A007908 -id:A007908 - cp's OEIS frontend

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

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

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

Original entry on oeis.org

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

A262300 Let S(n,k) denote the number formed by concatenating the decimal numbers 1,2,3,...,k, but omitting n; a(n) is the smallest k for which S(n,k) is prime, or -1 if no term in S(n,*) is prime.

Original entry on oeis.org

2, 3, 7, 9, 11, 7, 11, 1873, 19, 14513, 13, 961

Offset: 1

N. J. A. Sloane and Jerrold B. Tunnell, Sep 27 2015

Sep 28 2015: David Broadhurst has found a(10) = 14513, a(12) = 961, a(14) = 653, a(16) = 5109, a(17) = 493, a(18) = 757, and a(20) = 1313. All these correspond to probable primes.
It is easy to check that a(19)=29.
So the sequence begins 2, 3, 7, 9, 11, 7, 11, 1873, 19, 14513, 13, 961, ???, 653, ???, 5109, 493, 757, 29, 1313, ...
a(13) is either -1 or greater than 40000. - Robert Price, Nov 03 2018

			a(5) = 11 because the smallest prime in S(5,*) (A262575) is 123467891011.
a(8) = 1873 (corresponding to the 6364-digit probable prime 1234567910111213...1873) was found by David Broadhurst on Sep 27 2015.
a(9) = 19 because the smallest prime in S(9,*) is 1234567810111213141516171819.
a(10) = 14513 (corresponding to the 61457-digit probable prime 123456789111213...14513) was found by David Broadhurst on Sep 28 2015.

Cf. A262299.
See A262571-A262582 for the sequences S(1,*) through S(12,*).
See also A007908 (which plays the role of S(0,*)).
For the primes in S(1,*) and S(2,*) see A089987, A262298.

A262300[n_] := Module[{k = 1}, While[! PrimeQ[FromDigits[Flatten[Map[IntegerDigits, Complement[Range[k], {n}]]]]], k++]; k];
Table[A262300[n], {n, 12}]  (* Robert Price, Oct 27 2018 *)

s(n, k) = my(s=""); for(x=1, k, if(x!=n, s=concat(s, x))); eval(Str(s))
a(n) = for(k=1, oo, my(s=s(n, k)); if(ispseudoprime(s), return(k))) \\ Felix Fröhlich, Oct 27 2018

a(8) was found by David Broadhurst, Sep 27 2015. On Sep 28 2015 David Broadhurst also found a(10), a(12), a(14), a(16), a(17), a(18), and a(20).

A048438 Take the first n numbers written in base 6, concatenate them, then convert from base 6 to base 10.

Original entry on oeis.org

1, 8, 51, 310, 1865, 67146, 2417263, 87021476, 3132773145, 112779833230, 4060073996291, 146162663866488, 5261855899193581, 189426812370968930, 6819365245354881495, 245497148832775733836, 8837897357979926418113, 318164304887277351052086, 11453914975941984637875115

Offset: 1

Patrick De Geest, May 15 1999

The first three primes in this sequence occur for n = 11 (a(11) = 4060073996291), n = 43 (a(43) = 4.3194...*10^68), n = 173 (a(n) = 1.3014...*10^372) (email from Kurt Foster, Oct 24 2015). - N. J. A. Sloane, Oct 25 2015

			a(8) = (1)(2)(3)(4)(5)(10)(11)(12) = 12345101112_6 = 87021476.

Vincenzo Librandi, Table of n, a(n) for n = 1..200

Cf. A014829.

Concatenation of first n numbers in other bases: 2: A047778, 3: A048435, 4: A048436, 5: A048437, 6: this sequence, 7: A048439, 8: A048440, 9: A048441, 10: A007908, 11: A048442, 12: A048443, 13: A048444, 14: A048445, 15: A048446, 16: A048447.

[n eq 1 select 1 else Self(n-1)*6^(1+Ilog(6, n))+n: n in [1..20]]; // Vincenzo Librandi, Dec 30 2012

If[STARTPOINT==1, n={}, n=Flatten[IntegerDigits[Range[STARTPOINT-1], 6]]]; Table[AppendTo[n, IntegerDigits[w, 6]]; n=Flatten[n]; FromDigits[n, 6], {w, STARTPOINT, ENDPOINT}] (* Dylan Hamilton, Aug 11 2010 *)
Table[FromDigits[Flatten[IntegerDigits[#,6]&/@Range[n]],6],{n,20}] (* Harvey P. Dale, Sep 29 2012 *)

A048439 Take the first n numbers written in base 7, concatenate them, then convert from base 7 to base 10.

