Sunday, 3 March 2013

SUNDAY, 3 MARCH 2013

Today is the $62^{nd}$ day of the year.

$62 = 1^2 + 5^2 + 6^2 = 2^2 + 3^2 + 7^2$ so it is the sum of three squares in two different ways, see A000408.

$62^2 + 62 + 1 = 3,907 which is prime, see A002384. As can be seen from the list below of the first$24$members of this sequence, there are a lot of numbers that are prime when you square them, add the number and the add one.$1, 2, 3, 5, 6, 8, 12, 14, 15, 17, 20, 21, 24, 27, 33, 38, 41, 50, 54, 57, 59, 62, 66, 69$Consider the sentence that starts, "'T' is the first, ... letter of this sentence". If this sentence is to be self-referential then we need to add in the other Ts in it. There is clearly a second T at the beginning of the word 'the' so we add that. Ignoring spaces and punctuation the second T is located at the fourth character. Thus the sentence now reads "'T' is the first, fourth, ... letter of this sentence". Then there is a third T at character eleven and a fourth T at character sixteen so we can add those details to the sentence so that it becomes "'T' is the first, fourth, eleventh, sixteenth ... letter of this sentence". The sequence of positions of the first$24$Ts is$1, 4, 11, 16, 24, 29, 33, 35, 39, 45, 47, 51, 56, 58, 62, 64, 69, 73, 78, 80, 84, 89, 94, 99\$
This is known as Aronson's sequence, see A005224.