms from Max Alekseyev, Sep 11 2009

A071589 Numbers n such that Reversal(n) > n.

Original entry on oeis.org

12, 13, 14, 15, 16, 17, 18, 19, 23, 24, 25, 26, 27, 28, 29, 34, 35, 36, 37, 38, 39, 45, 46, 47, 48, 49, 56, 57, 58, 59, 67, 68, 69, 78, 79, 89, 102, 103, 104, 105, 106, 107, 108, 109, 112, 113, 114, 115, 116, 117, 118, 119, 122, 123, 124, 125, 126, 127, 128, 129

Offset: 1

Benoit Cloitre, Jun 01 2002

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A004086 (digit reversal), A071590 (reversal < n), A002113 (reversal = n).

Cf. A161601 (binary reversal > n).

Select[Range[200], # < FromDigits[Reverse[IntegerDigits[#]]] &] (* T. D. Noe, Mar 14 2012 *)

for(i=2,300,n=(i); s=ceil(log(n)/log(10)); if((sum(i=0,s,10^(s-i-1)*(floor(n/10^i*1.)-10*floor(n/10^(i+1)*1.))))>i,print1((i),",")))

A297270 Numbers whose base-10 digits have greater down-variation than up-variation; see Comments.

Original entry on oeis.org

10, 20, 21, 30, 31, 32, 40, 41, 42, 43, 50, 51, 52, 53, 54, 60, 61, 62, 63, 64, 65, 70, 71, 72, 73, 74, 75, 76, 80, 81, 82, 83, 84, 85, 86, 87, 90, 91, 92, 93, 94, 95, 96, 97, 98, 100, 110, 120, 130, 140, 150, 160, 170, 180, 190, 200, 201, 210, 211, 220, 221

Offset: 1

Clark Kimberling, Jan 16 2018

Suppose that n has base-b digits b(m), b(m-1), ..., b(0). The base-b down-variation of n is the sum DV(n,b) of all d(i)-d(i-1) for which d(i) > d(i-1); the base-b up-variation of n is the sum UV(n,b) of all d(k-1)-d(k) for which d(k) < d(k-1). The total base-b variation of n is the sum TV(n,b) = DV(n,b) + UV(n,b). See the guide at A297330.

Differs first from A071590 at 1101, which is in A071590, but not in here because UV(1101) = DV(1101). - R. J. Mathar, Jan 23 2018

			6151413121 in base-10:  6,1,5,1,4,1,3,1,2,1, having DV = 15, UV = 10, so that 6151413121 is in the sequence.

Clark Kimberling, Table of n, a(n) for n = 1..10000

Cf. A297330, A297271, A297272.

g[n_, b_] := Map[Total, GatherBy[Differences[IntegerDigits[n, b]], Sign]];
x[n_, b_] := Select[g[n, b], # < 0 &]; y[n_, b_] := Select[g[n, b], # > 0 &];
b = 10; z = 2000; p = Table[x[n, b], {n, 1, z}]; q = Table[y[n, b], {n, 1, z}];
w = Sign[Flatten[p /. {} -> {0}] + Flatten[q /. {} -> {0}]];
Take[Flatten[Position[w, -1]], 120]   (* A297270 *)
Take[Flatten[Position[w, 0]], 120]    (* A297271 *)
Take[Flatten[Position[w, 1]], 120]    (* A297272 *)

A071955 a(n) = remainder when n is reduced mod reverse(n).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 12, 13, 14, 15, 16, 17, 18, 19, 0, 9, 0, 23, 24, 25, 26, 27, 28, 29, 0, 5, 9, 0, 34, 35, 36, 37, 38, 39, 0, 13, 18, 9, 0, 45, 46, 47, 48, 49, 0, 6, 2, 18, 9, 0, 56, 57, 58, 59, 0, 13, 10, 27, 18, 9, 0, 67, 68, 69, 0, 3, 18, 36, 27, 18, 9, 0, 78, 79, 0, 9

Offset: 1

Joseph L. Pe, Jun 16 2002

a(n)=0 if n is palindromic - Labos Elemer, Jan 28 2005

			a(85) = 85 mod 58 = 27.

Rémy Sigrist, Table of n, a(n) for n = 1..10000

Cf. A002113, A071590, A103164, A103168.

Table[Mod[n, FromDigits[Reverse[IntegerDigits[n]]]], {n, 1, 256}] (* Labos Elemer, Jan 28 2005 *)
Table[Mod[n, FromDigits[Reverse[IntegerDigits[n]]]], {n, 1, 100}]

a(n, base=10) = my (r=fromdigits(Vecrev(digits(n, base)), base)); n%r \\ Rémy Sigrist, Apr 05 2020

Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, Jul 14 2007

A103167 a(n) = 2^n mod reverse(2^n).

