A030105 -id:A030105 - cp's OEIS frontend

A004086 Read n backwards (referred to as R(n) in many sequences).

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 11, 21, 31, 41, 51, 61, 71, 81, 91, 2, 12, 22, 32, 42, 52, 62, 72, 82, 92, 3, 13, 23, 33, 43, 53, 63, 73, 83, 93, 4, 14, 24, 34, 44, 54, 64, 74, 84, 94, 5, 15, 25, 35, 45, 55, 65, 75, 85, 95, 6, 16, 26, 36, 46, 56, 66, 76, 86, 96, 7, 17, 27, 37, 47

Offset: 0

N. J. A. Sloane

Also called digit reversal of n.

Leading zeros (after the reversal has taken place) are omitted. - N. J. A. Sloane, Jan 23 2017

Indranil Ghosh, Table of n, a(n) for n = 0..50000 (first 1001 terms from Franklin T. Adams-Watters)
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 [math.CO], Dec 08 2020.
Michael Penn, A digit moving number puzzle., YouTube video, 2022.

Cf. A004185, A004186, A009996, A073138, A188649, A221714.

Cf. A030101, A030102, A030103, A030104, A030105, A030107, A030108.

a004086 = read . reverse . show  -- Reinhard Zumkeller, Apr 11 2011

|.&.": i.@- 1e5 NB. Stephen Makdisi, May 14 2018

read transforms; A004086 := digrev; #cf "Transforms" link at bottom of page
A004086:=proc(n) local s,t; if n<10 then n else s:=irem(n,10,'t'); while t>9 do s:=s*10+irem(t,10,'t') od: s*10+t fi end; # M. F. Hasler, Jan 29 2012

Table[FromDigits[Reverse[IntegerDigits[n]]], {n, 0, 75}]
IntegerReverse[Range[0,80]](* Requires Mathematica version 10 or later *) (* Harvey P. Dale, May 13 2018 *)

dig(n) = {local(m=n,r=[]); while(m>0,r=concat(m%10,r);m=floor(m/10));r}
A004086(n) = {local(b,m,r);r=0;b=1;m=dig(n);for(i=1,matsize(m)[2],r=r+b*m[i];b=b*10);r} \\ Michael B. Porter, Oct 16 2009

A004086(n)=fromdigits(Vecrev(digits(n))) \\ M. F. Hasler, Nov 11 2010, updated May 11 2015, Sep 13 2019

def A004086(n):
    return int(str(n)[::-1]) # Chai Wah Wu, Aug 30 2014

For n > 0, a(a(n)) = n iff n mod 10 != 0. - Reinhard Zumkeller, Mar 10 2002
a(n) = d(n,0) with d(n,r) = r if n=0, otherwise d(floor(n/10), r*10+(n mod 10)). - Reinhard Zumkeller, Mar 04 2010
a(10*n+x) = x*10^m + a(n) if 10^(m-1) <= n < 10^m and 0 <= x <= 9. - Robert Israel, Jun 11 2015

Extended by Ray Chandler, Dec 30 2004

A255691 Convert n to base 6, move the most significant digit to the least significant one and convert back to base 10.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 1, 7, 13, 19, 25, 31, 2, 8, 14, 20, 26, 32, 3, 9, 15, 21, 27, 33, 4, 10, 16, 22, 28, 34, 5, 11, 17, 23, 29, 35, 1, 7, 13, 19, 25, 31, 37, 43, 49, 55, 61, 67, 73, 79, 85, 91, 97, 103, 109, 115, 121, 127, 133, 139, 145, 151, 157, 163, 169, 175

Offset: 0

Paolo P. Lava, Mar 02 2015

a(6*n) = 1.

Fixed points of the transform are listed in A048331.

			16 in base 6 is 24: moving the most significant digit as the least significant one we have 42 that is 26 in base 10.

