A111345 - OEIS

%I #7 Feb 01 2017 15:31:47

%S 4375,19684,7077889,7962625,34012225,100663297,129140164,452984833,

%T 459165025,544195585,644972545,918330049,5159780353,7346640385,

%U 8589934593,13947137605,14495514625,23219011585,27518828545,28991029249

%N Pierpont 5-almost primes. 5-almost primes of form (2^K)*(3^L)+1.

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

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

%F a(n) is in this sequence iff there exist nonnegative integers K and L such that Omega((2^K)*(3^L)+1) = 5.

%e a(1) = 4375 = (2^1)*(3^7)+1 = 5 * 5 * 5 * 5 * 7.

%e a(2) = 19684 = (2^0)*(3^9)+1 = 2 * 2 * 7 * 19 * 37.

%e a(3) = 7077889 = (2^18)*(3^3)+1 = 7 * 13 * 13 * 31 * 193 (prime factors each have all odd digits).

%e a(4) = 7962625 = (2^15)*(3^5)+1 = 5 * 5 * 5 * 11 * 5791 (again, coincidentally, prime factors each have all odd

%e digits).

%e a(7) = 129140164 = (2^0)*(3^17)+1 = 2 * 2 * 103 * 307 * 1021.

%e a(15) = 8589934593 = (2^33)*(3^0)+1 = 3 * 3 * 67 * 683 * 20857.

%e a(21) = 34359738369 = (2^35)*(3^0)+1 = 3 * 11 * 43 * 281 * 86171.

%e a(30) = 793437161473 = (2^11)*(3^18)+1 = 11 * 11 * 11 * 43 * 13863281.

%e a(32) = 847288609444 = (2^0)*(3^25)+1 = 2 * 2 * 61 * 151 * 22996651.

%e a(47) = 68630377364884 = (2^0)*(3^29)+1 = 2 * 2 * 523 * 6091 * 5385997.

%e a(48) = 70368744177665 = (2^46)*(3^0)+1 = 5 * 277 * 1013 * 1657 * 30269.

%e a(81) = 50031545098999708 = (2^0)*(3^35)+1 = 2 * 2 * 61 * 547 * 374857981681.

%e a(89) = 144115188075855873 = (2^57)*(3^0)+1 = 3 * 3 * 571 * 174763 * 160465489.

%e a(99) = 450283905890997364 = (2^0)*(3^37)+1 = 2 * 2 * 18427 * 107671 * 56737873.

%e a(113) = 4611686018427387905 = (2^62)*(3^0)+1 = 5 * 5581 * 8681 * 49477 * 384773.

%o (PARI) list(lim)=my(v=List(), L=lim\1-1); for(e=0, logint(L, 3), my(t=3^e); while(t<=L, if(bigomega(t+1)==5, listput(v, t+1)); t*=2)); Set(v) \\ _Charles R Greathouse IV_, Feb 01 2017

%Y Intersection of A014614 and A055600.

%Y A005109 gives the Pierpont primes, which are primes of the form (2^K)*(3^L)+1.

%Y A113432 gives the Pierpont semiprimes, 2-almost primes of the form (2^K)*(3^L)+1.

%Y A112797 gives the Pierpont 3-almost primes, of the form (2^K)*(3^L)+1.

%Y A111344 gives the Pierpont 4-almost primes, of the form (2^K)*(3^L)+1.

%Y A111346 gives the Pierpont 6-almost primes, of the form (2^K)*(3^L)+1.

%Y A113739 gives the Pierpont 7-almost primes, of the form (2^K)*(3^L)+1.

%Y A113740 gives the Pierpont 8-almost primes, of the form (2^K)*(3^L)+1.

%Y A113741 gives the Pierpont 9-almost primes, of the form (2^K)*(3^L)+1.

%K easy,nonn

%O 1,1

%A _Jonathan Vos Post_, Nov 08 2005

%E Extended by _Ray Chandler_, Nov 08 2005