A000422 -id:A000422 - cp's OEIS frontend

A272617 Concatenation of the numbers from n down to 1 with numbers from 1 to n.

11, 2112, 321123, 43211234, 5432112345, 654321123456, 76543211234567, 8765432112345678, 987654321123456789, 1098765432112345678910, 11109876543211234567891011, 121110987654321123456789101112, 1312111098765432112345678910111213, 14131211109876543211234567891011121314

Offset: 1

José de Jesús Camacho Medina, May 03 2016

Conjecture: a(1) is the only prime number.

No other prime terms up to a(8000). - Giovanni Resta, May 07 2016

			a(1) = concatenate("1", "1") = 11.
a(2) = concatenate("2", "1", "1", "2") = 2112.
a(3) = concatenate("3", "2", "1", "1", "2", "3") = 321123.

Andrew Howroyd, Table of n, a(n) for n = 1..100

Cf. A259937, A007942.

FromDigits@Flatten@IntegerDigits@Join[Reverse@#, #] & /@ Table[Range@n, {n, 20}]

a(n)={fromdigits(concat(concat(Vecrev(vector(n,i,digits(i)))), concat(vector(n,i,digits(i)))))} \\ Andrew Howroyd, Dec 23 2019

a(n) = A000422(n) concatenated with A007908(n).

Terms a(11) and beyond from Andrew Howroyd, Dec 23 2019

A280987 {Concatenation n, n-1, n-2, ...3,2,1} mod sigma(n).

Original entry on oeis.org

0, 0, 1, 2, 3, 9, 1, 6, 4, 1, 9, 21, 1, 9, 9, 16, 9, 24, 1, 33, 17, 1, 9, 21, 0, 9, 1, 41, 21, 33, 17, 6, 33, 19, 33, 25, 25, 21, 1, 1, 33, 81, 17, 21, 45, 1, 33, 85, 49, 69, 57, 77, 27, 81, 1, 81, 1, 1, 21, 57, 59, 81, 33, 60, 21, 33, 45, 51, 81, 1, 9, 66, 41, 9, 97, 1, 81, 81, 1, 57, 117, 73, 33, 145

Offset: 1

Indranil Ghosh, Jan 12 2017

			For n = 11, A000422(n) mod sigma(n) = 1110987654321 mod 12 = 9. S0 a(11) = 9.

Indranil Ghosh, Table of n, a(n) for n = 1..10000

Cf. A000203, A000422, A114795.

Table[Mod[FromDigits[Flatten[IntegerDigits/@Range[n,1,-1]]],DivisorSigma[ 1,n]],{n,90}] (* Harvey P. Dale, Jul 01 2020 *)

def sigma(n):
    s=0
    for i in range(1,n+1):
        if n%i==0:
            s+=i
    return s
def C(n):
    s=""
    for i in range(n,0,-1):
        s+=str(i)
    return int(s)
for i in range(1,101):
    print(i, C(i)%sigma(i))

from sympy import divisor_sigma
def A280987(n): return int(''.join(map(str, range(n, 0, -1)))) % divisor_sigma(n) # David Radcliffe, Aug 08 2025

a(n) = A000422(n) mod A000203(n).

A333183 Number of digits in concatenation of first n positive even integers.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 97, 100, 103, 106, 109, 112, 115, 118, 121, 124, 127, 130, 133, 136, 139, 142, 145, 148, 151, 154

Offset: 1

Alexander Goebel, Mar 10 2020

Connected with A019520 and A038396, similar to how A058183 applies to both A007908 and A000422 to count the digits in them, as the order of the digits does not matter (2468 returns the same result as 8642).

			For example, a(5) = 6 because 246810 (the concatenation of the first five positive even integers) has six digits.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A058183, A007908, A000422, A019520, A038396.

