A104759 -id:A104759 - cp's OEIS frontend

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

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 *)

A138960 a(n) = smallest prime divisor of A138957(n).

Original entry on oeis.org

1, 2, 3, 2, 3, 2, 127, 2, 3, 857, 3, 3, 18503, 3, 3, 43, 3, 3, 17, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 7, 3, 3, 1051, 3, 3, 67103, 3, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 5, 3, 3, 5, 3, 3, 5, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2

Offset: 1

Artur Jasinski, Apr 04 2008

For largest prime divisors see A138961.

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

a = {}; b = {}; Do[w = RealDigits[n]; w = First[w]; Do[AppendTo[a, w[[Length[w] - k + 1]]], {k, 1, Length[w]}]; p = FromDigits[a]; AppendTo[b, First[First[FactorInteger[p]]]], {n, 1, 38}]; b
A137957[n_] := FromDigits[Flatten[Reverse /@ IntegerDigits[Range[n]]]];
Table[FactorInteger[A137957[n]][[1, 1]], {n, 39}] (* Robert Price, May 10 2019 *)

a(39)-a(69) from Robert Price, May 10 2019

A104757 Decimal expansion of solution to x^(9^x)=9.

Original entry on oeis.org

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

Offset: 1

Zak Seidov, Mar 23 2005

			x=1.179067994130546711326627268870672714247363863698857908913601448...

Cf. A103561, A104750, A104751, A104752, A104753, A104754, A104755, A104756, A104757, A104758, A104759, A104760, A104761.

RealDigits[ FindRoot[x^(9^x) == 9, {x, 1}, WorkingPrecision -> 2^7] [[1, 2]]][[1]] (* Robert G. Wilson v, Mar 26 2005 *)

More terms from Robert G. Wilson v, Mar 26 2005

A138961 a(n) = largest prime divisor of A138957(n).

Original entry on oeis.org

1, 3, 41, 617, 823, 643, 9721, 14593, 3803, 14405693, 10939223, 4156374407, 2663693, 5603770631, 1221751714624799, 287108811653770498027, 74103167823547, 11843077531813991, 726216405947772436185983423, 769725127, 18274551225153265813469

Offset: 1

Artur Jasinski, Apr 04 2008

For smallest prime divisors see A138960.

Robert Price, Table of n, a(n) for n = 1..60

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

a = {}; b = {}; Do[w = RealDigits[n]; w = First[w]; Do[AppendTo[a, w[[Length[w] - k + 1]]], {k, 1, Length[w]}]; p = FromDigits[a]; AppendTo[b, First[Last[FactorInteger[p]]]], {n, 1, 18}]; b
A137957[n_] := FromDigits[Flatten[Reverse /@ IntegerDigits[Range[n]]]];
Table[First[Last[FactorInteger[A137957[n]]]], {n, 39}] (* Robert Price, May 10 2019 *)

A138962 a(1) = 1, a(n) = the smallest prime divisor of A138793(n).

Original entry on oeis.org

1, 3, 3, 29, 3, 3, 19, 3, 3, 457, 3, 3, 16087, 3, 3, 35963, 3, 3, 167, 3, 3, 7, 3, 3, 13, 3, 3, 953, 3, 3, 7, 3, 3, 548636579, 3, 3, 19, 3, 3, 71, 3, 3, 13, 3, 3, 89, 3, 3, 114689, 3, 3, 17, 3, 3, 12037, 3, 3, 7, 3, 3

Offset: 1

Artur Jasinski, Apr 04 2008

nonn

a(61) > 10^11. - Robert Price, Mar 22 2015

Daniel Suteu, Table of n, a(n) for n = 1..333

Cf. A020639, A138793, A104759, A000422, A116504, A007908, A116505, A104759, A075019, A075020, A075021, A075022, A138789, A138790, A138960, A138962.

