A075021 -id:A075021 - cp's OEIS frontend

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.

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

A075023 a(n) = the smallest prime divisor of A173426(n) = concatenation of (1, 2, 3,..., n, n-1, ..., 1) for n > 1; a(1) = 1.

Original entry on oeis.org

1, 11, 3, 11, 41, 3, 239, 11, 3, 12345678910987654321, 7, 3, 1109, 7, 3, 71, 7, 3, 251, 7, 3, 70607, 7, 3, 989931671244066864878631629, 7, 3, 149, 7, 3, 827, 7, 3, 197, 7, 3, 39907897297, 7, 3, 17047, 7, 3, 191, 7, 3, 967, 7, 3, 139121, 7, 3, 109, 7, 3, 5333, 7, 3

Offset: 1

Amarnath Murthy, Sep 01 2002

			a(5) = 41 as 123454321 = 41*41*271*271.
a(25) = 989931671244066864878631629 is the smaller factor of the semiprime A173426(24) = a(25) * A075023(25).
A173426(37) = 39907897297 * P58 * P59, where Pxx are primes with xx digits, therefore a(37) = 39907897297.

FactorDB, (121*10^(4*n-19) - 1002*10^(4*n-28) - 2*10^(2*n-9) + 879*10^10 + 121)/99^2.

Cf. A075019, A075020, A075021, A075022, A075024.

A075023(n)=A020639(A173426(n)) \\ Efficient code for computing the least prime factor should be developed in A020639. For n = 37, use \g3 (debugging level 3) to see the lpf within milliseconds, while factorization would take hours. - M. F. Hasler, Jul 29 2015

a(n) = A020639(A173426(n)). a(3n) = 3 for all n > 0. a(3n-1) = 7 for 3 < n < 34. - M. F. Hasler, Jul 29 2015

More terms from Sascha Kurz, Jan 03 2003

Terms beyond a(24) from M. F. Hasler, Jul 29 2015

A075024 a(n) is the largest prime divisor of the number A173426(n) = concatenate(1,2,...,n-1,n,n-1,...,2,1); a(1) = 1.

Original entry on oeis.org

1, 11, 37, 101, 271, 37, 4649, 137, 333667, 12345678910987654321, 17636684157301569664903, 2799473675762179389994681, 2354041513534224607850261, 2068140300159522133, 498056174529497, 112240064764214229701, 4188353169004802474320231191377

Offset: 1

Amarnath Murthy, Sep 01 2002

Also for 1 < n < 10, a(n) is the common prime divisor for all A010785(m) which consist of n digits. - Alexander R. Povolotsky, Jun 05 2014, corrected by M. F. Hasler, Jul 30 2015

According to the definition (and given terms), this is the greatest prime factor (A006530) of A173426 and not of A002477, as an earlier formula asserted and which may have been an assumption of the preceding comment. - M. F. Hasler, Jul 29 2015

			a(5) = 271 as 123454321 = 41*41*271*271.
a(25) = 12471243489559387823527232424981012432152516319410549 is the larger factor of the semiprime A173426(24) = A075023(25) * a(n).

Amiram Eldar, Table of n, a(n) for n = 1..58 (terms 1..36 from M. F. Hasler)
FactorDB, (121*10^(4*n-19) - 1002*10^(4*n-28) - 2*10^(2*n-9) + 879*10^10 + 121)/99^2.

Cf. A002477, A006530, A010785, A075019, A075020, A075021, A075022, A075023, A173426.

Table[FactorInteger[FromDigits[Join[Flatten[IntegerDigits/@Range[ n]], Flatten[ IntegerDigits/@Range[n-1,1,-1]]]]][[-1,1]],{n,20}] (* Harvey P. Dale, May 20 2016 *)

a(n) = {if (n == 1, return (1)); s = ""; for (i=1, n, s = concat(s, Str(i));); forstep (i=n-1, 1, -1, s = concat(s, Str(i));); f = factor(eval(s)); f[#f~, 1];} \\ Michel Marcus, Jun 05 2014

A075024(n)=A006530(A173426(n)) \\ A006530 should provide efficient code and also covers the case n=1. - M. F. Hasler, Jul 29 2015

a(n) = A006530(A173426(n)). - Michel Marcus, Jun 05 2014, corrected by M. F. Hasler, Jul 29 2015

More terms from Sascha Kurz, Jan 03 2003
a(16)-a(17) from Michel Marcus, Jun 05 2014
More terms from M. F. Hasler, Jul 29 2015

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

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

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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A075023 a(n) = the smallest prime divisor of A173426(n) = concatenation of (1, 2, 3,..., n, n-1, ..., 1) for n > 1; a(1) = 1.

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A075024 a(n) is the largest prime divisor of the number A173426(n) = concatenate(1,2,...,n-1,n,n-1,...,2,1); a(1) = 1.

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

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