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A180590
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Numbers k such that k! is the sum of two triangular numbers.
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2
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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
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OFFSET
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1,3
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COMMENTS
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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.
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LINKS
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EXAMPLE
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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.
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MATHEMATICA
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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
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PROG
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(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))
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CROSSREFS
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KEYWORD
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nonn,more
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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