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))
CROSSREFS
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