b = {}; a = {}; Do[w = RealDigits[n]; w = First[w]; Do[AppendTo[a, w[[k]]], {k, 1, Length[w]}]; p = FromDigits[Reverse[a]]; AppendTo[b, First[First[FactorInteger[p]]]], {n, 1, 31}]; b (* Artur Jasinski, Apr 04 2008 *)
lst = {}; Table[First[First[FactorInteger[FromDigits[Reverse[lst = Join[lst,IntegerDigits[n]]]]]]], {n, 1, 60}] (* Robert Price, Mar 22 2015 *)

f(n) = my(D = Vec(concat(apply(s->Str(s), [1..n])))); eval(concat(vector(#D, k, D[#D-k+1]))); \\ A138793
a(n) = my(k=f(n)); forprime(p=2, 10^6, if(k%p == 0, return(p))); if(n == 1, 1, vecmin(factor(k)[,1])); \\ Daniel Suteu, May 27 2022

a(n) = A020639(A138793(n)). - Daniel Suteu, May 27 2022

a(32)-a(60) from Robert Price, Mar 22 2015

A138794 a(n) = A138793(n+1)-A138793(n).

Original entry on oeis.org

20, 300, 4000, 50000, 600000, 7000000, 80000000, 900000000, 1000000000, 1100000000000, 210000000000000, 31000000000000000, 4100000000000000000, 510000000000000000000, 61000000000000000000000

Offset: 1

Artur Jasinski, Mar 30 2008

First differences of A138793

Cf. A000422, A116504, A007908, A116505, A104759, A138789, A138790, A138793, A075019, A075020, A075021, A075022.

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, 31}]; c = {}; Do[AppendTo[c, b[[n + 1]] - b[[n]]], {n, 1, Length[b] - 1}]; c (*Artur Jasinski*)

a(n) = A138793(n+1)-A138793(n)

A138963 a(1) = 1, a(n) = the largest prime divisor of A138793(n).

Original entry on oeis.org

1, 7, 107, 149, 953, 218107, 402859, 4877, 379721, 4349353, 169373, 182473, 1940144339383, 2184641, 437064932281, 5136696159619, 67580875919190833, 1156764458711, 464994193118899, 4617931439293, 1277512103328491957510030561, 8177269604099

Offset: 1

Artur Jasinski, Apr 04 2008

For the smallest prime divisors of A138793 see A138962.

Daniel Suteu and Robert Price, Table of n, a(n) for n = 1..63 (terms a(1)..a(45) from Robert Price)

Cf. A006530, A138793, A138962.

Cf. A104759, A000422, A116504, A007908, A116505, A104759, A075019, A075020, A075021, A075022, A138789, A138790, A138958, A138959, A138960.

b = {}; a = {}; Do[w = RealDigits[n]; w = First[w]; Do[AppendTo[a, w[[k]]], {k, 1, Length[w]}]; p = FromDigits[Reverse[a]]; AppendTo[b, First[Last[FactorInteger[p]]]], {n, 1, 31}]; b (* Artur Jasinski, Apr 04 2008 *)
lst = {}; Table[First[Last[FactorInteger[FromDigits[Reverse[lst = Join[lst,IntegerDigits[n]]]]]]], {n, 1, 10}] (* Robert Price, Mar 22 2015 *)

f(n) = my(D = Vec(concat(apply(s->Str(s), [1..n])))); eval(concat(vector(#D, k, D[#D-k+1]))); \\ A138793
a(n) = if(n == 1, 1, vecmax(factor(f(n))[,1])); \\ Daniel Suteu, May 26 2022

a(n) = A006530(A138793(n)). - Daniel Suteu, May 26 2022

A371720 a(n) = m^^m mod 10^len(m), where m = A038399(n) and ^^ indicates tetration or hyper-4.

Original entry on oeis.org

1, 11, 811, 3811, 63811, 763811, 3763811, 5103763811, 515103763811, 19515103763811, 6819515103763811, 8146819515103763811, 3808146819515103763811, 7213808146819515103763811, 9807213808146819515103763811, 4939807213808146819515103763811

Offset: 1

Marco Ripà, Apr 04 2024

For any n, a(n) == a(n + 1) (mod 10^len(A038399(n))), where len(k) := number of digits in k. Assuming len(a(n)) > 1, this is a general property of every concatenated sequence with fixed rightmost digits (such as A014925, A061839, A092845, and A104759), as shown in Ripà's book "La strana coda della serie n^n^...^n".

