A000422 -id:A000422 - cp's OEIS frontend

A110757 a(n) = number of divisors of N, where N = reverse concatenation of 1,2,3,...,n.

1, 4, 4, 4, 8, 4, 4, 12, 18, 8, 4, 8, 8, 16, 48, 16, 96, 576, 16, 32, 16, 32, 16, 32, 64, 256, 96, 32, 128, 256, 8, 64, 32, 128, 384, 144, 16, 8, 64, 32, 256, 64, 8, 192, 96, 32, 128, 128, 8, 64, 8, 128, 1280, 2560, 8, 24, 16, 64, 8, 8, 32, 384, 48, 64, 128, 128

Offset: 1

Amarnath Murthy, Aug 11 2005

			a(3) = tau(321) = 4.

Tyler Busby, Table of n, a(n) for n = 1..109

Cf. A000005, A000422, A176024.

Cf. A110756, A110758, A110759, A110760.

s = ""; Do[s = ToString[n] <> s; Print[DivisorSigma[0, ToExpression[s]]], {n, 1, 45}] (* Ryan Propper, Sep 23 2005 *)
Table[DivisorSigma[0,FromDigits[Flatten[IntegerDigits/@Range[n,1,-1]]]],{n,50}] (* The program takes a long time to run. *) (* Harvey P. Dale, Jun 06 2018 *)

a(n) = A000005(A000422(n)). - Jinyuan Wang, May 23 2020

More terms from Ryan Propper, Sep 23 2005

a(46)-a(66) from Jinyuan Wang, May 23 2020

A137233 Number of n-digit even numbers.

Original entry on oeis.org

5, 45, 450, 4500, 45000, 450000, 4500000, 45000000, 450000000, 4500000000, 45000000000, 450000000000, 4500000000000, 45000000000000, 450000000000000, 4500000000000000, 45000000000000000, 450000000000000000, 4500000000000000000, 45000000000000000000, 450000000000000000000

Offset: 1

Ctibor O. Zizka, Mar 08 2008

From Kival Ngaokrajang, Oct 18 2013: (Start)
a(n) is also the total number of double rows identified numbers in n digit.
For example:
  n = 1: 01 23 45 67 89 = 5 double rows;
  n = 2: 1011 1213 1415 1617 1819...9899 = 45 double rows;
  n = 3: 100101 102103 104105...998999 = 450 double rows;
The number of double rows is also A030656. (End)
a(n) is also the number of n-digit integers with an even number of even digits (A356929); a(5) = 45000 is the answer to the question 2 of the Olympiade Mathématique Belge in 2004 (link). - Bernard Schott, Sep 06 2022
a(n) is also the number of n-digit integers with an odd number of odd digits (A054684). - Bernard Schott, Nov 07 2022

			a(2) = 45 because there are 45 2-digit even numbers.

Olympiade Mathématique Belge, OMB 2004, Finale Maxi, Question 2.
Index to sequences related to Olympiads.
Index entries for linear recurrences with constant coefficients, signature (10).

Cf. A000422, A002275, A002276, A011577, A014923, A014925, A016313, A019518, A037487, A053052, A057138, A090843, A097166, A099914, A099915.

Cf. A030656, A052268, A054684, A356929.

Vec(5*x*(1-x)/(1-10*x) + O(x^22)) \\ Elmo R. Oliveira, Jul 23 2025

def A137233(n): return 9*10**(n-1)+1>>1 # Chai Wah Wu, Nov 11 2022

a(n) = 9*10^(n-1)/2 if n > 1. - R. J. Mathar, May 23 2008
From Elmo R. Oliveira, Jul 23 2025: (Start)
G.f.: 5*x*(1-x)/(1-10*x).
E.g.f.: (-9 + 10*x + 9*exp(10*x))/20.
a(n) = 10*a(n-1) for n > 2.
a(n) = A052268(n)/2 for n >= 2. (End)

Corrected and extended by R. J. Mathar, May 23 2008

More terms from Elmo R. Oliveira, Jul 23 2025

A050679 Positions of 2's in A048288.

Original entry on oeis.org

2, 3, 4, 6, 7, 11

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 & Problems connection.

Cf. A000422, A046461, A048288, A050680, A050681, A050682.

Select[Range[2,12],PrimeOmega[FromDigits[Flatten[IntegerDigits[Range[#,1,-1]]]]]==2 &] (* Jayanta Basu, May 30 2013 *)

Offset changed to 1 by Jinyuan Wang, Sep 04 2021

A050680 Positions of 3's in A048288.

Original entry on oeis.org

5, 10, 12, 13, 31, 38, 43, 49, 51, 55, 59, 60, 76, 83, 85, 97, 117, 118

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, A046462, A048288, A050679, A050681, A050682.

a(13)-a(16) from Sean A. Irvine, Aug 17 2021

Offset changed to 1 by and a(17)-a(18) from Jinyuan Wang, Sep 04 2021

A050681 Positions of 4's in A048288.

Original entry on oeis.org

8, 14, 16, 19, 21, 23, 37, 57, 67, 70, 79, 88, 106

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, A046463, A048288, A050679, A050680, A050682.

a(9)-a(13) from Sean A. Irvine, Aug 17 2021

Offset changed to 1 by Jinyuan Wang, Sep 04 2021

A114795 {concatenation n, n-1, n-2, ...3,2,1} mod n.

