A095370 -id:A095370 - cp's OEIS frontend

A095371 Number of distinct prime factors of record setting repunits (A328899).

0, 1, 2, 5, 7, 8, 10, 13, 14, 20, 21, 22, 26, 29, 32, 33, 34, 35, 40, 44, 55, 56, 63

Offset: 1

Labos Elemer, Jun 04 2004

Conjecture: a(24) = 73, a(25) = 94, a(26) = 99, a(27) >= 107, a(28) >= 127, a(29) >= 136, a(30) >= 140, a(31) >= 151, a(32) >= 159, a(33) >= 163, a(34) >= 178, a(35) >= 184, a(36) >= 213, a(37) >= 214. - Chai Wah Wu, Nov 01 2019

Cf. A067063, A003020, A001221, A002275, A095370, A004023.

r[n_] := (10^n-1)/9; L = {}; bst = -1; Do[v = PrimeNu[r[n]]; If[v > bst, bst = v; AppendTo[L, v]], {n, 60}]; L
(* or, based on the b-file of A095370: *)
w = Last /@ Cases[Import["https://oeis.org/A095370/b095370.txt", "Table"], {Integer, _Integer}]; L = {}; bst = -1; Do[ If[j > bst, AppendTo[L, bst = j]], {j, w}]; L (* _Giovanni Resta, Oct 30 2019 *)

Data corrected by Ray Chandler and N. J. A. Sloane, May 03 2017
Name edited by Giovanni Resta, Oct 30 2019
a(20)-a(21) from Chai Wah Wu, Oct 30 2019
a(22)-a(23) from Chai Wah Wu, Nov 01 2019

A095417 Sum of all decimal digits of distinct prime factors for n-th repunit.

Original entry on oeis.org

0, 2, 13, 4, 15, 26, 37, 25, 41, 36, 45, 47, 62, 67, 57, 67, 60, 92, 19, 76, 116, 99, 23, 105, 63, 120, 124, 134, 133, 110, 140, 155, 141, 130, 132, 168, 137, 103, 184, 176, 182, 226, 188, 217, 168, 145, 186, 220, 245, 170, 221, 228, 232, 267, 216, 225, 179, 259

Offset: 1

Labos Elemer, Jun 22 2004

			n=60: concatenated distinct-prime factor-set for 60th-repunit is:
371113313741611012112412712161354190919901279612906161418890139526741,
its digit sum=a(60)=244.

Max Alekseyev, Table of n, a(n) for n = 1..352 (first 322 terms from Giovanni Resta)

Cf. A095402, A002275, A095370, A095413-A095418.

a[1]=0; a[n_] := Total[ Flatten[ IntegerDigits /@ First /@ FactorInteger[(10^n - 1)/9]]]; Array[a, 60] (* Giovanni Resta, Jul 19 2018 *)

a(n) = A095402(A002275(n)).

Data corrected by Giovanni Resta, Jul 19 2018

A095415 Length of repunits of which the prime factor-digit-excess computed by A095414 equals 0.

Original entry on oeis.org

2, 3, 5, 7, 9, 11, 13, 17, 19, 23, 27, 31, 47, 59, 67, 71, 83, 113, 127, 139, 163, 197, 211, 229, 251, 263, 311, 317, 347, 421, 457, 461

Offset: 1

Labos Elemer, Jun 22 2004

541, 701, 857 are also terms. Conjecture: Except for the number 4, A046413 is a subsequence. Conjecture: except for the prime powers 9 and 27, all terms are prime. - Chai Wah Wu, Nov 03 2019

Sequence continues as 467?, 479?, 509?, 541, 557?, 571?, 577?, 593?, 599?, 617?, 643?, 647?, 661?, 673?, 683?, 691?, 701, 727?, 743?, 751?, 757?, 769?, 773?, 821?, 857, 863?, 887?, 911?, 967?, 971?, 977?, 991?, where ? marks uncertain/candidate terms. - Max Alekseyev, Apr 29 2022

A004023 is a subsequence.

Cf. A002275, A004023, A095370, A095407, A095413-A095418.

d[1] = -1; d[n_] := Total[ IntegerLength /@ First /@ FactorInteger[(10^n - 1)/9]] - n; Select[ Range[67], d[#] == 0 &] (* Giovanni Resta, Jul 16 2018 *)

Solutions to A095414(x) = 0.

