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

Author

Labos Elemer, Jun 04 2004; corrected Jun 09 2004

Keywords

• nonn

Comments

Factoring certain repunits is especially difficult.

Examples

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

References

• 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.

Links

• 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'

Crossrefs

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

Cf. A046053 (total number of prime factors).

Programs

Mathematica

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

PARI

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

Formula

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

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

Extensions

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