A116504 -id:A116504

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A000422

Concatenation of numbers from n down to 1.

+10
68

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

(list; graph; refs; listen; history; text; internal format)

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.

KEYWORD

nonn,base

AUTHOR

R. Muller

EXTENSIONS

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

STATUS

approved

A116505

Number of distinct prime divisors of the concatenation of 1..n.

+10
18

0, 2, 2, 2, 3, 3, 2, 4, 3, 3, 6, 4, 3, 3, 3, 3, 4, 5, 6, 6, 8, 6, 4, 5, 4, 6, 5, 5, 4, 7, 3, 5, 6, 2, 7, 5, 4, 4, 6, 8, 5, 7, 4, 4, 9, 7, 5, 7, 6, 9, 3, 3, 4, 9, 5, 4, 6, 4, 4, 6, 3, 7, 4, 9, 6, 8, 3, 7, 7, 6, 5, 5, 3, 9, 5, 4, 5, 6, 6, 7, 4, 7, 6, 3, 5, 7, 6, 5, 9, 8, 6, 6, 7, 5, 6, 5, 2, 9, 5, 9

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Dario Alpern's factorization program was used for n > 43.

LINKS

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

Dario Alpern, Factorization using the Elliptic Curve Method

EXAMPLE

123456 = 2*2*2*2*2*2*3*643, with distinct prime divisors 2, 3 and 643. Hence, a(6) = 3.

MATHEMATICA

Table[PrimeNu[FromDigits[Flatten[IntegerDigits[Range[n]]]]], {n, 30}] (* Jan Mangaldan, Jul 07 2020 *)

PROG

(PARI) {a=""; for(n=1, 43, a=concat(a, n); print1(omega(eval(a)), ", "))}

CROSSREFS

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

KEYWORD

nonn,base

AUTHOR

Parthasarathy Nambi, Mar 20 2006

EXTENSIONS

Edited and extended by Klaus Brockhaus, Mar 29 2006

Terms 59-100 from Sean A. Irvine, Nov 04 2009

STATUS

approved

A138789

a(n) = number of distinct prime divisors of A104759(n).

+10
18

0, 2, 2, 2, 3, 2, 2, 3, 3, 2, 2, 4, 4, 2, 5, 4, 2, 3, 7, 4, 4, 3, 3, 2, 5, 3, 5, 2, 3, 6, 6, 3, 3, 5, 5, 7, 3, 3, 4, 3, 3, 4, 4, 5, 3, 5, 7, 3, 5, 6, 6, 5, 4, 5, 2, 6, 4, 6, 4, 4, 7, 6, 5, 5, 6, 9, 5, 5, 7, 5, 5, 5, 5, 6, 6, 4, 4, 4, 5, 8, 7, 6, 4, 5, 4, 4, 9

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Indices where number 1 occured in this sequence see A138790

LINKS

Sean A. Irvine, Table of n, a(n) for n = 1..165

FORMULA

a(n) = A001221(A104759(n)). - Michel Marcus, Jun 30 2024

MATHEMATICA

lst = {}; Do[lst = Join[lst, IntegerDigits[n]], {n, 1, 50}]; Table[Length[FactorInteger[FromDigits[Reverse[lst[[Range[1, n]]]]]]], {n, 1, Length[lst]}] (* Robert Price, Mar 24 2015 *)

CROSSREFS

Cf. A001221, A104759, A000422, A116504, A007908, A116505.

KEYWORD

nonn,base

AUTHOR

Artur Jasinski, Mar 30 2008

EXTENSIONS

Entire sequence corrected by Robert Price, Mar 24 2015

More terms from Sean A. Irvine, Jul 21 2024

STATUS

approved

A075019

a(1) = 1; for n > 1, a(n) = the smallest prime divisor of the number C(n) formed from the concatenation of 1,2,3,... up to n.

+10
17

1, 2, 3, 2, 3, 2, 127, 2, 3, 2, 3, 2, 113, 2, 3, 2, 3, 2, 13, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 29, 2, 3, 2, 3, 2, 71, 2, 3, 2, 3, 2, 7, 2, 3, 2, 3, 2, 23, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 10386763, 2, 3, 2, 3, 2, 397, 2, 3, 2, 3, 2, 37907, 2, 3, 2, 3, 2, 73, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 37, 2, 3, 2

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Least prime factor of A007908(n). For 1 < n <= 5000, a(n) < A007908(n), but this should fail infinitely often (assuming standard heuristics). - Charles R Greathouse IV, Apr 10 2014

From Robert Israel, Aug 28 2015: (Start)

a(n) = 2 iff n is even.

a(n) = 3 iff n == 3 or 5 (mod 6).

a(n) = 5 iff n == 25 (mod 30). (End)

LINKS

Robert G. Wilson v, Table of n, a(n) for n = 1..1000 (first 120 terms from Robert Israel)

EXAMPLE

a(5)= 3, 3 is the smallest prime divisor of 12345.