Original entry on oeis.org

1, 9, 66, 466, 3267, 22875, 1120882, 54923226, 2691238083, 131870666077, 6461662637784, 316621469251428, 15514451993319985, 760208147672679279, 37250199235961284686, 1825259762562102949630, 89437728365543044531887, 4382448689911609182062481

Offset: 1

Patrick De Geest, May 15 1999

The first two primes in this sequence occur for n = 10 (a(10) = 131870666077) and n = 37 (a(37) = 569432644200356239518976257368822195317881440478377541397) (email from Kurt Foster, Oct 24 2015). What is the next prime? - N. J. A. Sloane, Oct 25 2015

After a(37), there are no more primes through a(4000) = 2.2670...*10^14538. - Jon E. Schoenfield, Jan 19 2018

			a(8): (1)(2)(3)(4)(5)(6)(10)(11) = 1234561011_7 = 54923226.

Vincenzo Librandi, Table of n, a(n) for n = 1..200

Cf. A014830.

Concatenation of first n numbers in other bases: 2: A047778, 3: A048435, 4: A048436, 5: A048437, 6: A048438, 7: this sequence, 8: A048440, 9: A048441, 10: A007908, 11: A048442, 12: A048443, 13: A048444, 14: A048445, 15: A048446, 16: A048447.

[n eq 1 select 1 else Self(n-1)*7^(1+Ilog(7, n))+n: n in [1..20]]; // Vincenzo Librandi, Dec 30 2012

a[n_]:= FromDigits[Flatten@IntegerDigits[Range@n, 7], 7]; Array[a, 20] (* Vincenzo Librandi, Dec 30 2012 *)

A048440 Take the first n numbers written in base 8, concatenate them, then convert from base 8 to base 10.

Original entry on oeis.org

1, 10, 83, 668, 5349, 42798, 342391, 21913032, 1402434057, 89755779658, 5744369898123, 367639673479884, 23528939102712589, 1505852102573605710, 96374534564710765455, 6167970212141488989136, 394750093577055295304721, 25264005988931538899502162

Offset: 1

Patrick De Geest, May 15 1999

83 is the only prime in this sequence among the first 3000 terms (email from Kurt Foster, Oct 24 2015). - N. J. A. Sloane, Oct 25 2015

			a(9): (1)(2)(3)(4)(5)(6)(7)(10)(11) = 12345671011_8 = 1402434057.

Harvey P. Dale, Table of n, a(n) for n = 1..390

Cf. A014831.

Concatenation of first n numbers in other bases: 2: A047778, 3: A048435, 4: A048436, 5: A048437, 6: A048438, 7: A048439, 8: this sequence, 9: A048441, 10: A007908, 11: A048442, 12: A048443, 13: A048444, 14: A048445, 15: A048446, 16: A048447.

[n eq 1 select 1 else Self(n-1)*8^(1+Ilog(8, n))+n: n in [1..20]]; // Vincenzo Librandi, Dec 30 2012

If[STARTPOINT==1, n={}, n=Flatten[IntegerDigits[Range[STARTPOINT-1], 8]]]; Table[AppendTo[n, IntegerDigits[w, 8]]; n=Flatten[n]; FromDigits[n, 8], {w, STARTPOINT, ENDPOINT}] (* Dylan Hamilton, Aug 11 2010 *)
Table[FromDigits[Flatten[IntegerDigits[#,8]&/@Range[n]],8],{n,20}] (* Harvey P. Dale, Dec 07 2012 *)

from functools import reduce
def A048440(n): return reduce(lambda i,j:(i<<3*(1+(j.bit_length()-1)//3))+j,range(n+1)) # Chai Wah Wu, Feb 26 2023

A048441 Take the first n numbers written in base 9, concatenate them, then convert from base 9 to base 10.