L:= [1$4, seq(i $(9/2*10^(i-1),i=2..3)]:
ListTools:-PartialSums(L); # Robert Israel, Apr 05 2020

Accumulate[IntegerLength[2Range[80]]] (* Alonso del Arte, Mar 12 2020, after Harvey P. Dale *)

a(n) = sum(k=1, n, #Str(2*k)); \\ Michel Marcus, Apr 02 2020

(1 to 80).map{n: Int => new java.math.BigInteger(((2 to (2 * n) by 2).map(.toString)).mkString("")).toString.length} // _Alonso del Arte, Mar 12 2020

a(n) = A058183(n) - Sum_{1..A058183(n)} A000035(A058183(n)).

a(n) = Sum_{i=1..n} (1+floor(log_10(2*i))). - Robert Israel, Apr 05 2020

A348792 Numbers k such that the reverse concatenation of the first k binary numbers A098780(k) is prime.

Original entry on oeis.org

2, 3, 4, 7, 11, 13, 25, 97, 110, 1939

Offset: 1

N. J. A. Sloane, Dec 03 2021

			a(4) = 7 is because the binary number 111 110 101 100 11 10 1 (with no spaces), which is 128413 in decimal, is prime.

Cf. A000422, A007908, A098780, A176024.

q:= n-> isprime(Bits[Join](['Bits[Split](i)[]'$i=1..n])):
select(q, [$1..200])[];  # Alois P. Heinz, Dec 03 2021

f[n_] := FromDigits[Flatten @ IntegerDigits[Range[n, 1, -1], 2], 2]; Select[Range[120], PrimeQ[f[#]] &] (* Amiram Eldar, Dec 03 2021 *)

from sympy import isprime
def afind(limit):
    s, k = "", 1
    for k in range(1, limit+1):
        s += bin(k)[2:][::-1]
        t = int(s[::-1], 2)
        if isprime(t):
            print(k, end=", ")
afind(200) # Michael S. Branicky, Dec 03 2021

a(8)-a(10) from Amiram Eldar, Dec 03 2021

A360505 Concatenate the ternary strings for n, n-1, n-2, ..., 2, 1.

Original entry on oeis.org

1, 21, 1021, 111021, 12111021, 2012111021, 212012111021, 22212012111021, 10022212012111021, 10110022212012111021, 10210110022212012111021, 11010210110022212012111021, 11111010210110022212012111021, 11211111010210110022212012111021, 12011211111010210110022212012111021

Offset: 1

N. J. A. Sloane, Feb 17 2023

Similar to A360502, but here we count down from n to 1 in base 3 and concatenate the strings to get a(n).

This is the ternary analog of A000422.

Index entries for sequences related to Most Wanted Primes video

Cf. A000422, A007809, A360502, A360506.

a[n_] := FromDigits @ Flatten @ IntegerDigits[Range[n, 1, -1], 3]; Array[a, 15] (* Amiram Eldar, Feb 18 2023 *)

a(n) = strjoin(concat([digits(k, 3) | k <- Vecrev([1..n])])) \\ Rémy Sigrist, Feb 18 2023

from sympy.ntheory import digits
def a(n): return int("".join("".join(map(str, digits(k, 3)[1:])) for k in range(n, 0, -1)))
print([a(n) for n in range(1, 16)]) # Michael S. Branicky, Feb 18 2023

# faster version for initial segment of sequence
from sympy.ntheory import digits
from itertools import count, islice
def agen(s=""): yield from (int(s:="".join(map(str, digits(n, 3)[1:]))+s) for n in count(1))
print(list(islice(agen(), 15))) # Michael S. Branicky, Feb 18 2023

A078192 a(n) = floor(concatenation of n down to 1 divided by n).

Original entry on oeis.org

1, 10, 107, 1080, 10864, 109053, 1093474, 10956790, 109739369, 1098765432, 100998877665, 10092582304526, 1009316229819563, 100937222213403880, 10094208074065843621, 1009463320069436728395, 100950083124771234567901, 10095341745173395054869684

Offset: 1

Amarnath Murthy, Nov 21 2002

			a(6) = floor(654321/6) = 109053.

