A000422 -id:A000422 - cp's OEIS frontend

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

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

A050678 Index of first occurrence of n in A048288.

Original entry on oeis.org

82, 2, 5, 8, 9, 15, 17, 30, 26, 18, 91, 53

Offset: 1

Patrick De Geest, Aug 15 1999

Patrick De Geest, Reversed Smarandache Concatenated Numbers, Prime factors from n down to 1
M. Fleuren, Factors and primes of Smarandache sequences.
M. Fleuren, Smarandache Factors and Reverse factors
Carlos Rivera, Puzzle 8. Primes by Listing, The Prime Puzzles and Problems Connection.

Cf. A000422, A050676, A048288.

Join[{1},Table[i=1; While[PrimeOmega[FromDigits[Flatten[IntegerDigits[Range[i,1,-1]]]]]!=n,i++]; i,{n,2,10}]] (* Jayanta Basu, May 30 2013 *)

a(11)-a(12) from Sean A. Irvine, Aug 17 2021

Edited by N. J. A. Sloane, Sep 04 2021

A052246 Concatenation of integers from n down to 0.

Original entry on oeis.org

0, 10, 210, 3210, 43210, 543210, 6543210, 76543210, 876543210, 9876543210, 109876543210, 11109876543210, 1211109876543210, 131211109876543210, 14131211109876543210, 1514131211109876543210, 161514131211109876543210, 17161514131211109876543210

Offset: 0

Henry Bottomley, Feb 01 2000

Cf. A000422.

a:= proc(n) a(n):= `if`(n=0, 0, parse(cat(n, a(n-1)))) end:
seq(a(n), n=0..22);  # Alois P. Heinz, Jan 12 2021

nn=20;With[{c=Range[nn,0,-1]},Table[FromDigits[Flatten[ IntegerDigits/@ Take[ c,-n]]],{n,nn}]] (* Harvey P. Dale, Feb 01 2013 *)

a(n)=if(n==0, 0, eval(Str(n, a(n-1)))); \\ Joerg Arndt, Sep 14 2014

a(0)=0, a(n) = n*10^len(a(n-1)) + a(n-1), where len(k) = number of digits in k and len(0)=1.

a(n) = A000422(n)*10.

A053052 Append n to the previous term, reverse alternate terms.

Original entry on oeis.org

1, 21, 123, 4321, 12345, 654321, 1234567, 87654321, 123456789, 10987654321, 1234567891011, 121110987654321, 12345678910111213, 1413121110987654321, 123456789101112131415

Offset: 1

Felice Russo, Feb 25 2000

Felice Russo, A set of new Smarandache functions, sequences and conjectures in number theory, American Research Press 2000

Cf. A000422, A007908.

A078262 Sum of the forward and reverse concatenations of 1 to n.

Original entry on oeis.org

2, 33, 444, 5555, 66666, 777777, 8888888, 99999999, 1111111110, 23333333231, 2345555545332, 244567776755433, 25466789897765534, 2647689001998775635, 274869910212099785736, 28497092031222200795837, 2950719304132232301805938, 305172940514233242402816039

Offset: 1

Amarnath Murthy, Nov 24 2002

			a(3) = 123 + 321 = 444, a(10) = 12345678910 + 10987654321 = 23333333231.

Harvey P. Dale, Table of n, a(n) for n = 1..369

Cf. A000422, A007908, A078263.

a:= n-> parse(cat(1+n-i$i=1..n))+parse(cat($1..n)):
seq(a(n), n=1..18);  # Alois P. Heinz, Jun 25 2025

Table[With[{c1=FromDigits[Flatten[IntegerDigits/@Range[n]]],c2= FromDigits[ Flatten[ IntegerDigits/@Range[n,1,-1]]]},c1+c2],{n,20}] (* Harvey P. Dale, Jul 03 2020 *)

a(n) = A000422(n) + A007908(n). - Sean A. Irvine, Jun 25 2025

More terms from Sascha Kurz, Jan 04 2003

A078569 Numbers n such that A078568(n) is prime.

Original entry on oeis.org

1, 5, 25, 53, 370

Offset: 1

Jason Earls, Nov 29 2002

Some of the larger entries may only correspond to probable primes.

Cf. A002275, A000422, A078568.

A095249 Reverse concatenation of first n positive integers modulo forward concatenation of first n positive integers.

Original entry on oeis.org

0, 9, 75, 619, 4941, 37041, 246919, 1234575, 9, 10987654321, 1110987654321, 121110987654321, 775432077543108, 178553219976533007, 27956332009875522906, 3805734210999774512805, 481583522109989673502704, 58259362312008979572492603, 6836037241301907969471482502

Offset: 1

Amarnath Murthy, Jun 17 2004

			a(4) = 619 = 4321 mod 1234.

Vincenzo Librandi, Table of n, a(n) for n = 1..375

Cf. A000422, A007908.

a:= n-> irem(parse(cat(n-i$i=0..n-1)), parse(cat(i$i=1..n))):
seq(a(n), n=1..20);  # Alois P. Heinz, Nov 19 2018

r = f = ""; Do[r = ToString[n] <> r; f = f <> ToString[n]; Print[Mod[ToExpression[r], ToExpression[f]]], {n, 1, 30}] (* Ryan Propper, Aug 26 2005 *)
nn=30;Module[{r,f},Table[f=Flatten[IntegerDigits/@Range[n]]; r=Flatten[ IntegerDigits/@Range[n,1,-1]];Mod[FromDigits[r],FromDigits[f]],{n,nn}]] (* Harvey P. Dale, Mar 04 2013 *)

a(n) = A000422(n) mod A007908(n). - Michel Marcus, Nov 19 2018

Corrected and extended by Ryan Propper, Aug 26 2005

A138794 a(n) = A138793(n+1)-A138793(n).

