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 A180590 Numbers k such that k! is the sum of two triangular numbers. 2
 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, 66, 68, 72, 79, 80, 81, 85, 86, 95, 96, 99, 102 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS 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). Based on an email from R. K. Guy to the Sequence Fans Mailing List, Sep 10 2010. See A152089 for further links. LINKS Factor Database, Factors of the numbers 4z!+1 EXAMPLE 0! = 1! = T(0) + T(1); 2! = T(1) + T(1); 3! = T(0) + T(3) = T(2) + T(2); 4! = T(2) + T(6); 5! = T(0) + T(15) = T(5) + T(14); 7! = T(45) + T(89); 8! = T(89) + T(269); 9! = T(210) + T(825); 10! = T(665) + T(2610) = T(1770) + T(2030); 13! = T(71504) + T(85680); 15! = T(213384) + T(1603064) = T(299894) + T(1589154); 16! = T(3631929) + T(5353005); 17! = T(12851994) + T(23370945) = T(17925060) + T(19750115); etc. MATHEMATICA 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 PROG (Python) from math import factorial from itertools import count, islice from sympy import factorint def A180590_gen(): # generator of terms     return filter(lambda n:all(p & 3 != 3 or e & 1 == 0 for p, e in factorint(4*factorial(n)+1).items()), count(0)) A180590_list = list(islice(A180590_gen(), 15)) # Chai Wah Wu, Jun 27 2022 CROSSREFS A171099 gives the number of solutions. Cf. A000142, A000217, A001481, A076680. Complement of A152089. Sequence in context: A175020 A050728 A230999 * A289342 A286302 A299767 Adjacent sequences:  A180587 A180588 A180589 * A180591 A180592 A180593 KEYWORD nonn,more AUTHOR Robert G. Wilson v, Sep 10 2010 EXTENSIONS Edited by N. J. A. Sloane, Sep 24 2010 69 eliminated (see A152089) by N. J. A. Sloane, Sep 24 2010 Extended by Georgi Guninski and D. S. McNeil, Sep 24 2010 a(35)-a(38) from Georgi Guninski, Oct 12 2010 STATUS approved

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Last modified October 2 11:46 EDT 2022. Contains 357205 sequences. (Running on oeis4.)