MAPLE

C:= 1: A[1]:= 1:

for n from 2 to 100 do

C:= C*10^(1+ilog10(n))+n;

F:= map(t -> t[1], ifactors(C, 'easy')[2]);

if hastype(F, integer) then A[n]:= min(select(type, F, integer))

else A[n]:= min(numtheory:-factorset(C))

od:

seq(A[n], n=1..100); # Robert Israel, Aug 28 2015

MATHEMATICA

a = {}; b = {}; Do[w = RealDigits[n]; w = First[w]; Do[AppendTo[a, w[[k]]], {k, Length[w]}]; p = FromDigits[a]; AppendTo[b, First[First[FactorInteger[ p]]]], {n, 25}]; b (* Artur Jasinski, Apr 04 2008 *)

PROG

(PARI) lpf(n)=forprime(p=2, 1e3, if(n%p==0, return(p))); factor(n)[1, 1]

print1(N=1); for(n=2, 100, N=N*10^#Str(n)+n; print1(", "lpf(N))) \\ Charles R Greathouse IV, Apr 10 2014

CROSSREFS

Cf. A000422, A007908, A075020, A104759, A116504, A116505, A138789, A138790, A138960, A138961, A138962.

KEYWORD

base,nonn

AUTHOR

Amarnath Murthy, Sep 01 2002

EXTENSIONS

More terms from Sascha Kurz, Jan 03 2003

STATUS

approved

A138790

a(n) = indices n for which A138793(n) is prime.

+10
17

61, 946

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

Indices where number 1 occured in A138789.

There are no more primes for n<=5000.

a(3) > 20000. - Robert Price, Mar 24 2015

LINKS

Table of n, a(n) for n=1..2.

EXAMPLE

a(1) = 61 because the number 160695...654321 is prime.

MATHEMATICA

b = {}; a = {}; Do[w = RealDigits[n]; w = First[w]; Do[AppendTo[a, w[[k]]], {k, 1, Length[w]}]; p = FromDigits[Reverse[a]]; If[PrimeQ[p], Print[n]; AppendTo[b, p]], {n, 1, 2000}]; b (* Artur Jasinski, Mar 30 2008 *)

Select[Range[1, 1000], PrimeQ[lst = {}; Do[lst = Join[lst, IntegerDigits[n]], {n, 1, #}]; FromDigits[Reverse[lst]]] &] (* Robert Price, Mar 24 2015 *)

CROSSREFS

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

KEYWORD

nonn,bref,hard,more,base

AUTHOR

Artur Jasinski, Mar 30 2008, Mar 31 2008

STATUS

approved

A138793

a(n) = concatenation of reversed digits of natural numbers from n down to 1.

+10
17

1, 21, 321, 4321, 54321, 654321, 7654321, 87654321, 987654321, 1987654321, 1101987654321, 211101987654321, 31211101987654321, 4131211101987654321, 514131211101987654321, 61514131211101987654321, 7161514131211101987654321

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Note that leading zeros are not omitted when writing down digits in reversed order. So 10 reversed becomes 01. - N. J. A. Sloane, Jan 23 2017

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 1..100

MAPLE

read(transforms): A138793 := proc(n) return digrev(parse(cat($(1..n)))): end: seq(A138793(n), n=1..17); # Nathaniel Johnston, Jun 26 2011

MATHEMATICA

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}]; b (* Artur Jasinski, Mar 30 2008 *)

lst = {}; Table[FromDigits[Reverse[lst = Join[lst, IntegerDigits[n]]]], {n, 1, 15}] (* Robert Price, Mar 22 2015 *)

PROG

(Magma) [Seqint(&cat[Reverse(Intseq(k)): k in [1..n]]): n in [1..16]]; // Bruno Berselli, May 27 2011

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

CROSSREFS

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

KEYWORD

nonn,easy,base

AUTHOR

Artur Jasinski, Mar 30 2008, Apr 04 2008

STATUS

approved

A075022

a(1) = 1; for n>1, a(n) = the largest prime divisor of the number C(n) formed from the concatenation of 1,2,3,... up to n.

+10
10

1, 3, 41, 617, 823, 643, 9721, 14593, 3803, 1234567891, 630803, 2110805449, 869211457, 205761315168520219, 8230452606740808761, 1231026625769, 584538396786764503, 801309546900123763, 833929457045867563

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

N. J. A. Sloane, Table of n, a(n) for n = 1..56 [First 50 terms from Zak Seidov]

EXAMPLE

a(4) = 617 since 1234 = 2*617.

MATHEMATICA

a = {}; b = {}; Do[w = RealDigits[n]; w = First[w]; Do[AppendTo[a, w[[k]]], {k, 1, Length[w]}]; p = FromDigits[a]; AppendTo[b, First[Last[FactorInteger[p]]]], {n, 1, 25}]; b (* Artur Jasinski, Apr 04 2008 *)

Table[FactorInteger[FromDigits[Flatten[IntegerDigits/@Range[n]]]][[-1, 1]], {n, 20}] (* Harvey P. Dale, Aug 31 2015 *)

CROSSREFS

Cf. A075019, A075020, A075021.