Original entry on oeis.org

20, 300, 4000, 50000, 600000, 7000000, 80000000, 900000000, 1000000000, 1100000000000, 210000000000000, 31000000000000000, 4100000000000000000, 510000000000000000000, 61000000000000000000000

Offset: 1

Artur Jasinski, Mar 30 2008

First differences of A138793

Cf. A000422, A116504, A007908, A116505, A104759, A138789, A138790, A138793, A075019, A075020, A075021, A075022.

b = {}; a = {}; Do[w = RealDigits[n]; w = First[w]; Do[AppendTo[a, w[[k]]], {k, 1, Length[w]}]; p = FromDigits[Reverse[a]]; AppendTo[b, p], {n, 1, 31}]; c = {}; Do[AppendTo[c, b[[n + 1]] - b[[n]]], {n, 1, Length[b] - 1}]; c (*Artur Jasinski*)

a(n) = A138793(n+1)-A138793(n)

A138963 a(1) = 1, a(n) = the largest prime divisor of A138793(n).

Original entry on oeis.org

1, 7, 107, 149, 953, 218107, 402859, 4877, 379721, 4349353, 169373, 182473, 1940144339383, 2184641, 437064932281, 5136696159619, 67580875919190833, 1156764458711, 464994193118899, 4617931439293, 1277512103328491957510030561, 8177269604099

Offset: 1

Artur Jasinski, Apr 04 2008

For the smallest prime divisors of A138793 see A138962.

Daniel Suteu and Robert Price, Table of n, a(n) for n = 1..63 (terms a(1)..a(45) from Robert Price)

Cf. A006530, A138793, A138962.

Cf. A104759, A000422, A116504, A007908, A116505, A104759, A075019, A075020, A075021, A075022, A138789, A138790, A138958, A138959, A138960.

b = {}; a = {}; Do[w = RealDigits[n]; w = First[w]; Do[AppendTo[a, w[[k]]], {k, 1, Length[w]}]; p = FromDigits[Reverse[a]]; AppendTo[b, First[Last[FactorInteger[p]]]], {n, 1, 31}]; b (* Artur Jasinski, Apr 04 2008 *)
lst = {}; Table[First[Last[FactorInteger[FromDigits[Reverse[lst = Join[lst,IntegerDigits[n]]]]]]], {n, 1, 10}] (* Robert Price, Mar 22 2015 *)

f(n) = my(D = Vec(concat(apply(s->Str(s), [1..n])))); eval(concat(vector(#D, k, D[#D-k+1]))); \\ A138793
a(n) = if(n == 1, 1, vecmax(factor(f(n))[,1])); \\ Daniel Suteu, May 26 2022

a(n) = A006530(A138793(n)). - Daniel Suteu, May 26 2022

A226195 Numbers n = x0 x1 x2...x9 such that xi is the number of digits greater than i in n.

Original entry on oeis.org

1, 10, 21, 22, 100, 210, 220, 311, 321, 332, 333, 1000, 2100, 2200, 3110, 3210, 3320, 3330, 4111, 4211, 4321, 4331, 4422, 4432, 4443, 4444, 10000, 21000, 22000, 31100, 32100, 33200, 33300, 41110, 42110, 43210, 43310, 44220, 44320, 44430, 44440, 51111, 52111

Offset: 1

Michel Lagneau, May 30 2013

This sequence contains 1142 terms.
Extension of the autobiographical numbers (or curious numbers) (A046043).
The concatenated numbers from n down to 1 (A000422(1) - A000422(9)): 1, 21, 321, ..., 987654321 are in the sequence.
The sequence includes A000461(n) (the n-digit number consisting of all n's) for n=1..9, i.e., 1, 22, 333, 4444, ..., 999999999.
The powers of 10 (A011557(0)..A011557(9)) are in the sequence.
The numbers of the form a(n)*10^p are also in the sequence.

			x0 x1 x2 x3 = 4211 is in the sequence because, for i = 0, 1, 2, 3:
xi > 0 (4 times) => x0 = 4;
xi > 1 (2 times) => x1 = 2;
xi > 2 (1 time) => x2 = 1;
xi > 3 (1 time) => x3 = 1.

Michel Lagneau, Table of n, a(n) for n = 1..1142

Cf. A000422, A000461, A011557, A046043.

T:=array(1..10):for n from 1 to 100000 do:nn:=length(n):for a from 1 to nn do:T[a] :=0:od:x:=convert(n,base,10): for k from 1 to nn do:for i from 1 to nn do: if k-1

cp's OEIS Frontend

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

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A050678 Index of first occurrence of n in A048288.

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A052246 Concatenation of integers from n down to 0.

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A053052 Append n to the previous term, reverse alternate terms.

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A078262 Sum of the forward and reverse concatenations of 1 to n.

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A078569 Numbers n such that A078568(n) is prime.

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A095249 Reverse concatenation of first n positive integers modulo forward concatenation of first n positive integers.

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A138794 a(n) = A138793(n+1)-A138793(n).

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A138963 a(1) = 1, a(n) = the largest prime divisor of A138793(n).

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