Data corrected and extended by Giovanni Resta, Jul 16 2018

a(29)-a(32) confirmed by Max Alekseyev, Apr 29 2022

A095416 Length of smallest repunit of which the prime factor-digit-excess computed by A095414 equals n.

Original entry on oeis.org

2, 4, 6, 12, 24, 32, 30, 80, 96, 60, 84, 126, 120, 200, 168, 264, 210, 252

Offset: 0

Labos Elemer, Jun 22 2004

a(18), a(19) > 322. a(20) = 300. - Giovanni Resta, Jul 19 2018

a(A095371(n)-1) >= A328899(n). a(18) <= 440, a(19) <= 336, a(21) <= 624, a(22) <= 560, a(23) <= 480, a(24) <= 540, a(25) <= 720, a(26) <= 612, a(27) <= 420, a(28) <= 600, a(30) = 1050, a(31) <= 660, a(32) <= 1400, a(33) <= 900, a(34) <= 1020, a(35) <= 1500, a(36) <= 1380, a(37) <= 840, a(48) <= 1260, a(50) <= 1680. - Chai Wah Wu, Nov 03 2019

			n=60: concatenated p-set for 60th-repunit is:
371113313741611012112412712161354190919901279612906161418890139526741,
its length=69, so excess=9, 60 is the smallest such repunit

Cf. A002275, A095370, A095407, A095413-A095418.

a(n) = Min{x; A095414(x)=n}.

Edited by Charles R Greathouse IV, Aug 03 2010
Data corrected and extended by Giovanni Resta, Jul 19 2018
a(0) from Chai Wah Wu, Nov 03 2019

A328899 Numbers k such that the k-th repunit (10^k-1)/9 sets a new record for the number of distinct prime factors.

Original entry on oeis.org

1, 2, 3, 6, 12, 18, 24, 30, 42, 60, 84, 96, 120, 168, 180, 210, 240, 252, 300, 360, 420, 630, 660

Offset: 1

Giovanni Resta, Oct 30 2019

The corresponding numbers of distinct prime factors are in A095371.
a(20) > 322.
From Chai Wah Wu, Oct 30 2019: (Start)
Since A095371(19) = 40, to show that a(20) > 323 we use the fact that (10^323-1)/9 is a product of 4 primes and a 271-digit composite number C. We then use a computer search to show that C has no prime factor <= floor(C^(1/(41-4))) = 19858291. This implies that (10^323-1)/9 has less than 41 distinct prime factors.
Applying this same approach to 337 and 353 (the only numbers between 323 and 359 for which the complete factorization of the corresponding repunit is not known) and using the factorization of (10^360-1)/9 with 44 distinct prime factors show that a(20) = 360 and A095371(20) = 44.
This approach also shows that a(21) = 420 and A095371(21) = 55. (End)
a(24) <= 840. Conjecture: a(24) = 840, a(25) = 1260, a(26) = 1680, a(27) = 1980, a(28) = 2520, a(29) = 3360, a(30) = 3780, a(31) = 3960, a(32) = 4620, a(33) = 5040, a(34) = 6300, a(35) = 7560, a(36) = 9240, a(37) = 10080. - Chai Wah Wu, Nov 01 2019

Cf. A001221, A002275, A095370, A095371.

r[n_] := (10^n - 1)/9; L = {}; bst = -1; Do[v = PrimeNu[r[n]]; If[v > bst, bst = v; AppendTo[L, n]], {n, 60}]; L
(* or, based on the b-file of A095370: *)
w = Last /@ Cases[Import["https://oeis.org/A095370/b095370.txt", "Table"], {_Integer, _Integer}]; L={}; bst=-1; Do[If[w[[j]] > bst, AppendTo[L, j]; bst = w[[j]]], {j, Length@w}]; L

a(20)-a(21) from Chai Wah Wu, Oct 30 2019

a(22)-a(23) from Chai Wah Wu, Nov 01 2019

A086565 Smallest k such that (10^k - 1)/9 has n distinct prime divisors. Or a(n)= smallest value of k such that A000042(k) has exactly n distinct prime divisors.

