A000422 -id:A000422 - cp's OEIS frontend

A116505 Number of distinct prime divisors of the concatenation of 1..n.

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

Offset: 1

Parthasarathy Nambi, Mar 20 2006

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

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

Dario Alpern, Factorization using the Elliptic Curve Method

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

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

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

Edited and extended by Klaus Brockhaus, Mar 29 2006

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

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 *)

A048288 Number of prime factors counted with multiplicity of the reverse concatenation of numbers from 1 to n.

Original entry on oeis.org

0, 2, 2, 2, 3, 2, 2, 4, 5, 3, 2, 3, 3, 4, 6, 4, 7, 10, 4, 5, 4, 5, 4, 5, 6, 9, 9, 5, 7, 8, 3, 6, 5, 7, 9, 8, 4, 3, 6, 5, 8, 6, 3, 8, 7, 5, 7, 7, 3, 6, 3, 7, 12, 14, 3, 5, 4, 6, 3, 3, 5, 9, 6, 6, 7, 7, 4, 8, 8, 4, 9, 5, 7, 8, 10, 3, 7, 6, 4, 9, 10, 1, 3, 8, 3

Offset: 1

Paul Jasper (jasperpaul(AT)hotmail.com)

			21 = 3*7 so a(2) = 2; 321 = 3*107 so a(3) = 2; 4321 = 29*149 so a(4) = 2; etc.
a(1)=0 since 1 has no prime factors.

Sean A. Irvine, Table of n, a(n) for n = 1..106
Patrick De Geest, Reversed Smarandache Concatenated Numbers
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, A001222, A046460, A046461, A046468, A050679, A050680, A050681, A050682.

Join[{0},Table[PrimeOmega[FromDigits[Flatten[IntegerDigits[Range[i,1,-1]]]]],{i,2,36}]] (* Jayanta Basu, May 30 2013 *)

a(n) = A001222(A000422(n)). - Michel Marcus, Jun 14 2021

Offset and a(19) corrected and more terms from Sean A. Irvine, Jun 13 2021

Edited by N. J. A. Sloane, Sep 04 2021

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

A038399 Concatenate first n nonzero Fibonacci numbers in reverse order.

Original entry on oeis.org

1, 11, 211, 3211, 53211, 853211, 13853211, 2113853211, 342113853211, 55342113853211, 8955342113853211, 1448955342113853211, 2331448955342113853211, 3772331448955342113853211, 6103772331448955342113853211, 9876103772331448955342113853211

Offset: 1

M. I. Petrescu (mipetrescu(AT)yahoo.com)

Mihaly Bencze [Beneze], L. Tutescu, Some Notions and Questions in Number Theory, Sequence 7.

Reinhard Zumkeller, Table of n, a(n) for n = 1..96

Cf. A000045, A000422, A019523.

a038399 n = a038399_list !! (n-1)
a038399_list = h "" $ tail a000045_list where
   h xs (f:fs) = (read ys :: Integer) : h ys fs
     where ys = show f ++ xs
-- Reinhard Zumkeller, Mar 01 2014

Module[{nn=20,fibs},fibs=Fibonacci[Range[nn]];Table[FromDigits[ Flatten[ IntegerDigits/@ Reverse[Take[fibs,n]]]],{n,nn}]] (* Harvey P. Dale, Aug 30 2016 *)

a(n) = my(t=fibonacci(n)); forstep(k=n-1, 1, -1, t=t*10^#Str(fibonacci(k))+fibonacci(k)); t; \\ Michel Marcus, Apr 06 2024

More terms from Andrew Gacek (andrew(AT)dgi.net), Feb 21 2000
Offset changed by Reinhard Zumkeller, Mar 01 2014
More terms from Harvey P. Dale, Aug 30 2016

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

cp's OEIS Frontend

A116505 Number of distinct prime divisors of the concatenation of 1..n.

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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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A048288 Number of prime factors counted with multiplicity of the reverse concatenation of numbers from 1 to n.

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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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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.

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