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Semiprimes in A054567.
4

%I #10 Sep 08 2022 08:45:23

%S 6,69,106,265,334,411,589,799,1041,1174,1315,1959,2329,3394,4659,5221,

%T 5815,7099,8146,8511,10869,16449,21979,23181,23794,25681,26326,31774,

%U 33949,35439,36961,38515,40101,43369,45051,48511,50289,52099,54874

%N Semiprimes in A054567.

%C This sequence, A113692, contains semiprimes from the center straight left along the x-axis in the semiprime spiral of A113688-A113689. A113690 contains semiprimes from the center straight right along the x-axis in the semiprime spiral. A113691 contains semiprimes from the center straight down the y-axis in the semiprime spiral. A113693 contains semiprimes from the center straight up the y-axis in the semiprime spiral.

%H Vincenzo Librandi, <a href="/A113692/b113692.txt">Table of n, a(n) for n = 1..1000</a>

%F {a(n)} = Intersection of A001358 and A054567. Semiprimes of the form 4*k^2 - 7*k + 4.

%e a(4) = 4*9^2 - 7*9 + 4 = 265 = 5 * 53.

%e a(5) = 4*10^2 - 7*10 + 4 = 334 = 2 * 167.

%e a(6) = 4*11^2 - 7*11 + 4 = 411 = 3 * 137.

%e a(4), a(5) and a(6) are horizontally adjacent in the semiprime spiral, hence part of a clump and not isolated semiprimes as in A113688. a(9), a(10) and a(11) are another such horizontal string of 3 adjacent semiprimes.

%e a(46) = 4*151^2 - 7*151 + 4 = 90151 = 17 * 5303 is the greatest member under 10^5 (it is coincidence that this integer ends, base 10, with the same 151 that is the index of the quadratic).

%t Select[Table[4*n^2 - 7*n + 4, {n, 200}], PrimeOmega[#] == 2&] (* _Vincenzo Librandi_, Sep 22 2012 *)

%o (Magma) IsSemiprime:= func<n | &+[d[2]: d in Factorization(n)] eq 2>; [s: n in [2..120] | IsSemiprime(s) where s is 4*n^2 - 7*n + 4]; // _Vincenzo Librandi_, Sep 22 2012

%Y Cf. A001358, A054567, A113688-A113699.

%K easy,nonn

%O 1,1

%A _Jonathan Vos Post_, Nov 05 2005