A116505 -id:A116505 - cp's OEIS frontend

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

R. Muller

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

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

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

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

See A176024 for primes.

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

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 *)

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

A000422(n,p=1,L=1)=sum(k=1,n,k*p*=L+(k==L&&!L*=10)) \\ M. F. Hasler, Nov 02 2016

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

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)

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

A116504 Number of distinct prime divisors of the concatenation of n,...,1.

Original entry on oeis.org

0, 2, 2, 2, 3, 2, 2, 3, 3, 3, 2, 3, 3, 4, 5, 4, 6, 8, 4, 5, 4, 5, 4, 5, 6, 7, 5, 5, 7, 8, 3, 6, 5, 7, 8, 6, 4, 3, 6, 5, 8, 6, 3, 7, 6, 5, 7, 7, 3, 6, 3, 7, 9, 9, 3, 4, 4, 6, 3, 3, 5, 8, 5, 6, 7, 7, 4, 8, 8, 4, 8, 4, 7, 8, 10, 3, 7, 6, 4, 7, 7, 1, 3, 8, 3, 8, 5, 4, 5, 7, 11, 9, 6

Offset: 1

Parthasarathy Nambi, Mar 20 2006

			87654321 = 3*3*1997*4877, distinct prime divisors are 3, 1997 and 4877, hence a(8) = 3.

Sean A. Irvine, Table of n, a(n) for n = 1..106

Cf. A116505.

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

b = {}; a = {}; Do[w = RealDigits[n]; w = First[w]; Do[PrependTo[a, w[[Length[w] - k + 1]]], {k, 1, Length[w]}]; p = FromDigits[a]; m = FactorInteger[p]; AppendTo[b, Length[m]], {n, 1, 30}]; b (* Artur Jasinski, Mar 30 2008 *)
Table[PrimeNu[FromDigits[Flatten[IntegerDigits/@Range[n,1,-1]]]],{n,95}] (* Harvey P. Dale, Oct 03 2015 *)

{a="";for(n=1,58,a=concat(n,a);print1(omega(eval(a)),","))}

Edited and extended by Klaus Brockhaus, Mar 29 2006
Terms a(59)-a(93) from Sean A. Irvine, Nov 04 2009
a(90) corrected by Sean A. Irvine, Nov 02 2024

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.

Original entry on oeis.org

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

Offset: 1

Amarnath Murthy, Sep 01 2002

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)

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

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

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

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

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 *)

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

More terms from Sascha Kurz, Jan 03 2003

A138789 a(n) = number of distinct prime divisors of A104759(n).

Original entry on oeis.org

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

Offset: 1

Artur Jasinski, Mar 30 2008

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

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

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 *)

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

Entire sequence corrected by Robert Price, Mar 24 2015

More terms from Sean A. Irvine, Jul 21 2024

A138793 a(n) = concatenation of reversed digits of natural numbers from n down to 1.

Original entry on oeis.org

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

Offset: 1

Artur Jasinski, Mar 30 2008, Apr 04 2008

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

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

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

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

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

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 *)

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

A138790 Numbers k such that A138793(k) is prime.

Original entry on oeis.org

61, 946

Offset: 1

Artur Jasinski, Mar 30 2008, Mar 31 2008

There are no more primes for k <= 5000.

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

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

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

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 *)

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.

Original entry on oeis.org

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

Offset: 1

Amarnath Murthy, Sep 01 2002

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

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

Cf. A075019, A075020, A075021.

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

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 *)

More terms from Sascha Kurz, Jan 03 2003

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.

Original entry on oeis.org

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

Offset: 1

Amarnath Murthy, Sep 01 2002

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

Cf. A075019.

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

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 *)

More terms from Sascha Kurz, Jan 03 2003

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.

Original entry on oeis.org

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

Offset: 1

Amarnath Murthy, Sep 01 2002

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

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

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

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

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 *)

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

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

More terms from Sascha Kurz, Jan 03 2003

Name edited by Felix Fröhlich, May 26 2022

A138957 Concatenation of the reversed digits of numbers from 1 to n.

Original entry on oeis.org

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

Offset: 1

Artur Jasinski, Apr 04 2008, Apr 05 2008

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

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

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 *)

cp's OEIS Frontend

A000422 Concatenation of numbers from n down to 1.

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A116504 Number of distinct prime divisors of the concatenation of n,...,1.

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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.

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A138789 a(n) = number of distinct prime divisors of A104759(n).

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A138793 a(n) = concatenation of reversed digits of natural numbers from n down to 1.

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A138790 Numbers k such that A138793(k) is prime.

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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.

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