Moreover, assuming n > 1, since A038399(n) is congruent to 11 (mod 20), the convergence speed of A038399(n)^^b (say, V(A038399(n), b) = {2, 1, 1, 1, ...}) is 2 at height 1 and becomes a unit value for any integer b > 1 (see Links). Hence, a(n) is given by A038399(n)^^len(A038399(n) - 1) (mod 10^len(A038399(n))), and also by A038399(n)^^len(A038399(n)) (mod 10^len(A038399(n))) since A038399(n)^^len(A038399(n)) == A038399(n)^^len(A038399(n) - 1) (mod 10^len(A038399(n))) holds for any n.

			a(8) is given by the rightmost 10 digits of 2113853211^^2113853211 and thus a(8) = 5103763811.
a(9) == a(8) (mod 10^10), i.e., the digits of a(9) end with the digits of a(8) (and then a(9) has 2 more preceding).

Marco Ripà, La strana coda della serie n^n^...^n, Trento, UNI Service, Nov 2011, page 60. ISBN 978-88-6178-789-6

Marco Ripà, The congruence speed formula, Notes on Number Theory and Discrete Mathematics, 2021, 27(4), 43-61.
Marco Ripà and Luca Onnis, Number of stable digits of any integer tetration, Notes on Number Theory and Discrete Mathematics, 2022, 28(3), 441-457.
Wikipedia, Tetration.

Cf. A000045 (Fibonacci), A038399, A171882 (tetration), A317824, A317903, A317905.

a(n) = A038399(n)^^(len(A038399(n)) - 1) mod 10^len(A038399(n)), where len(A038399(n)) = ceiling(log_10(A038399(n) + 1)).

A078998 Choose a(n) so that a(1)+a(2)+...+a(n) = concatenation of n first natural numbers.

Original entry on oeis.org

1, 11, 111, 1111, 11111, 111111, 1111111, 11111111, 111111111, 12222222121, 1222222212101, 122222221210101, 12222222121010101, 1222222212101010101, 122222221210101010101, 12222222121010101010101

Offset: 1

Benoit Cloitre, Jan 12 2003

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

b = {}; a = {}; Do[w = RealDigits[n]; w = First[w]; Do[AppendTo[a, w[[k]]], {k, 1, Length[w]}]; p = FromDigits[a]; AppendTo[b, p], {n, 1, 30}]; c = {}; Do[AppendTo[c, b[[n + 1]] - b[[n]]], {n, 1, Length[b] - 1}]; c (* Artur Jasinski, Mar 30 2008 *)

a(1)=1; for n>1, a(n) = A007908(n)-A007908(n-1)

A138795 a(n) = (A138793(n+1)-A138793(n))/10^n.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 8, 9, 1, 110, 2100, 31000, 410000, 5100000, 61000000, 710000000, 8100000000, 91000000000, 20000000000, 1200000000000, 22000000000000, 320000000000000, 4200000000000000, 52000000000000000, 620000000000000000

Offset: 1

Artur Jasinski, Mar 30 2008

First differences of A138793 divided by 10^n

Cf. A000422, A116504, A007908, A116505, A104759, A138789, A138790, A138793, A075019, A075020, A075021, A075022, A138794.

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}]; c = {}; Do[AppendTo[c, (b[[n + 1]] - b[[n]])/(10^n)], {n, 1, Length[b] - 1}]; c (*Artur Jasinski*)

a(n) = A138793(n+1)-A138793(n)

cp's OEIS Frontend

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

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A138960 a(n) = smallest prime divisor of A138957(n).

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A104757 Decimal expansion of solution to x^(9^x)=9.

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A138961 a(n) = largest prime divisor of A138957(n).

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A138962 a(1) = 1, a(n) = the smallest prime divisor of A138793(n).

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A138794 a(n) = A138793(n+1)-A138793(n).

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A138963 a(1) = 1, a(n) = the largest prime divisor of A138793(n).

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A371720 a(n) = m^^m mod 10^len(m), where m = A038399(n) and ^^ indicates tetration or hyper-4.

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A078998 Choose a(n) so that a(1)+a(2)+...+a(n) = concatenation of n first natural numbers.

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