Saturday 4 June 2016

Golden Semiprimes

Today I am 24534 days old and when checking this number on OEIS I came across the following:


Not having heard of the term before, I investigated what defined a golden semiprime. Below is the OEIS reference:


In today's case of 24534, the sum of the digits is 18 and the factorisation is 2×3^2×29×47, so n/(sum of digits of n) is 29 x 47. Now the absolute value of  29 x ø - 47 is approximately 0.077 and thus the criterion for a golden semiprime is satisfied. As can be seen from the list in OEIS A108542, such semiprimes are relatively sparse.

The significance of the concept is that it identifies that the fact that the two factors making up the semiprime are very nearly in the golden ratio. Of course the factors cannot be in the exact ratio because ø is not rational. Again, I've learned something quite interesting by keeping track of my daily numbers.

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