Original entry on oeis.org

0, 0, 0, 16, 9, 18, 128, 256, 82, 1024, 2048, 4096, 2356, 16384, 32768, 1980, 131072, 262144, 524288, 1048576, 2097152, 159390, 319770, 16777216, 10108899, 20228688, 134217728, 268435456, 98713642, 1073741824, 2147483648, 4294967296, 2681134876, 17179869184

Offset: 1

Labos Elemer, Jan 28 2005

Remainder if 2^n is divided by the reverse of 2^n.

			a(5) = 2^5 mod reverse(2^5) = 32 mod reverse(32) = 32 mod 23 = 9.

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

Cf. A002113, A071590, A103164-A103167.

Table[Mod[FromDigits[Reverse[IntegerDigits[2^n]]], 2^n], {n, 1, 256}]
Table[PowerMod[2,n,IntegerReverse[2^n]],{n,40}] (* Harvey P. Dale, Jan 30 2022 *)

def a(n): t = 2**n; return t%int(str(t)[::-1])
print([a(n) for n in range(1, 35)]) # Michael S. Branicky, Dec 12 2021

A103166 a(n) = reverse(2^n) mod 2^n.

Original entry on oeis.org

0, 0, 0, 13, 23, 46, 53, 140, 215, 105, 210, 2808, 2918, 15593, 21187, 63556, 7987, 179118, 358137, 466945, 420750, 4034914, 8068838, 10946113, 23445533, 46880176, 22406063, 117663950, 219078635, 1060248229, 2021396468, 2632727628, 2954399858, 13837158803

Offset: 1

Labos Elemer, Jan 28 2005

Remainder if (2^n written backwards) is divided by 2^n.

			a(4) = reverse(2^4) mod 2^4 = reverse(16) mod 16 = 61 mod 16 = 13.

Cf. A002113, A071590, A103164-A103167.

Table[Mod[FromDigits[Reverse[IntegerDigits[2^n]]], 2^n], {n, 1, 256}]

def a(n): t = 2**n; return int(str(t)[::-1])%t
print([a(n) for n in range(1, 35)]) # Michael S. Branicky, Dec 12 2021

A210589 Numbers which, when divided by their first digit, have their last digit as remainder.

Original entry on oeis.org

10, 20, 21, 30, 31, 32, 40, 41, 42, 43, 50, 51, 52, 53, 54, 60, 61, 62, 63, 64, 65, 70, 71, 72, 73, 74, 75, 76, 80, 81, 82, 83, 84, 85, 86, 87, 90, 91, 92, 93, 94, 95, 96, 97, 98, 100, 110, 120, 130, 140, 150, 160, 170, 180, 190, 200, 201, 210, 211, 220, 221, 230, 231, 240, 241

Offset: 1

Eric Angelini (idea) and M. F. Hasler, Mar 23 2012

Coincides with A071590 up to the 79th term, A071590(79)=310 is not in this sequence.
Charles R Greathouse IV observes that this is an automatic sequence in the terminology of Allouche & Shallit.
See A210582 for the obvious "symmetric" counterpart: first digit = x mod last digit. - M. F. Hasler, Jan 14 2014

Vincenzo Librandi, Table of n, a(n) for n = 1..7000
Charles R Greathouse IV, in reply to E. Angelini, Re: Divided by first digit, have last digit as remainder, SeqFan list, Mar 21 2012
Index entries for 10-automatic sequences.

[ n: n in [1..249] | n mod d[#d] eq d[1] where d is Intseq(n) ]; // Bruno Berselli, Mar 23 2012

ldrQ[n_]:=Module[{idn=IntegerDigits[n],f,l},f=First[idn];l=Last[idn];Mod[n,f]==l]; Select[Range[10000],ldrQ]  (* Harvey P. Dale, Mar 21 2012 *)

is_A210589(x)=x%(x\10^(#Str(x)-1))==x%10

def ok(n): s = str(n); return n > 0 and n%int(s[0]) == int(s[-1])
print([k for k in range(242) if ok(k)]) # Michael S. Branicky, Oct 20 2021

cp's OEIS Frontend

A103168 a(n) is the remainder when (n written backwards) is divided by n.

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A161602 Positive integers k that are greater than the value of the reversal of k's binary representation.

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A071589 Numbers n such that Reversal(n) > n.

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A297270 Numbers whose base-10 digits have greater down-variation than up-variation; see Comments.

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A071955 a(n) = remainder when n is reduced mod reverse(n).

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A103167 a(n) = 2^n mod reverse(2^n).

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A103166 a(n) = reverse(2^n) mod 2^n.

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A210589 Numbers which, when divided by their first digit, have their last digit as remainder.

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