A138793 -id:A138793 - cp's OEIS frontend

A281254 Concatenate the decimal numbers n n-2 n-4 ...5 3 1 2 4 ... n-5 n-3 n-1 if n is odd, or n n-2 n-4 ... 6 4 2 1 3 5 ... n-5 n-3 n-1 if n is even.

1, 21, 312, 4213, 53124, 642135, 7531246, 86421357, 975312468, 10864213579, 1197531246810, 121086421357911, 13119753124681012, 1412108642135791113, 151311975312468101214, 16141210864213579111315, 1715131197531246810121416

Offset: 1

Robert G. Wilson v, Jan 18 2017

The old name was "Alternately concatenate the decimal digits from back to front 1...n such that n is always to the left.".
a(40714), with 192464 decimal digits, is the sequence's first (probable) prime. - Hans Havermann, Dec 17 2019
The terms at even indices are the same as the even-indexed terms of A053063. - Hans Havermann, Jan 16 2020

			a(13) = 13119753124681012.

Indranil Ghosh, Table of n, a(n) for n = 1..340

See A053063 for another version.

Cf. A000422, A007908, A138793, A138957, A281253.

b:= proc(n) b(n):= `if`(n=1, 1, parse(cat(a(n-1), n))) end:
a:= proc(n) a(n):= `if`(n=1, 1, parse(cat(n, b(n-1)))) end:
seq(a(n), n=1..20);  # Alois P. Heinz, Jan 18 2017

f[n_] := Fold[ If[ Mod[n +#2, 2] == 0, #2, #1]*10^IntegerLength@ If[ Mod[n +#2, 2] == 0, #1, #2] +If[ Mod[n +#2, 2] == 1, #2, #1] &, 0, Range@ n]; Array[f, 17]

def a(n):
    if n==1:
        return ["1"]
    return [str(n)]+a(n-1)[::-1]
def A281254(n):
    return "".join(a(n)) # Indranil Ghosh, Jan 23 2017

Edited by N. J. A. Sloane, Dec 07 2019

A061963 Numbers n such that n divides the (left) concatenation of all numbers <= n written in base 10 (most significant digit on right).

Original entry on oeis.org

1, 3, 9, 189, 753, 987, 6739, 10953, 51963, 171897, 224081, 635031, 1135001, 4437459

Offset: 1

Larry Reeves (larryr(AT)acm.org), May 24 2001

This sequence differs from A029527 in that all least significant zeros are kept during concatenation.
Left concatenation, reverse order (i.e., digit-wise reversal of the concatenation 123...n), as in A138793.
No more terms < 10^7.
All terms must be odd.

			n = 13 is not a term since 31211101987654321 is not divisible by 13. (Note that the order of the digits of 13, 12 and 10 is reversed.)
See A061955 for further examples.

Index entries for related sequences

Cf. A029447-A029470, A029471-A029494, A029495-A029518, A029519-A029542, A061931-A061954, A061955-A061978.

Cf. A138793.

k = 2; lst = {}; rid = 1; While[k < 1001, exp = Floor[ Log10[rid]] + 1 + If[Mod[k, 10] == 1, IntegerExponent[k - 1, 10], 0]; rid = rid + FromDigits@ Reverse@ IntegerDigits@ k*(10^exp); If[ Mod[rid, k] == 0, Print@ k; AppendTo[lst,k]]; k++]; lst (* and to test any single value n *) fQ[n_] := Mod[ FromDigits@ Reverse@ Flatten@ IntegerDigits@ Range@ n, n] == 0 (* Robert G. Wilson v, Sep 12 2011 *)
Select[Range[5*10^6],Divisible[FromDigits[Reverse[Flatten[ IntegerDigits/@ Range[ #]]]], #]&] (* Harvey P. Dale, Apr 10 2017 *)

isok(n) = my(s = ""); forstep (k=n, 1, -1, sk = digits(k); forstep (j=#sk, 1, -1, s = concat(s, sk[j]))); (eval(s) % n) == 0; \\ Michel Marcus, Jan 28 2017

Edited and updated by Larry Reeves (larryr(AT)acm.org), Apr 12 2002

a(12)-a(14) from Lars Blomberg, Aug 19 2011

A281066 Concatenation R(n)R(n-1)R(n-2)...R(2)R(1) read mod n, where R(x) is the digit-reversal of x (with leading zeros not omitted).

Original entry on oeis.org

0, 1, 0, 1, 1, 3, 3, 1, 0, 1, 4, 9, 5, 7, 6, 1, 6, 9, 17, 1, 15, 15, 19, 9, 21, 1, 18, 13, 28, 21, 26, 17, 15, 3, 16, 9, 30, 3, 15, 1, 1, 33, 10, 37, 36, 43, 22, 33, 19, 21, 48, 45, 2, 45, 26, 49, 27, 33, 33, 21, 48, 25, 36, 49, 36, 15, 22, 5, 27, 11, 42, 9, 2, 73, 21, 17, 59, 57, 5, 1

Offset: 1

Robert G. Wilson v, Jan 14 2017

			a(13) = 31211101987654321 (mod 13) = 5.

Indranil Ghosh, Table of n, a(n) for n = 1..20008

Cf. A004086, A061963, A095221, A114795, A138793.

f[n_] := Mod[ FromDigits@ Fold[ Join[ Reverse@ IntegerDigits@#2, #1] &, {}, Range@ n], n]; Array[f, 80]

a(n) = my(s = ""); forstep (k=n,1,-1, sk = digits(k); forstep (j=#sk, 1, -1, s = concat(s, sk[j]))); eval(s) % n; \\ Michel Marcus, Jan 28 2017

def A281066(n):
    s=""
    for i in range(n, 0, -1):
        s+=str(i)[::-1]
    return int(s)%n # Indranil Ghosh, Jan 28 2017

a(n) = A138793(n) (mod n).

