A000422 Concatenation of numbers from n down to 1.

1, 21, 321, 4321, 54321, 654321, 7654321, 87654321, 987654321, 10987654321, 1110987654321, 121110987654321, 13121110987654321, 1413121110987654321, 151413121110987654321, 16151413121110987654321, 1716151413121110987654321, 181716151413121110987654321

Offset

1,2

Comments

The first prime term in this sequence is a(82) (see A176024). - Artur Jasinski, Mar 30 2008

For n < 10^4, a(n)/A000217(n) is an integer for n = 1, 2, and 18. The integers are 1, 7 (prime), and 1062667552123515268933651, respectively. - Derek Orr, Sep 04 2014

References

F. Smarandache, "Properties of the Numbers", University of Craiova Archives, 1975; Arizona State University Special Collections, Tempe, AZ

Links

T. D. Noe, Table of n, a(n) for n = 1..150

R. W. Stephan, Factors and primes in two Smarandache sequences

Bertrand Teguia Tabuguia, Explicit formulas for concatenations of arithmetic progressions, arXiv:2201.07127 [math.CO], 2022.

Eric Weisstein's World of Mathematics, Consecutive Number Sequences

Index entries for sequences related to Most Wanted Primes video

Formula

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

a(n) = Sum_{k=1..n} k*10^(A058183(k) - (1+floor(log10(k)))). - Alexander Goebel, Mar 07 2020

From Serge Batalov, Dec 08 2021: (Start)

a(n) = ((n*9-1)*10^n+1)/9^2 for n < 10,

a(n) = ((n*99-1)*10^(2*n-19)-89)/99^2*10^10 + (8*10^10+1)/9^2 for 10 <= n < 100,

a(n) = ((n*999-1)*10^(3*n-299)-989)/999^2*10^191 + c2 for 10^2 <= n < 10^3,

a(n) = ((n*9999-1)*10^(4*n-3999)-9989)/9999^2*10^2892 + c3 for 10^3 <= n < 10^4,

a(n) = ((n*99999-1)*10^(5*n-49999)-99989)/99999^2*10^38893 + c4 for 10^4 <= n < 10^5,

a(n) = ((n*999999-1)*10^(6*n-599999)-999989)/999999^2*10^488894 + c5 for 10^5 <= n < 10^6,

where

c2 = (98*10^191 + 879*10^10 + 121)/99^2 = a(99),

c3 = (998*10^2701 - 989)/999^2*10^191 + c2 = a(999),

c4 = (9998*10^36001 - 9989)/9999^2*10^2892 + c3 = a(9999),

c5 = (99998*10^450001 - 99989)/99999^2*10^38893 + c4 = a(99999).

(End)

Maple

a[1]:= 1:
for n from 2 to 100 do
a[n]:= n*10^(1+ilog10(a[n-1])) + a[n-1]
od:
seq(a[n], n=1..100); # Robert Israel, Sep 05 2014
# second Maple program:
a:= proc(n) a(n):= `if`(n=1, 1, parse(cat(n, a(n-1)))) end:
seq(a(n), n=1..22);  # Alois P. Heinz, Jan 12 2021

Mathematica

b = {}; a = {}; Do[w = RealDigits[n]; w = First[w]; Do[PrependTo[a, w[[Length[w] - k + 1]]], {k, 1, Length[w]}]; p = FromDigits[a]; AppendTo[b, p], {n, 1, 30}]; b (* Artur Jasinski, Mar 30 2008 *)
Table[FromDigits[Flatten[IntegerDigits/@Range[n, 1, -1]]], {n, 20}] (* Harvey P. Dale, Jul 06 2019 *)

Prog

(PARI) a(n)=my(t=n); forstep(k=n-1, 1, -1, t=t*10^#Str(k)+k); t \\ Charles R Greathouse IV, Jul 15 2011
(PARI) A000422(n, p=1, L=1)=sum(k=1, n, k*p*=L+(k==L&&!L*=10)) \\ M. F. Hasler, Nov 02 2016
(Python)
def a(n): return int("".join(map(str, range(n, 0, -1))))
print([a(n) for n in range(1, 19)]) # Michael S. Branicky, Dec 08 2021

Crossrefs

Cf. A007908, A058183, A104759, A116504, A116505, A138789, A138790, A138793.

See A176024 for primes.

Sequence in context: A104759 A138793 A014925 * A060554 A057610 A036737

Adjacent sequences: A000419 A000420 A000421 * A000423 A000424 A000425

Keyword

nonn,base

Author

R. Muller

Extensions

Edited by N. J. A. Sloane, Dec 03 2021

Status

approved