I just got an idea of sequences that can be created out of each and every positive integer. I would like to start it with an example :
For the number 6 , the sequence would be 2,6,12,20,30,42,56 ........... it is because 6 = 1 x 6 or 6 = 2 x 3 . We could observe that the lowest gap between the factors is in the case of 2 x 3 . Hence if 2 is considered as the variable 'n' , 2 x 3 can also be written as n x (n+1) . Hence , when we substitute 1 in place of 'n' ,it is 2 , if substituted 2 the 2nd term would be 6 , 3rd term would be 12 and so on . Hence the number sequence of 6 is 2,6,12,20,30,42,56 .........
Yes , the next i would like to describe the number polynomial . Again , let me describe it with the number 6 . Hence we know that the factors of 6 are 1,2,3,6 . The polynomial of 6 would be f(x) = 1x^2 + 2x^6 + 3x^12 + 6x^42. Hence the idea is simple . The first term's coefficient would be the first divisor and the power of the term would be the 1st term of the number sequence of 6 i.e. 2 . The second term of the polynomial would be 2 as the second factor is 2 while the power of the term would be the second term in the number sequence of 6 i.e. 6 . Hence the last of the polynomial would have the coefficient as 6 , the last divisor with the power being the 6th term in the sequence of 42.
I hope you will like this idea . Thanks for reading my note and like & reshare the note if you like it !!!