Original entry on oeis.org

1, 11, 102, 922, 8303, 74733, 672604, 6053444, 490328973, 39716646823, 3217048392674, 260580919806606, 21107054504335099, 1709671414851143033, 138483384602942585688, 11217154152838349440744, 908589486379906304700281, 73595748396772410680722779

Offset: 1

Patrick De Geest, May 15 1999

The first two primes in this sequence occur for n = 2 (a(2) = 11) and n = 14 (a(14) = 1709671414851143033) (email from Kurt Foster, Oct 24 2015). - N. J. A. Sloane, Oct 25 2015

			a(9) = (1)(2)(3)(4)(5)(6)(7)(8)(10) = 1234567810_9 = 490328973.

Vincenzo Librandi, Table of n, a(n) for n = 1..200

Cf. A014832, A055150.

Concatenation of first n numbers in other bases: 2: A047778, 3: A048435, 4: A048436, 5: A048437, 6: A048438, 7: A048439, 8: A048440, 9: this sequence, 10: A007908, 11: A048442, 12: A048443, 13: A048444, 14: A048445, 15: A048446, 16: A048447.

[n eq 1 select 1 else Self(n-1)*9^(1+Ilog(9, n))+n: n in [1..20]]; // Vincenzo Librandi, Dec 30 2012

If[STARTPOINT==1, n={}, n=Flatten[IntegerDigits[Range[STARTPOINT-1], 9]]]; Table[AppendTo[n, IntegerDigits[w, 9]]; n=Flatten[n]; FromDigits[n, 9], {w, STARTPOINT, ENDPOINT}] (* Dylan Hamilton, Aug 11 2010 *)
f[n_]:= FromDigits[Flatten@IntegerDigits[Range@n, 9], 9]; Array[f, 20] (* Vincenzo Librandi, Dec 30 2012 *)

{ cuo=0;
for(ixp=1, 18,
casi = ixp; cvst=0;
while(casi != 0,
cvd = casi%9; cvst=10*cvst + cvd + 1; casi = (casi - cvd) / 9 );
while(cvst !=0, ptch = cvst%10;
cuo=cuo*9+ptch-1; cvst = (cvst - ptch) / 10 ); print1(cuo, ", "))}
\\ Douglas Latimer, Apr 27 2012

More terms from Douglas Latimer, May 10 2012

A048442 Take the first n numbers written in base 11, concatenate them, then convert from base 11 to base 10.

Original entry on oeis.org

1, 13, 146, 1610, 17715, 194871, 2143588, 23579476, 259374245, 2853116705, 345227121316, 41772481679248, 5054470283189021, 611590904265871555, 74002499416170458170, 8954302429356625438586, 1083470593952151678068923, 131099941868210353046339701

Offset: 1

Patrick De Geest, May 15 1999

			a(11) = (1)(2)(3)(4)(5)(6)(7)(8)(9)(A)(10) = 123456789A10_11 = 345227121316.

Vincenzo Librandi, Table of n, a(n) for n = 1..200

Cf. A014881.

Concatenation of first n numbers in other bases: 2: A047778, 3: A048435, 4: A048436, 5: A048437, 6: A048438, 7: A048439, 8: A048440, 9: A048441, 10: A007908, 11: this sequence, 12: A048443, 13: A048444, 14: A048445, 15: A048446, 16: A048447.

[n eq 1 select 1 else Self(n-1) * 11^(1+Ilog(11, n)) + n: n in [1..20]]; // Vincenzo Librandi, Dec 30 2012

If[STARTPOINT==1, n={}, n=Flatten[IntegerDigits[Range[STARTPOINT-1], 11]]]; Table[AppendTo[n, IntegerDigits[w, 11]]; n=Flatten[n]; FromDigits[n, 11], {w, STARTPOINT, ENDPOINT}] (* Dylan Hamilton, Aug 11 2010 *)
f[n_]:= FromDigits[Flatten@IntegerDigits[Range@n, 11], 11]; Array[f, 20] (* Vincenzo Librandi, Dec 30 2012 *)

{ cuo=0;
for(ixp=1, 18, casi = ixp; cvst=0;
while(casi != 0,
cvd = casi%11; cvst=100*cvst + cvd + 1; casi = (casi - cvd) / 11 );
while(cvst !=0, ptch = cvst%100;
cuo=cuo*11+ptch-1; cvst = (cvst - ptch) / 100 ); print1(cuo, ", "))}
\\ Douglas Latimer, May 09 2012

1 more term from Douglas Latimer, May 10 2012

A048443 Take the first n numbers written in base 12, concatenate them, then convert from base 12 to base 10.

