Problem 647
Linear Transformations of Polygonal Numbers
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 $\frac 1 2 n(n(k-2)+4-k)$ where $n\ge1$. For example when $k=3$ we get $\frac 1 2 n(n+1)$ 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 $\sum_kF_k(10^3)=14993$ where the sum is over all odd $k=3,5,7,\ldots$.
Find $\sum_kF_k(10^{12})$ where the sum is over all odd $k=3,5,7,\ldots$.
多边形数线性变换
我们能够找到一组正整数$A$和$B$，使得对于任意三角形数$T_n$，其线性变换$AT_n+B$仍然是三角形数。对于所有满足$\max(A,B)\le N$的这类正整数对$(A,B)$，记$(A+B)$之和为$F_3(N)$。
在三角形数之上，我们可以考虑更一般的多边形数，比如$k$边形数。第$n$个$k$边形数的公式为$\frac 1 2 n(n(k-2)+4-k)$，其中$n\ge1$。例如，取$k=3$，上述公式就简化为$\frac 1 2 n(n+1)$，就是三角形数的公式。
对于五边形数、七边形数，进而任意$k$为奇数的$k$边形数，我们都能找到如上所述的数对。比如说，当$k=5$时，我们能够找到一组正整数$A$和$B$，使得对于任意五边形数$P_n$，其线性变换$AP_n+B$仍然是五边形数。同样地，对于所有满足$\max(A,B)\le N$的这类正整数对$(A,B)$，记$(A+B)$之和为$F_5(N)$。
类似地，我们可以对任意奇数$k$定义函数$F_k(N)$。已知，仅对奇数$k=3,5,7,\ldots$求和时，$\sum_kF_k(10^3)=14993$。
仅对奇数$k=3,5,7,\ldots$求和，求$\sum_kF_k(10^{12})$。