Paolo P. Lava, Table of n, a(n) for n = 0..1000

Cf. A006257, A030105, A255588-A255594, A255689-A255690, A255692-A255693.

with(numtheory): P:=proc(q,h) local a,b,k,n; print(0);
for n from 1 to q do
a:=convert(n,base,h); b:=[]; for k from 1 to nops(a)-1 do b:=[op(b),a[k]]; od; a:=[a[nops(a)],op(b)];
a:=convert(a,base,h,10); b:=0; for k from nops(a) by -1 to 1 do b:=10*b+a[k]; od;
print(b); od; end: P(10^4,6);

roll[n_, b_] := Block[{w = IntegerDigits[n, b]}, Append[Rest@ w, First@ w]]; b = 6; FromDigits[#, b] & /@ (roll[#, b] & /@ Range[0, 65]) (* Michael De Vlieger, Mar 04 2015 *)
Table[FromDigits[RotateLeft[IntegerDigits[n,6]],6],{n,0,70}] (* Harvey P. Dale, Mar 02 2023 *)

A055952 n + reversal of base 6 digits of n (written in base 10).

Original entry on oeis.org

0, 2, 4, 6, 8, 10, 7, 14, 21, 28, 35, 42, 14, 21, 28, 35, 42, 49, 21, 28, 35, 42, 49, 56, 28, 35, 42, 49, 56, 63, 35, 42, 49, 56, 63, 70, 37, 74, 111, 148, 185, 222, 49, 86, 123, 160, 197, 234, 61, 98, 135, 172, 209, 246, 73, 110, 147, 184, 221, 258, 85, 122, 159, 196

Offset: 0

Henry Bottomley, Jul 18 2000

If n has an even number of digits in base 6 then a(n) is a multiple of 7.

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A056964.

Table[n + IntegerReverse[n, 6], {n, 0, 100}] (* Paolo Xausa, Aug 08 2024 *)

a(n) = n + A030105(n).

A255591 Convert n to base 6, move least significant digit to most significant digit and convert back to base 10.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 1, 7, 13, 19, 25, 31, 2, 8, 14, 20, 26, 32, 3, 9, 15, 21, 27, 33, 4, 10, 16, 22, 28, 34, 5, 11, 17, 23, 29, 35, 6, 42, 78, 114, 150, 186, 7, 43, 79, 115, 151, 187, 8, 44, 80, 116, 152, 188, 9, 45, 81, 117, 153, 189, 10, 46, 82, 118, 154, 190, 11

Offset: 0

Paolo P. Lava, Mar 02 2015

a(6*n) = n.

Fixed points of the transform are listed in A048331.

			16 in base 6 is 24: moving the least significant digit to the most significant one we have 42 that is 26 in base 10.

Indranil Ghosh, Table of n, a(n) for n = 0..46656 (terms 0..1000 from Paolo P. Lava)

Cf. A030105, A038572, A048331, A255588-A255590, A255592-A255594, A048787, A255689-A255693.

with(numtheory): P:=proc(q,h) local a,b,k,n; print(0);
for n from 1 to q do
a:=convert(n,base,h); b:=[]; for k from 2 to nops(a) do b:=[op(b),a[k]]; od; a:=[op(b),a[1]];
a:=convert(a,base,h,10); b:=0; for k from nops(a) by -1 to 1 do b:=10*b+a[k]; od;
print(b); od; end: P(10^4,6);

roll[n_, b_] := Block[{w = IntegerDigits[n, b]}, Prepend[Most@ w, Last@ w]]; b = 6; FromDigits[#, b] & /@ (roll[#, b] & /@ Range[0, 66]) (* Michael De Vlieger, Mar 04 2015 *)

def A255591(n):
    x=A007092(n)
    return int(x[-1]+x[:-1],6) # Indranil Ghosh, Feb 03 2017

A346113 Base-10 numbers k whose number of divisors equals the number of divisors in R(k), where k is written in all bases from base-2 to base-10 and R(k), the digit reversal of k, is read as a number in the same base.