Original entry on oeis.org

1, 14, 171, 2056, 24677, 296130, 3553567, 42642812, 511713753, 6140565046, 73686780563, 10610896401084, 1527969081756109, 220027547772879710, 31683966879294678255, 4562491230618433668736, 656998737209054448298001, 94607818158103840554912162

Offset: 1

Patrick De Geest, May 15 1999

			a(12) = (1)(2)(3)(4)(5)(6)(7)(8)(9)(A)(B)(10) = 123456789AB10_12 = 10610896401084.

Vincenzo Librandi, Table of n, a(n) for n = 1..200

Cf. A014882.

Concatenation of first n numbers in other bases: 2: A047778, 3: A048435, 4: A048436, 5: A048437, 6: A048438, 7: A048439, 8: A048440, 9: A048441, 10: A007908, 11: A048442, 12: this sequence, 13: A048444, 14: A048445, 15: A048446, 16: A048447.

[n eq 1 select 1 else Self(n-1)*12^(1+Ilog(12, n))+n: n in [1..20]]; // Vincenzo Librandi, Dec 30 2012

If[STARTPOINT==1, n={}, n=Flatten[IntegerDigits[Range[STARTPOINT-1], 12]]]; Table[AppendTo[n, IntegerDigits[w, 12]]; n=Flatten[n]; FromDigits[n, 12], {w, STARTPOINT, ENDPOINT}] (* Dylan Hamilton, Aug 11 2010 *)
f[n_]:= FromDigits[Flatten@IntegerDigits[Range@n, 12], 12]; Array[f, 20] (* Vincenzo Librandi, Dec 30 2012 *)

A048444 Take the first n numbers written in base 13, concatenate them, then convert from base 13 to base 10.

Original entry on oeis.org

1, 15, 198, 2578, 33519, 435753, 5664796, 73642356, 957350637, 12445558291, 161792257794, 2103299351334, 355457590375459, 60072332773452585, 10152224238713486880, 1715725896342579282736, 289957676481895898782401, 49002847325440406894225787

Offset: 1

Patrick De Geest, May 15 1999

No primes in the first 31000 terms. - Giovanni Resta, Jun 08 2018

			a(12) = (1)(2)(3)(4)(5)(6)(7)(8)(9)(A)(B)(C) = 123456789ABC_13 = 2103299351334.

Vincenzo Librandi, Table of n, a(n) for n = 1..200

Cf. A014896.

Concatenation of first n numbers in other bases: 2: A047778, 3: A048435, 4: A048436, 5: A048437, 6: A048438, 7: A048439, 8: A048440, 9: A048441, 10: A007908, 11: A048442, 12: A048443, 13: this sequence, 14: A048445, 15: A048446, 16: A048447.

[n eq 1 select 1 else Self(n-1)*13^(1+Ilog(13, n))+n: n in [1..20]]; // Vincenzo Librandi, Dec 30 2012

If[STARTPOINT==1, n={}, n=Flatten[IntegerDigits[Range[STARTPOINT-1], 13]]]; Table[AppendTo[n, IntegerDigits[w, 13]]; n=Flatten[n]; FromDigits[n, 13], {w, STARTPOINT, ENDPOINT}] (* Dylan Hamilton, Aug 11 2010 *)
f[n_]:= FromDigits[Flatten@IntegerDigits[Range@n, 13], 13]; Array[f, 20] (* Vincenzo Librandi, Dec 30 2012 *)

cp's OEIS Frontend

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

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

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A262300 Let S(n,k) denote the number formed by concatenating the decimal numbers 1,2,3,...,k, but omitting n; a(n) is the smallest k for which S(n,k) is prime, or -1 if no term in S(n,*) is prime.

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A048438 Take the first n numbers written in base 6, concatenate them, then convert from base 6 to base 10.

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A048439 Take the first n numbers written in base 7, concatenate them, then convert from base 7 to base 10.

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A048440 Take the first n numbers written in base 8, concatenate them, then convert from base 8 to base 10.

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A048441 Take the first n numbers written in base 9, concatenate them, then convert from base 9 to base 10.

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