A046413 - OEIS

%I #56 Aug 26 2024 16:57:08

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

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

%C 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

%C 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

%C 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

%C 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

%C 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

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

%H Patrick De Geest, <a href="http://www.worldofnumbers.com/repunits.htm">Repunits prime factors</a>.

%H Makoto Kamada, <a href="https://stdkmd.net/nrr/repunit">Factorizations of 11...11 (Repunit)</a>.

%H Yousuke Kiode, <a href="https://repunit-koide.jimdofree.com/">Factorizations of Repunit Numbers</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Repunit.html">Repunit</a>.

%H <a href="http://factordb.com/index.php?query=%2810%5E509-1%29%2F9">Status of (10^509-1)/9 in factordb.com</a>.

%H <a href="http://factordb.com/index.php?query=%2810%5E557-1%29%2F9">Status of (10^557-1)/9 in factordb.com</a>.

%H <a href="http://factordb.com/index.php?query=%2810%5E647-1%29%2F9">Status of (10^647-1)/9 in factordb.com</a>.

%e 7 is a term because 1111111 = 239*4649.

%t ange[60],PrimeOmega[FromDigits[PadRight[{},#,1]]]==2&] (* The program generates the first 8 terms of the sequence. *) (* _Harvey P. Dale_, Aug 26 2024 *)

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

%K nonn,base,more,hard

%O 1,1

%A _Patrick De Geest_, Jul 15 1998

%E More terms from _Rick L. Shepherd_, Mar 11 2003

%E a(13)-a(16) from _Robert G. Wilson v_, Apr 26 2010