The Math Game

[Jubal]

9999934275812

[Glaurung]

9999934275812

10^13 - (2^2 * 16,431,047)

360

[Othko97]

10*6^2

e

[Glaurung]

1/(0!) + 1/(1!) + 1/(2!) + 1/(3!) + 1/(4!) + 1/(5!) + 1/(6!) + ... + 1/(n!) + ...
This one was also interesting - thanks, Othko.

-1

[Othko97]

(1/(0!) + 1/(1!) + 1/(2!) + 1/(3!) + 1/(4!) + 1/(5!) + 1/(6!) + ... + 1/(n!) + ...)^(i * pi)

pi

[Glaurung]

4 * (1 - 1/3 + 1/5 - 1/7 + ... + {(-1)ⁿ / (2n + 1)} + ...

And one for this year: 2016
Hint: this last happened in 1953, and won't happen again until 2080.

[Othko97]

Hmm, I didn't realise this was a triangular year.

1+2+3+5+6+7+8+9+10+11+12+13+14+15+16+17+18+19+20+21+22+23+24+25+26+27+28+29+30+31+32+33+34+35+36+37+38+39+40+41+42+43+44+45+46+47+48+49+50+51+52+53+54+55+56+57+58+59+60+61+62+63

Also of note is that 2016 = 741+1275 = (1+2+3+4+5+6+7+8+9+10+11+12+13+14+15+16+17+18+19+20+21+22+23+24+25+26+27+28+29+30+31+32+33+34+35+36+37+38) + (1+2+3+5+6+7+8+9+10+11+12+13+14+15+16+17+18+19+20+21+22+23+24+25+26+27+28+29+30+31+32+33+34+35+36+37+38+39+40+41+42+43+44+45+46+47+48+49+50)


(pi^2)/6

[comrade_general]

[Glaurung]

1/1² + 1/2² + 1/3² + 1/4² + ... + 1/n² + ...
or zeta(2), if you prefer.

I've run out of interesting ideas, so here's something more straightforward:

34

[Othko97]

2(2^4+1)

120

[Jubal]

(12x11)-(3+4+5)

[Glaurung]

Ermm... next number?

[Jubal]

Oops

Have 8573, just because.

[Cuddly Khan]

9000+600+80+4-1111

9001

[Jubal]

Minimum integer value of the Vegeta Rule

Spoiler

8462193