Cf. A000422, A077147.

a:= n-> floor(parse(cat(1+n-i$i=1..n))/n):
seq(a(n), n=1..19);  # Alois P. Heinz, Jun 20 2025

Table[Floor[FromDigits[Flatten[IntegerDigits/@Reverse[Range[n]]]]/n],{n,20}] (* Harvey P. Dale, Jul 22 2012 *)

a(n) = floor(A000422(n) / n). - Sean A. Irvine, Jun 20 2025

More terms from Jon E. Schoenfield, May 08 2010

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)

A095252 a(n) = floor(sqrt{concatenation n,(n-1),...,3,2,1}).

Original entry on oeis.org

1, 4, 17, 65, 233, 808, 2766, 9362, 31426, 104822, 1054033, 11005043, 114547418, 1188747707, 12305003905, 127088210000, 1310019623181, 13480213329659, 138498072735374, 1420979139942389, 14560878466498901, 149036972532711329, 1523880579042109897

Offset: 1

Amarnath Murthy, Jun 17 2004

a(n) = floor(sqrt(A000422(n))). - Stefan Steinerberger, May 05 2007

			a(4) = floor(sqrt(4321)) = 65.

Cf. A068995.

a:= n-> floor(sqrt(parse(cat((n-i)$i=0..n-1)))):
seq(a(n), n=1..30);  # Alois P. Heinz, Aug 26 2015

Table[Floor[Sqrt[FromDigits[Flatten[Table[IntegerDigits[i],{i,n,1,-1}]]]]],{n,1,25}] (* Stefan Steinerberger, May 05 2007 *)
Table[Floor[Sqrt[FromDigits[Flatten[IntegerDigits/@Range[n,1,-1]]]]],{n,25}] (* Harvey P. Dale, Jul 22 2024 *)

More terms from Stefan Steinerberger, May 05 2007

A138795 a(n) = (A138793(n+1)-A138793(n))/10^n.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 8, 9, 1, 110, 2100, 31000, 410000, 5100000, 61000000, 710000000, 8100000000, 91000000000, 20000000000, 1200000000000, 22000000000000, 320000000000000, 4200000000000000, 52000000000000000, 620000000000000000

Offset: 1

Artur Jasinski, Mar 30 2008

First differences of A138793 divided by 10^n

Cf. A000422, A116504, A007908, A116505, A104759, A138789, A138790, A138793, A075019, A075020, A075021, A075022, A138794.

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, 61}]; c = {}; Do[AppendTo[c, (b[[n + 1]] - b[[n]])/(10^n)], {n, 1, Length[b] - 1}]; c (*Artur Jasinski*)

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

A283514 Concatenation of the numbers from n down to 3.

Original entry on oeis.org

3, 43, 543, 6543, 76543, 876543, 9876543, 109876543, 11109876543, 1211109876543, 131211109876543, 14131211109876543, 1514131211109876543, 161514131211109876543, 17161514131211109876543, 1817161514131211109876543, 191817161514131211109876543

Offset: 3

XU Pingya, Apr 14 2017

The a(3) = 3, a(4) = 43, a(7) = 76543, a(46) = 464544...876543 are primes in this sequence (for n < 2823).

Eric Weisstein's World of Mathematics, Consecutive Number Sequences

Cf. A000422, A176024, A284891.

cp's OEIS Frontend

A272617 Concatenation of the numbers from n down to 1 with numbers from 1 to n.

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A280987 {Concatenation n, n-1, n-2, ...3,2,1} mod sigma(n).

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A333183 Number of digits in concatenation of first n positive even integers.

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A348792 Numbers k such that the reverse concatenation of the first k binary numbers A098780(k) is prime.

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A360505 Concatenate the ternary strings for n, n-1, n-2, ..., 2, 1.

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A078192 a(n) = floor(concatenation of n down to 1 divided by n).

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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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