cp's OEIS Frontend

A099497 Numbers k such that A007925(k) = k^(k+1) - (k+1)^k is a semiprime.

%I A099497 #25 Jul 08 2023 18:30:14
%S A099497 5,7,8,11,17,18,21,23,25,27,32,47,51,56,59,165
%N A099497 Numbers k such that A007925(k) = k^(k+1) - (k+1)^k is a semiprime.
%C A099497 a(15)=59 confirmed by the factorization of 59^60 - 60^59, which is the product of the 52-digit prime 1994803969065168661575061125592557043358338451845483 and the 55-digit prime 8529249434913526091880095870250840825853220069057672947.
%C A099497 The next term is >= 182. - _Hugo Pfoertner_, Jul 18 2019
%H A099497 Dario Alpern, <a href="https://www.alpertron.com.ar/ECM.HTM">Factorization using the Elliptic Curve Method</a>.
%H A099497 factordb.com, <a href="http://factordb.com/index.php?query=182%5E183-183%5E182">Status of 182^183-183^182</a>.
%H A099497 Jason Papadopoulos, <a href="https://sourceforge.net/projects/msieve/">MSIEVE: A Library for Factoring Large Integers</a>.
%e A099497 a(1) = 5 because 5^6 - 6^5 = 7849 = 47*167 is a semiprime.
%e A099497 a(1) = 5 because 5^6 - 6^5 = 47*167
%e A099497 a(2) = 7 because 7^8 - 8^7 = 23*159463
%e A099497 a(3) = 8 because 8^9 - 9^8 = 257*354751
%e A099497 a(4) = 11 because 11^12 - 12^11 = 33479*71549927
%e A099497 a(5) = 17 because 17^18 - 18^17 = 443881*26757560905578361
%e A099497 a(6) = 18 because 18^19 - 19^18 = 100417*6015993258685545623
%e A099497 a(7) = 21 because 21^22 - 22^21 = 10745792197529*9973660056412561
%e A099497 a(8) = 23 because 23^24 - 24^23 = 92798617729*4576344458074395243073
%e A099497 a(9) = 25 because 25^26 - 26^25 = 1627*1219220786258356172077730898121187
%e A099497 a(10) = 27 because 27^28 - 28^27 = 12298336501553*877252504725615101634783073
%e A099497 a(11) = 32 because 32^33 - 33^32 = 3506869732968391733353*12220478717670771804763962407
%e A099497 a(12) = 47 because 47^48 - 48^47 = 11*15621013371424880252957237277868559270462038147831682437840584991339231377934499
%e A099497 a(13) = 51 because 51^52 - 52^51 = 10562756058978342869988055703171*5575962824795589360993690554534422732411612977322491058843
%e A099497 a(14) = 56 because 56^57 - 57^56 = 5*843980334169667457302970806376511482920948635540290643213973523914715036518308339240201775858865907
%e A099497 a(15) = 59 because 59^60 - 60^59 = 1994803969065168661575061125592557043358338451845483*8529249434913526091880095870250840825853220069057672947
%e A099497 a(16) = 165 because 165^166 - 166^165 = 7633959407*16307690786821361595026621717879347561301150483781862339651556401266189322630373265190696672506475741217560239791446654891805648807872536646884416611251422684856600732984767987061831649144878649678190762809385448362714901584206533854093359279076584767352259587745683931159999248465944943129517543272252180930134912057221968601458271001580745436226192252814407
%Y A099497 Cf. A007925 (n^(n+1)-(n+1)^n), A072179 (k^(k+1)-(k+1)^k is prime), A099498 (semiprimes of the form k^(k+1)-(k+1)^k).
%K A099497 nonn,hard,more
%O A099497 1,1
%A A099497 _Hugo Pfoertner_, Oct 19 2004, Aug 13 2007
%E A099497 165 from Herman Jamke (hermanjamke(AT)fastmail.fm), Jan 12 2008