A046413 Numbers k such that the repunit of length k (11...11, with k 1's) has exactly 2 prime factors.

3, 4, 5, 7, 11, 17, 47, 59, 71, 139, 211, 251, 311, 347, 457, 461

Offset: 1

Author

Patrick De Geest, Jul 15 1998

Keywords

• nonn
• base
• more
• hard

Comments

347, 457, 461 and 701 are also terms. The only other possible terms up to 1000 are 263, 311, 509, 557, 617, 647 and 991; repunits of these lengths are known to be composite but the linked sources do not provide their factors. - Rick L. Shepherd, Mar 11 2003

The Yousuke Koide reference now shows the repunit of length 263 partially factored; 263 is no longer a possible candidate for this sequence. - Ray Chandler, Sep 06 2005

The repunit of length 263 has 3 prime factors; the repunit of length 617 has one known prime factor and a large composite. Possible terms > 1000 are 1117, 1213, 1259, 1291, 1373, 1447, 1607, 1637, 1663, 1669, 1759, 1823, 1949, 1987, 2063 & 2087. - Robert G. Wilson v, Apr 26 2010

All terms are either primes or squares of primes in A004023. In particular, the only composite below a million is 4. - Charles R Greathouse IV, Nov 21 2014

a(17) >= 509. The only confirmed term below 2500 is 701. As of July 2019, no factorization is known for the potential terms 509, 557, 647, 991, 1117, 1259, 1447, 1607, 1637, 1663, 1669, 1759, 1823, 1949, 1987, 2063, 2087, 2111, 2203, 2269, 2309, 2341, 2467, 2503, 2521, ... Unless the least prime factors of the respective composites have fewer than ~80 decimal digits and are thus accessible by massive ECM computations, there is no chance for an extension using current publicly available factorization methods. See links to factordb.com for the status of the factorization of the smallest unknown terms. - Hugo Pfoertner, Jul 30 2019

Examples

			7 is a term because 1111111 = 239*4649.
		

References

• Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 60.

Links

• Patrick De Geest, Repunits prime factors.
• Shyam Sunder Gupta, Repunit Numbers, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 11, 327-352.
• Makoto Kamada, Factorizations of 11...11 (Repunit).
• Yousuke Kiode, Factorizations of Repunit Numbers.
• Eric Weisstein's World of Mathematics, Repunit.
• Status of (10^509-1)/9 in factordb.com.
• Status of (10^557-1)/9 in factordb.com.
• Status of (10^647-1)/9 in factordb.com.

Crossrefs

Cf. A000042, A004022 (repunit primes), A046053, A102782.

Programs

Mathematica

Select[Range[60],PrimeOmega[FromDigits[PadRight[{},#,1]]]==2&] (* The program generates the first 8 terms of the sequence. *) (* Harvey P. Dale, Aug 26 2024 *)

Extensions

More terms from Rick L. Shepherd, Mar 11 2003

a(13)-a(16) from Robert G. Wilson v, Apr 26 2010