Cf. A104759, A000422, A116504, A007908, A116505, A104759, A138789, A138790, A138960, A138961, A138962.

KEYWORD

base,nonn

AUTHOR

Amarnath Murthy, Sep 01 2002

EXTENSIONS

More terms from Sascha Kurz, Jan 03 2003

STATUS

approved

A075020

a(1) = 1; for n>1, a(n) = the smallest prime divisor of the number C(n) formed from the reverse concatenation of 1,2,3,... up to n.

+10
9

1, 3, 3, 29, 3, 3, 19, 3, 3, 7, 3, 3, 17, 3, 3, 23, 3, 3, 17, 3, 3, 13, 3, 3, 11, 3, 3, 23, 3, 3, 7, 3, 3, 89, 3, 3, 29, 3, 3, 11, 3, 3, 52433, 3, 3, 23, 3, 3, 71, 3, 3, 7, 3, 3

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

Table of n, a(n) for n=1..54.

EXAMPLE

a(4)= 29, 29 is the smallest prime divisor of 4321 =29*149

MATHEMATICA

b = {}; a = {}; Do[w = RealDigits[n]; w = First[w]; Do[AppendTo[a, w[[Length[w] - k + 1]]], {k, 1, Length[w]}]; p = FromDigits[Reverse[a]]; AppendTo[b, First[First[FactorInteger[p]]]], {n, 1, 21}]; b (* Artur Jasinski, Apr 04 2008 *)

CROSSREFS

Cf. A075019.

Cf. A104759, A000422, A116504, A007908, A116505, A104759, A138789, A138790, A138960, A138961, A138962.

KEYWORD

base,nonn

AUTHOR

Amarnath Murthy, Sep 01 2002

EXTENSIONS

More terms from Sascha Kurz, Jan 03 2003

STATUS

approved

A075021

a(1) = 1; for n>1, a(n) = the largest prime divisor of the number C(n) formed from the concatenation of n, n-1, n-2, n-3, ... down to 1.

+10
8

1, 7, 107, 149, 953, 218107, 402859, 4877, 379721, 54421, 370329218107, 5767189888301, 237927839, 1728836281, 136133374970881, 1190788477118549, 677181889, 399048049, 40617114482123, 629639170774346584751, 2605975408790409767, 65372140114441

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

Daniel Suteu, Table of n, a(n) for n = 1..106

FORMULA

a(n) = A006530(A000422(n)). - Daniel Suteu, May 26 2022

EXAMPLE

a(4)= 149 as 149 is the largest prime divisor of 4321 =29*149

MATHEMATICA

b = {}; a = {}; Do[w = RealDigits[n]; w = First[w]; Do[AppendTo[a, w[[Length[w] - k + 1]]], {k, 1, Length[w]}]; p = FromDigits[Reverse[a]]; AppendTo[b, First[Last[FactorInteger[p]]]], {n, 1, 21}]; b (* Artur Jasinski, Apr 04 2008 *)

Table[FactorInteger[FromDigits[Flatten[IntegerDigits/@Range[n, 1, -1]]]] [[-1, 1]], {n, 20}] (* Harvey P. Dale, Dec 14 2020 *)

PROG

(PARI) a(n) = if(n==1, 1, vecmax(factor(eval(concat(apply(k->Str(n-k+1), [1..n]))))[, 1])); \\ Daniel Suteu, May 26 2022

CROSSREFS

Cf. A006530, A000422, A075019, A075020, A075022.

Cf. A104759, A116504, A007908, A116505, A104759, A138789, A138790, A138957, A138960, A138961, A138962.

KEYWORD

base,nonn

AUTHOR

Amarnath Murthy, Sep 01 2002

EXTENSIONS

More terms from Sascha Kurz, Jan 03 2003

Name edited by Felix Fröhlich, May 26 2022

STATUS

approved

A138957

Concatenation of the reversed digits of numbers from 1 to n.

+10
7

1, 12, 123, 1234, 12345, 123456, 1234567, 12345678, 123456789, 12345678901, 1234567890111, 123456789011121, 12345678901112131, 1234567890111213141, 123456789011121314151, 12345678901112131415161

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

There are no primes in this sequence for n<=7000

LINKS

Table of n, a(n) for n=1..16.

MATHEMATICA

b = {}; a = {}; Do[w = RealDigits[n]; w = First[w]; Do[AppendTo[a, w[[Length[w] - k + 1]]], {k, 1, Length[w]}]; p = FromDigits[a]; AppendTo[b, p], {n, 1, 21}]; b

(* or *)

Table[FromDigits[Flatten[Reverse/@IntegerDigits[Range[n]]]], {n, 20}] (* Harvey P. Dale, Oct 22 2013 *)

CROSSREFS

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

KEYWORD

nonn,base

AUTHOR

Artur Jasinski, Apr 04 2008, Apr 05 2008

STATUS

approved

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