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I'm exploring some arithmetic pattern f(a,b) defines as "the sum of all numbers from 1 to a that are not divisible by b" which goes like this:

$$f(4,4) = 1 + 2 + 3 = 6$$ $$f(8,4) = 1 + 2 + 3 + 5 + 6 + 7 = 24$$ $$f(12,4) = 1 + 2 + 3 + 5 + 6 + 7 + 9 + 10 + 11 = 54$$

But some f(a,b) is not unique since it can be expressed in different bases. For instance, f(12,b) can be generated from f(12,2), f(12,3), f(12,4), and f(12,6) where 2,3,4 and 6 divides 12, that is, a|d.

Like

$f(12,2) = 1 + 3 + 5 + 7 + 9 + 11 = 36$

$f(12,3) = 1 + 2 + 4 + 5 + 7 + 8 + 10 + 11 = 48$

$f(12,4) = 1 + 2 + 3 + 5 + 6 + 7 + 9 + 10 + 11 = 54$

$f(12,6) = 1 + 2 + 3 + 4 + 5 + 7 + 8 + 9 + 10 + 11 = 60$

$f(12,12) = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 = 66$

I noticed that it seems that their GCD is 6 but I don't know how it came to that.

I already tried other functions

GCD of $f(4,a|d) = 2$
GCD of $f(6,a|d) = 3$
GCD of $f(8,a|d) = 4$
GCD of $f(9,a|d) = 9$

GCD of $f(10,a|d) = 5$

GCD of $f(18,a|d) = 9$
GCD of $f(25,a|d) = 25$

and I came up with the following observations:

a. If $a$ is composite and even, then $[f(a,a|d)] = a/2$

b. If $a$ is composite and odd, then $[f(a,a|d)] = a$

Is there a way to find GCF of a certain function? Any idea would be much appreciated.

Jerwin Ablay
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  • Please use MathJax to improve readability. – Harish Chandra Rajpoot Nov 27 '22 at 14:35
  • You continue to post questions about this function without reference to the expression of it in terms of the triangular numbers. That expression will help you find properties. You have $f(a,b)=\frac 12a(a+1)-b\cdot \frac12\cdot \frac ba(\frac ba+1)$ – Ross Millikan Nov 27 '22 at 14:38

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