Original entry on oeis.org

0, 1, 0, 1, 1, 3, 3, 1, 0, 1, 6, 9, 2, 1, 6, 1, 4, 9, 0, 1, 12, 17, 4, 9, 21, 21, 0, 13, 11, 21, 20, 17, 6, 29, 1, 9, 8, 25, 36, 1, 0, 33, 41, 17, 36, 23, 26, 33, 34, 21, 30, 41, 38, 27, 6, 1, 36, 39, 7, 21, 31, 59, 54, 49, 21, 39, 57, 45, 60, 41, 31, 9, 37, 41, 21

Offset: 1

Amarnath Murthy, Nov 19 2005

			a(7) = 7654321 mod 7 = 3.

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

Cf. A000422.

f[n_] := Mod[FromDigits@ Flatten@ IntegerDigits@ Reverse@ Range@ n,
n]; Array[f, 75] (* or *)
f[n_] := Fold[Mod[#1*10^Floor[1 + Log10@#2] + #2, n] &, 0,
  Reverse@Range@n]; Array[f, 75] (* Robert G. Wilson v, Jan 10 2017 *)

def A114795(n):
    s=""
    for i in range(n,0,-1):
        s+=str(i)
    return int(s)%n # Indranil Ghosh, Jan 10 2017

a(n) = A000422(n) mod n. - Michel Marcus, Jan 10 2017

More terms from Ryan Murphy (murphy(AT)minegoboom.com) and Evonne Haines (avissean(AT)adelphia.net), Jan 21 2006

A060554 String together the first n numbers in an order which maximizes the result.

Original entry on oeis.org

1, 21, 321, 4321, 54321, 654321, 7654321, 87654321, 987654321, 98765432110, 9876543211110, 987654321211110, 98765432131211110, 9876543214131211110, 987654321514131211110, 98765432161514131211110

Offset: 1

Henry Bottomley, Apr 02 2001

Sean A. Irvine, Java program (github)

Cf. A000422, A060555.

from itertools import count, islice
def agen(): # generator of terms
    slst = []
    for n in count(1):
        s, greatest, argreatest = str(n), "/", None
        for i in range(len(slst)+1):
            t = "".join(slst[:i]) + s + "".join(slst[i:])
            if t > greatest: greatest, argreatest = t, i
        slst.insert(argreatest, s)
        yield int("".join(slst))
print(list(islice(agen(), 16))) # Michael S. Branicky, Nov 30 2022

Incorrect comment removed by Sean A. Irvine, Nov 30 2022

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.

Original entry on oeis.org

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

A060555 String together the first n numbers in an order which minimizes the result.

Original entry on oeis.org

1, 12, 123, 1234, 12345, 123456, 1234567, 12345678, 123456789, 10123456789, 1011123456789, 101111223456789, 10111121323456789, 1011112131423456789, 101111213141523456789, 10111121314151623456789

Offset: 1

Henry Bottomley, Apr 02 2001

Cf. A000422, A060554.

from itertools import count, islice
def agen(): # generator of terms
    slst = []
    for n in count(1):
        s, least, argleast = str(n), ":", None
        for i in range(len(slst)+1):
            t = "".join(slst[:i]) + s + "".join(slst[i:])
            if t < least: least, argleast = t, i
        slst.insert(argleast, s)
        yield int("".join(slst))
print(list(islice(agen(), 16))) # Michael S. Branicky, Nov 29 2022

Incorrect comment removed by Sean A. Irvine, Nov 30 2022

A078263 Product of the forward and reverse concatenations of 1 to n.

Original entry on oeis.org

1, 252, 39483, 5332114, 670592745, 80779853376, 9449772114007, 1082152022374638, 121932631112635269, 135650052221140070110, 1371589685334334871208531, 14951973660666886818972704952, 161989023197512459256340110001373, 1744593949735321085709329564827297794

Offset: 1

Amarnath Murthy, Nov 24 2002

			a(3) = 123*321 = 39483.

Cf. A000422, A007908, A078264.

a:= n-> parse(cat(1+n-i$i=1..n))*parse(cat($1..n)):
seq(a(n), n=1..14);  # Alois P. Heinz, Jun 25 2025

frc[n_]:=Module[{r1=FromDigits[Flatten[IntegerDigits/@Range[n]]], r2 = FromDigits[ Flatten[ IntegerDigits/@Range[n,1,-1]]]},r1*r2]; Array[ frc, 20] (* Harvey P. Dale, Aug 24 2015 *)

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

More terms from Sascha Kurz, Jan 04 2003

More terms from Sean A. Irvine, Jun 25 2025

cp's OEIS Frontend

A110757 a(n) = number of divisors of N, where N = reverse concatenation of 1,2,3,...,n.

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A137233 Number of n-digit even numbers.

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PARI

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A050679 Positions of 2's in A048288.

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A050680 Positions of 3's in A048288.

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A050681 Positions of 4's in A048288.

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A114795 {concatenation n, n-1, n-2, ...3,2,1} mod n.

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A060554 String together the first n numbers in an order which maximizes the result.

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Python

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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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