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Linear Transformations of Polygonal Numbers

Published on Sunday, 16th December 2018, 07:00 am; Solved by 295;
Difficulty rating: 30%

Problem 647

It is possible to find positive integers $A$ and $B$ such that given any triangular number, $T_n$, then $AT_n +B$ is always a triangular number. We define $F_3(N)$ to be the sum of $(A+B)$ over all such possible pairs $(A,B)$ with $\max(A,B)\le N$. For example $F_3(100) = 184$.

Polygonal numbers are generalisations of triangular numbers. Polygonal numbers with parameter $k$ we call $k$-gonal numbers. The formula for the $n$th $k$-gonal number is $\frac 12n\big(n(k-2)+4-k\big)$ where $n \ge 1$. For example when $k = 3$ we get $\frac 12n(n+1)$ the formula for triangular numbers.

The statement above is true for pentagonal, heptagonal and in fact any $k$-gonal number with $k$ odd. For example when $k=5$ we get the pentagonal numbers and we can find positive integers $A$ and $B$ such that given any pentagonal number, $P_n$, then $AP_n+B$ is always a pentagonal number. We define $F_5(N)$ to be the sum of $(A+B)$ over all such possible pairs $(A,B)$ with $\max(A,B)\le N$.

Similarly we define $F_k(N)$ for odd $k$. You are given $\sum_{k} F_k(10^3) = 14993$ where the sum is over all odd $k = 3,5,7,\ldots$.

Find $\sum_{k} F_k(10^{12})$ where the sum is over all odd $k = 3,5,7,\ldots$