Original entry on oeis.org

1, 9077, 10523, 10838, 30182, 58529, 73273, 77879, 83893, 244022, 303253, 303449, 304853, 329893, 332249, 334001, 334417, 335939, 336083, 346741, 374617, 391187, 504199, 512695, 516982, 595274, 680354, 687142, 758077, 780391, 792214, 854669, 946217, 948539, 995761, 1008487, 1377067, 1389341

Offset: 1

Scott R. Shannon, Jul 05 2021

There are 633 terms below 50 million and 1253 terms below 100 million. All of those have tau(k), the number of divisors of k, equal to 1, 2, 4, 8 or 16. The first term where tau(k) = 2 is n = 93836531, a prime, which is also the first term of A136634. All terms in A136634 will appear in this sequence, as will all terms in A228768(n) for n>=10. The first term with tau(k) = 4 is 9077, the first with tau(k) = 8 is 595274, and the first with tau(k) = 16 is 5170182. It is possible tau(k) must equal 2^i, with i>=0, although this is unknown.

All known terms are squarefree. - Michel Marcus, Jul 07 2021

			9077 is a term as the number of divisors of 9077 = tau(9077) = 4, and this equals the number of divisors of R(9077) when written and then read as a base-j number, with 2 <= j <= 10. See the table below for k = 9077.
.
  base | k_base         | R(k_base)      | R(k_base)_10  | tau(R(k_base)_10)
----------------------------------------------------------------------------------
   2   | 10001101110101 | 10101110110001 | 11185         | 4
   3   | 110110012      | 210011011      | 15421         | 4
   4   | 2031311        | 1131302        | 6002          | 4
   5   | 242302         | 203242         | 6697          | 4
   6   | 110005         | 500011         | 38887         | 4
   7   | 35315          | 51353          | 12533         | 4
   8   | 21565          | 56512          | 23882         | 4
   9   | 13405          | 50431          | 33157         | 4
  10   | 9077           | 7709           | 7709          | 4

Scott R. Shannon, Table of n, a(n) for n = 1..1253

Cf. A000005, A004086.
Cf. A030102, A030103, A030104, A030105, A030106, A030107, A030108.
Cf. A136634 (prime terms), A228768.
Subsequence of A062895.

Select[Range@100000,Length@Union@DivisorSigma[0,Join[{s=#},FromDigits[Reverse@IntegerDigits[s,#],#]&/@Range[2,10]]]==1&] (* Giorgos Kalogeropoulos, Jul 06 2021 *)

isok(k) = {my(t= numdiv(k)); for (b=2, 10, my(d=digits(k, b)); if (numdiv(fromdigits(Vecrev(d), b)) != t, return (0));); return(1);} \\ Michel Marcus, Jul 06 2021

A055953 n - reversal of base 6 digits of n (written in base 10).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 5, 0, -5, -10, -15, -20, 10, 5, 0, -5, -10, -15, 15, 10, 5, 0, -5, -10, 20, 15, 10, 5, 0, -5, 25, 20, 15, 10, 5, 0, 35, 0, -35, -70, -105, -140, 35, 0, -35, -70, -105, -140, 35, 0, -35, -70, -105, -140, 35, 0, -35, -70, -105, -140, 35, 0, -35, -70, -105, -140, 35, 0, -35, -70, -105, -140, 70, 35, 0, -35, -70

Offset: 0

Henry Bottomley, Jul 18 2000

a(n) is a multiple of 5.

Paolo Xausa, Table of n, a(n) for n = 0..10000

Cf. A030105, A056965.

Table[n - IntegerReverse[n, 6], {n, 0, 100}] (* Paolo Xausa, Aug 08 2024 *)

a(n) = n - A030105(n).

cp's OEIS Frontend

A004086 Read n backwards (referred to as R(n) in many sequences).

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A255691 Convert n to base 6, move the most significant digit to the least significant one and convert back to base 10.

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A055952 n + reversal of base 6 digits of n (written in base 10).

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A255591 Convert n to base 6, move least significant digit to most significant digit and convert back to base 10.

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Maple

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A346113 Base-10 numbers k whose number of divisors equals the number of divisors in R(k), where k is written in all bases from base-2 to base-10 and R(k), the digit reversal of k, is read as a number in the same base.

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A055953 n - reversal of base 6 digits of n (written in base 10).

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