Original entry on oeis.org

1, 2, 3, 9, 8, 6, 15, 12, 18, 45, 24, 32, 54, 30, 42, 64, 102, 72, 108, 154, 60, 84, 96, 140, 126, 200, 120, 204, 308, 168, 192, 280, 180, 210, 240, 252, 330

Offset: 0

Amarnath Murthy, Aug 31 2003

a(40) = 300; all other subsequent terms are > 322. - Ray Chandler, Apr 23 2017

a(38) = 336. - Max Alekseyev, Apr 29 2022

Cf. A000042, A001221, A002275, A095370.

a(n) = my(k=1); while(omega((10^k - 1)/9) !=n, k++); k; \\ Michel Marcus, Apr 23 2017

Corrected and extended by Sascha Kurz, Sep 22 2003
a(12)-a(15) from David Wasserman, Mar 28 2005
a(16)-a(27) from Donovan Johnson, Nov 17 2008
a(28)-a(35) from Ray Chandler, Apr 23 2017
a(36) from Max Alekseyev, Apr 29 2022

A095373 Integers k such that A095372(k) = 1 + 90*(-1+100^k)/99 is prime.

Original entry on oeis.org

2, 3, 8, 15, 26, 33, 146, 320, 1068, 1505

Offset: 1

Labos Elemer, Jun 07 2004

a(11) > 40000. - Michael S. Branicky, Jan 06 2025

			Corresponding primes are: 9091, 909091, 909090909090909091, 909090909090909090909090909091, ...

Cf. A095372, A002275, A095370, A095371.

{ta=Table[0, {100}], u=1}; Do[s=1+(90*(100^n-1)/99);If[PrimeQ[s], Print[{n, s}]; ta[[u]]=s;u=u+1], {n, 1, 320}] ta

a(9)-a(10) from Max Alekseyev, Jan 28 2012

A268582 Sphenic numbers having identical digits.

Original entry on oeis.org

66, 222, 555, 777, 2222, 3333, 5555, 7777, 22222, 33333, 55555, 77777, 2222222, 3333333, 5555555, 7777777, 22222222222, 33333333333, 55555555555, 77777777777, 1111111111111, 22222222222222222, 33333333333333333, 55555555555555555, 77777777777777777, 6666666666666666666

Offset: 1

Michel Lagneau, Feb 07 2016

Subsequence of A007304 (sphenic numbers: products of 3 distinct primes).

a(1)= A087331(4).

			222 is in the sequence because 222 = 2*3*37, product of 3 distinct primes.

Max Alekseyev, Table of n, a(n) for n = 1..89

Cf. A007304, A046053, A087331, A095370, A196104.

with(numtheory):
for n from 1 to 23 do:
  for b from 1 to 9 do:
    x:=(((10^n)- 1)/9)*b:y:=factorset(x):n1:=nops(y):
     if bigomega(x)=3 and n1=3
      then
      printf(`%d, `,x):
      else
     fi:
   od:
od:

Select[Flatten@ Map[Map[Function[k, FromDigits@ Table[k, {#}]], Range[1, 9]] &, Range@ 20], Length@ # == 3 && Times @@ Last /@ # == 1 &@ FactorInteger@ # &] (* Michael De Vlieger, Feb 07 2016 *)

cp's OEIS Frontend

A095371 Number of distinct prime factors of record setting repunits (A328899).

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Extensions

A095417 Sum of all decimal digits of distinct prime factors for n-th repunit.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A095415 Length of repunits of which the prime factor-digit-excess computed by A095414 equals 0.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Formula

Extensions

A095416 Length of smallest repunit of which the prime factor-digit-excess computed by A095414 equals n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

Extensions

A328899 Numbers k such that the k-th repunit (10^k-1)/9 sets a new record for the number of distinct prime factors.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Extensions

A086565 Smallest k such that (10^k - 1)/9 has n distinct prime divisors. Or a(n)= smallest value of k such that A000042(k) has exactly n distinct prime divisors.

Views

Author

Keywords

Comments

Crossrefs

Programs

PARI

Extensions

A095373 Integers k such that A095372(k) = 1 + 90*(-1+100^k)/99 is prime.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Extensions

A268582 Sphenic numbers having identical digits.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica