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

1,3

Comments

Factoring certain repunits is especially difficult.

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

P. 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'

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

Example

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

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 *)

Prog

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

Crossrefs

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

Cf. A046053 (total number of prime factors).

Sequence in context: A183413 A183380 A260587 * A046053 A368303 A368306

Adjacent sequences: A095367 A095368 A095369 * A095371 A095372 A095373

Keyword

nonn

Author

Labos Elemer, Jun 04 2004; corrected Jun 09 2004

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

Status

approved