A281253 Alternately concatenate the decimal digits from front to back 1...n such that n is always to the right.

Original entry on oeis.org

1, 12, 213, 3124, 42135, 531246, 6421357, 75312468, 864213579, 97531246810, 1086421357911, 119753124681012, 12108642135791113, 1311975312468101214, 141210864213579111315, 15131197531246810121416, 1614121086421357911131517

Offset: 1

Robert G. Wilson v, Jan 18 2017

a(n) is prime for n: 121, 1399 and no others < 3000.

			a(13) = 12108642135791113.

Indranil Ghosh, Table of n, a(n) for n = 1..340

Cf. A007908, A138957, A000422, A138793, A281254.

b:= proc(n) b(n):= `if`(n=1, 1, parse(cat(n, a(n-1)))) end:
a:= proc(n) a(n):= `if`(n=1, 1, parse(cat(b(n-1), n))) end:
seq(a(n), n=1..20);  # Alois P. Heinz, Jan 18 2017

f[n_] := Fold[ If[ Mod[n +#2, 2] == 0, #1, #2]*10^IntegerLength@ If[ Mod[n +#2, 2] == 0, #2, #1] +If[ Mod[n +#2, 2] == 0, #2, #1] &, 0, Range@ n]; Array[f, 17]

def a(n):
    if n==1:
        return ["1"]
    return a(n-1)[::-1]+[str(n)]
def A281253(n):
    return "".join(a(n)) # Indranil Ghosh, Jan 27 2017

A261138 The concatenation of 123456...n and the reverse of this number.

Original entry on oeis.org

11, 1221, 123321, 12344321, 1234554321, 123456654321, 12345677654321, 1234567887654321, 123456789987654321, 1234567891001987654321, 12345678910111101987654321, 123456789101112211101987654321, 1234567891011121331211101987654321, 12345678910111213144131211101987654321

Offset: 1

Umut Uludag, Aug 10 2015

Let R(n) denote the number obtained by formally reversing the digits of n, including any leading zeros that may appear; a(n) is the decimal concatenation of 1,2,...,n,R(n),R(n-1),...,R(3),R(2),R(1). - N. J. A. Sloane, Dec 01 2021
A palindromic version of A173426.
Has same start as A259937, but A259937 generates non-palindromic terms for n>9.
All terms are multiples of 11 (cf. A349805).

			For n=10 we concatenate 1,2,3,...,10,01,9,8,...3,2,1 getting 1234567891001987654321.

Cf. A002477, A173426, A259937, A261570, A349805.

with(StringTools);
myReverse := n -> Reverse(convert(n,string));
A349804:=proc(n) local i,L,R;
L:=""; R:="";
for i from n to 1 by -1 do
L:=Join( [convert(i,string), L],"");
R:=Join( [R, myReverse(convert(i,string))],"");
od:
parse(Join([L,R],""));
end proc; # N. J. A. Sloane, Dec 01 2021
# second Maple program:
a:= n-> (s-> parse(cat(s, seq(s[-i], i=1..length(s)))))(cat("", $1..n)):
seq(a(n), n=1..14); # Alois P. Heinz, Dec 01 2021

Table[d = Flatten[IntegerDigits /@ Range@ n]; FromDigits@ Flatten[{d, Reverse@ d}], {n, 13}] (* Michael De Vlieger, Aug 20 2015 *)

def A349804(n): return int((lambda x: x+x[::-1])(''.join(str(d) for d in range(1,n+1)))) # Chai Wah Wu, Dec 01 2021

a(n) = concatenate( A007908(n), A138793(n) retaining leading zeros).

More than the usual number of terms are shown in order to distinguish this from several similar sequences.

Edited by N. J. A. Sloane, Dec 11 2021

A078998 Choose a(n) so that a(1)+a(2)+...+a(n) = concatenation of n first natural numbers.

Original entry on oeis.org

1, 11, 111, 1111, 11111, 111111, 1111111, 11111111, 111111111, 12222222121, 1222222212101, 122222221210101, 12222222121010101, 1222222212101010101, 122222221210101010101, 12222222121010101010101

Offset: 1

Benoit Cloitre, Jan 12 2003

Cf. A000422, A116504, A007908, A116505, A104759, A138789, A138790, A138793.

b = {}; a = {}; Do[w = RealDigits[n]; w = First[w]; Do[AppendTo[a, w[[k]]], {k, 1, Length[w]}]; p = FromDigits[a]; AppendTo[b, p], {n, 1, 30}]; c = {}; Do[AppendTo[c, b[[n + 1]] - b[[n]]], {n, 1, Length[b] - 1}]; c (* Artur Jasinski, Mar 30 2008 *)

a(1)=1; for n>1, a(n) = A007908(n)-A007908(n-1)

cp's OEIS Frontend

A281254 Concatenate the decimal numbers n n-2 n-4 ...5 3 1 2 4 ... n-5 n-3 n-1 if n is odd, or n n-2 n-4 ... 6 4 2 1 3 5 ... n-5 n-3 n-1 if n is even.

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A061963 Numbers n such that n divides the (left) concatenation of all numbers <= n written in base 10 (most significant digit on right).

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A281066 Concatenation R(n)R(n-1)R(n-2)...R(2)R(1) read mod n, where R(x) is the digit-reversal of x (with leading zeros not omitted).

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A281253 Alternately concatenate the decimal digits from front to back 1...n such that n is always to the right.

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A261138 The concatenation of 123456...n and the reverse of this number.

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A078998 Choose a(n) so that a(1)+a(2)+...+a(n) = concatenation of n first natural numbers.

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