## Saturday, 16 March 2013

### SATURDAY, 16 MARCH 2013

Today is the $75^{th}$ day of the year.

$75 = 3 \times 5^2$

According to A036378 there are $75$ primes between $2^9 = 512$ and $2^{10} = 1024$
The $75$ primes are:
________ ________ ________
$1) 521$$2) 523$$3) 541$
$4) 547$$5) 557$$6) 563$
$7) 569$$8) 571$$9) 577$
$10) 587$$11) 593$$12) 599$
$13) 601$$14) 607$$15) 613$
$16) 617$$17) 619$$18) 631$
$19) 641$$20) 643$$21) 647$
$22) 653$$23) 659$$24) 661$
$25) 673$$26) 677$$27) 683$
$28) 691$$29) 701$$30) 709$
$31) 719$$32) 727$$33) 733$
$34) 739$$35) 743$$36) 751$
$37) 757$$38) 761$$39) 769$
$40) 773$$41) 787$$42) 797$
$43) 809$$44) 811$$45) 821$
$46) 823$$47) 827$$48) 829$
$49) 839$$50) 853$$51) 857$
$52) 859$$53) 863$$54) 877$
$55) 881$$56) 883$$57) 887$
$58) 907$$59) 911$$60) 919$
$61) 929$$62) 937$$63) 941$
$64) 947$$65) 953$$66) 967$
$67) 971$$68) 977$$69) 983$
$70) 991$$71) 997$$72) 1009$
$73) 1013$$74) 1019$$75) 1021$

Consider writing down a sequence using the following rules
1) Write down the first odd non-negative integer, $1$
2) Write down the next two even numbers, $2, 4$
3) Write down the next three odd numbers, $5, 7, 9$
4) Write down the next four even numbers $10, 12, 14, 16$
5) Write down the next five odd numbers $17, 19, 21, 23, 25$
6) Well, you get the idea.

The first $47$ members of this sequence looks like this:
$1, 2, 4, 5, 7, 9, 10, 12, 14, 16, 17, 19, 21, 23, 25, 26, 28, 30, 32, 34, 36, 37, 39, 41, 43, 45, 47, 49, 50, 52, 54, 56, 58, 60, 62, 64, 65, 67, 69, 71, 73, 75, 77, 79, 81$

As you can see $75$ is the $42^{nd}$ member, of what is known as the Connell Sequence, see A001614.

The formula for this sequence is
$a(n) = 2n - \lfloor \frac {1 + \sqrt {8n - 7} } {2} \rfloor$ where $\lfloor \rfloor$ indicates the floor function.

Thus we can calculate the $75^{th}$ member of this function:
$a(75) = (2 \times 75) - \lfloor \frac {1 + \sqrt {(8 \times 75) - 7} } {2} \rfloor$

$a(75) = 150 - \lfloor \frac {1 + \sqrt {600 - 7} } {2} \rfloor$

$a(75) = 150 - \lfloor \frac {1 + \sqrt {593} } {2} \rfloor$

$a(75) = 150 - \lfloor \frac {1 + 24.35159 } {2} \rfloor$

$a(75) = 150 - \lfloor \frac {25.35159 } {2} \rfloor$

$a(75) = 150 - \lfloor 12.67579 \rfloor$

$a(75) = 150 - 12$

$\underline {\underline {a(75) = 138}}$

As an aside note that the number at the end of each sub-sequence above is the square of the index of the sub-sequence, i.e the $5^{th}$ sub-sequence ends in $25$.