Friday 29 August 2014

[[{{ institutions of higher learning }}]] and {{ LPFs & GPFs together }}

add 1, multiply by 1, add 2, multiply by 2, etcetera
prime powers in mauve, LPFs in orange with bonus pink squares to reach the GPFs

Wednesday 27 August 2014

An angel at my picnic

The ruler function: 2^a(n) divides 2n. Or, a(n) = 2-adic valuation of 2n.  ( OEIS A001511 )
1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 1, 3, 1, 2, 1, 5, 1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 1, 3, 1, 2, 1, 6, 1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 1, 3, 1, 2, 1, 5, 1, 2, 1, 3, ...
Quarter-squares: floor(n/2)*ceiling(n/2). Equivalently, floor(n^2/4).   ( OEIS A002620 )
0, 0, 1, 2, 4, 6, 9, 12, 16, 20, 25, 30, 36, 42, 49, 56, 64, 72, 81, 90, 100, 110, 121, 132, 144, 156, 169, 182, 196, 210, 225, 240, 256, ...
Some examples :
for n=2, 3, 4, 5, 6, 7, 8, 9, 10  have  1*1,  1*2,  2*2,  2*3,  3*3,  3*4,  4*4,  4*5  and  5*5