A095371 -id:A095371 - cp's OEIS frontend

A095370 Number of distinct prime factors of the repunit (-1 + 10^n)/9.

0, 1, 2, 2, 2, 5, 2, 4, 3, 4, 2, 7, 3, 4, 6, 6, 2, 8, 1, 7, 7, 6, 1, 10, 5, 6, 5, 8, 5, 13, 3, 11, 6, 6, 7, 11, 3, 3, 6, 11, 4, 14, 4, 10, 9, 6, 2, 13, 4, 10, 8, 9, 4, 12, 8, 12, 6, 8, 2, 20, 7, 5, 13, 15, 7, 14, 3, 10, 6, 12, 2, 17, 3, 7, 12, 6, 8, 15, 6, 15, 10, 7, 3, 21, 7, 8, 10, 14, 5, 21, 12, 10

Offset: 1

Labos Elemer, Jun 04 2004; corrected Jun 09 2004

nonn

Factoring certain repunits is especially difficult.

			a(62)=5 because
11111111111111111111111111111111111111111111111111111111111111 =
11 * 2791 * 6943319 * 57336415063790604359 * 909090909090909090909090909091.
a(97)=3 because (10^97 - 1)/9 = 12004721 * 846035731396919233767211537899097169 * 109399846855370537540339266842070119107662296580348039.

Yates, S. "Peculiar Properties of Repunits." J. Recr. Math. 2, 139-146,1969.
Yates, S. "Prime Divisors of Repunits." J. Recr. Math. 8, 33-38, 1975.

Table of n, a(n) for n = 1..352
Patrick De Geest, Repunits and their prime factors
T. Granlund, Repunits.
M. Kamada, Factorization of 11...11(Repunits)
Y. Koide, Factorization of Repunit Numbers
W. M. Snyder, Factoring Repunits, Am. Math. Monthly 89, 462-466, 1982.
P. Yiu, Factorizations of repunits R_n for n<=50 Appendix Chap.18.5 pp. 173/360 in 'Recreational Mathematics'

Cf. A067063, A003020, A001221, A002275, A095371.

Cf. A046053 (total number of prime factors).

lst={};Do[p=(10^n-1)/9;AppendTo[lst,Length[FactorInteger[p]]],{n,0,2*4!}];lst (* Vladimir Joseph Stephan Orlovsky, Jun 15 2009 *)

a(n)=omega(10^n\9) \\ Charles R Greathouse IV, Sep 14 2015

a(n) = A001221(A002275(n)).

If 3|n, then a(n) = A102347(n); otherwise a(n) = A102347(n) - 1. - Max Alekseyev, Apr 25 2022

Terms to a(322) in b-file from Ray Chandler, Apr 22 2017

a(323)-a(352) in b-file from Max Alekseyev, Apr 26 2022

A095372 1+integers repeating "90" decimal digit pattern.

Original entry on oeis.org

1, 91, 9091, 909091, 90909091, 9090909091, 909090909091, 90909090909091, 9090909090909091, 909090909090909091, 90909090909090909091, 9090909090909090909091, 909090909090909090909091

Offset: 0

Labos Elemer, Jun 07 2004

These numbers arise for example as divisors of several repunits (A002275).
The aerated sequence A(n) = [1, 0, 91, 0, 9091, 0, 909091,...] is a divisibility sequence, i.e., A(n) divides A(m) whenever n divides m. It is the case P1 = 0, P2 = -11^2, Q = 10 of the 3-parameter family of 4th-order linear divisibility sequences found by Williams and Guy. - Peter Bala, Aug 22 2019
Except for a(0) = 1, these terms M are such that 21 * M = 1M1, where 1M1 denotes the concatenation of 1, M and 1. Actually 21 is A329914(1) and a(1) = A329915(1) = 91, and the terms >=91 form the set {M_21}; for example, 21 * 909091 = 1(909091)1. - Bernard Schott, Dec 01 2019

			Digit-pattern P=[ab..z] repeating integers equal formally with P*(-1+10^(Ln))/(-1+10^L), where L is the length of pattern;
a(9) divides A002275(38) repunit. See A095371.

H. C. Williams and R. K. Guy, Some fourth-order linear divisibility sequences, Intl. J. Number Theory 7 (5) (2011) 1255-1277.
H. C. Williams and R. K. Guy, Some Monoapparitic Fourth Order Linear Divisibility Sequences Integers, Volume 12A (2012) The John Selfridge Memorial Volume.
Index entries for linear recurrences with constant coefficients, signature (101,-100).

Cf. A002275, A095371.
Cf. A001562, A015585, A054416, A097209, A100047, A152577.
Cf. A329914, A329915.

Mathematica
```
Table[1+90*(100^n-1)/99, {n, 0, 20}]
```

a(n) = 1 + 90*(-1 + 100^n)/99 = (10^(2*n+1) + 1)/11. - Rick L. Shepherd, Aug 01 2004
From Colin Barker, Jul 03 2013: (Start)
a(n) = 101*a(n-1) - 100*a(n-2).
G.f.: -(10*x-1)/((x-1)*(100*x-1)). (End)
E.g.f.: exp(x)*(1 + 10*(exp(99*x) - 1)/11). - Elmo R. Oliveira, Mar 15 2025

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

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

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A095370 Number of distinct prime factors of the repunit (-1 + 10^n)/9.

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A095372 1+integers repeating "90" decimal digit pattern.

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

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

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

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