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A180590 Numbers k such that k! is the sum of two triangular numbers.

%I A180590 #27 Apr 25 2025 08:46:12
%S A180590 0,1,2,3,4,5,7,8,9,10,13,15,16,17,21,24,27,28,29,32,33,34,42,49,54,59,
%T A180590 66,68,72,79,80,81,85,86,95,96,99,102,109,118
%N A180590 Numbers k such that k! is the sum of two triangular numbers.
%C A180590 Numbers k such that there are nonnegative numbers x and y such that x*(x+1)/2 + y*(y+1)/2 = k!. Equivalently, (2x+1)^2 + (2y+1)^2 = 8k! + 2. A necessary and sufficient condition for this is that all the prime factors of 4k!+1 that are congruent to 3 (mod 4) occur to even powers (cf. A001481).
%C A180590 Based on an email from _R. K. Guy_ to the Sequence Fans Mailing List, Sep 10 2010.
%C A180590 See A152089 for further links.
%H A180590 Factor Database, <a href="http://factordb.com/index.php?query=4*z%21%2B1&amp;perpage=200">Factors of the numbers 4z!+1</a>
%e A180590 0! = 1! = T(0) + T(1);
%e A180590 2! = T(1) + T(1);
%e A180590 3! = T(0) + T(3) = T(2) + T(2);
%e A180590 4! = T(2) + T(6);
%e A180590 5! = T(0) + T(15) = T(5) + T(14);
%e A180590 7! = T(45) + T(89);
%e A180590 8! = T(89) + T(269);
%e A180590 9! = T(210) + T(825);
%e A180590 10! = T(665) + T(2610) = T(1770) + T(2030);
%e A180590 13! = T(71504) + T(85680);
%e A180590 15! = T(213384) + T(1603064) = T(299894) + T(1589154);
%e A180590 16! = T(3631929) + T(5353005);
%e A180590 17! = T(12851994) + T(23370945) = T(17925060) + T(19750115);
%e A180590 etc.
%t A180590 triQ[n_] := IntegerQ@ Sqrt[8 n + 1]; fQ[n_] := Block[{k = 1, lmt = Floor@Sqrt[2*n! ], nf = n!}, While[k < lmt && ! triQ[nf - k (k + 1)/2], k++ ]; r = (Sqrt[8*(nf - k (k + 1)/2) + 1] - 1)/2; Print[{k, r, n}]; If[IntegerQ@r, True, False]]; k = 1; lst = {}; While[k < 69, If[ fQ@ k, AppendTo[lst, k]]; k++ ]; lst
%o A180590 (Python)
%o A180590 from math import factorial
%o A180590 from itertools import count, islice
%o A180590 from sympy import factorint
%o A180590 def A180590_gen(): # generator of terms
%o A180590     return filter(lambda n:all(p & 3 != 3 or e & 1 == 0 for p, e in factorint(4*factorial(n)+1).items()),count(0))
%o A180590 A180590_list = list(islice(A180590_gen(),15)) # _Chai Wah Wu_, Jun 27 2022
%Y A180590 Complement of A152089. A171099 gives the number of solutions.
%Y A180590 Cf. A000142, A000217, A051533, A076680.
%K A180590 nonn,more
%O A180590 1,3
%A A180590 _Robert G. Wilson v_, Sep 10 2010
%E A180590 Edited by _N. J. A. Sloane_, Sep 24 2010
%E A180590 69 eliminated (see A152089) by _N. J. A. Sloane_, Sep 24 2010
%E A180590 Extended by _Georgi Guninski_ and _D. S. McNeil_, Sep 24 2010
%E A180590 a(35)-a(38) from _Georgi Guninski_, Oct 12 2010
%E A180590 a(39)-a(40) from _Tyler Busby_, Apr 24 2025