Here is a list of values for \(a\) that give an exponent of 4-500:
{-9, -5, -11/3, -3, -13/5, -7/3, -15/7, -2, -17/9, -9/5, -19/11, -5/3, -21/13, -11/7, -23/15, -3/2, -25/17, -13/9, -27/19, -7/5, -29/21, -15/11, -31/23, -4/3, -33/25, -17/13, -35/27, -9/7, -37/29, -19/15, -39/31, -5/4, -41/33, -21/17, -43/35, -11/9, -45/37, -23/19, -47/39, -6/5, -49/41, -25/21, -51/43, -13/11, -53/45, -27/23, -55/47, -7/6, -57/49, -29/25, -59/51, -15/13, -61/53, -31/27, -63/55, -8/7, -65/57, -33/29, -67/59, -17/15, -69/61, -35/31, -71/63, -9/8, -73/65, -37/33, -75/67, -19/17, -77/69, -39/35, -79/71, -10/9, -81/73, -41/37, -83/75, -21/19, -85/77, -43/39, -87/79, -11/10, -89/81, -45/41, -91/83, -23/21, -93/85, -47/43, -95/87, -12/11, -97/89, -49/45, -99/91, -25/23, -101/93, -51/47, -103/95, -13/12, -105/97, -53/49, -107/99, -27/25, -109/101, -55/51, -111/103, -14/13, -113/105, -57/53, -115/107, -29/27, -117/109, -59/55, -119/111, -15/14, -121/113, -61/57, -123/115, -31/29, -125/117, -63/59, -127/119, -16/15, -129/121, -65/61, -131/123, -33/31, -133/125, -67/63, -135/127, -17/16, -137/129, -69/65, -139/131, -35/33, -141/133, -71/67, -143/135, -18/17, -145/137, -73/69, -147/139, -37/35, -149/141, -75/71, -151/143, -19/18, -153/145, -77/73, -155/147, -39/37, -157/149, -79/75, -159/151, -20/19, -161/153, -81/77, -163/155, -41/39, -165/157, -83/79, -167/159, -21/20, -169/161, -85/81, -171/163, -43/41, -173/165, -87/83, -175/167, -22/21, -177/169, -89/85, -179/171, -45/43, -181/173, -91/87, -183/175, -23/22, -185/177, -93/89, -187/179, -47/45, -189/181, -95/91, -191/183, -24/23, -193/185, -97/93, -195/187, -49/47, -197/189, -99/95, -199/191, -25/24, -201/193, -101/97, -203/195, -51/49, -205/197, -103/99, -207/199, -26/25, -209/201, -105/101, -211/203, -53/51, -213/205, -107/103, -215/207, -27/26, -217/209, -109/105, -219/211, -55/53, -221/213, -111/107, -223/215, -28/27, -225/217, -113/109, -227/219, -57/55, -229/221, -115/111, -231/223, -29/28, -233/225, -117/113, -235/227, -59/57, -237/229, -119/115, -239/231, -30/29, -241/233, -121/117, -243/235, -61/59, -245/237, -123/119, -247/239, -31/30, -249/241, -125/121, -251/243, -63/61, -253/245, -127/123, -255/247, -32/31, -257/249, -129/125, -259/251, -65/63, -261/253, -131/127, -263/255, -33/32, -265/257, -133/129, -267/259, -67/65, -269/261, -135/131, -271/263, -34/33, -273/265, -137/133, -275/267, -69/67, -277/269, -139/135, -279/271, -35/34, -281/273, -141/137, -283/275, -71/69, -285/277, -143/139, -287/279, -36/35, -289/281, -145/141, -291/283, -73/71, -293/285, -147/143, -295/287, -37/36, -297/289, -149/145, -299/291, -75/73, -301/293, -151/147, -303/295, -38/37, -305/297, -153/149, -307/299, -77/75, -309/301, -155/151, -311/303, -39/38, -313/305, -157/153, -315/307, -79/77, -317/309, -159/155, -319/311, -40/39, -321/313, -161/157, -323/315, -81/79, -325/317, -163/159, -327/319, -41/40, -329/321, -165/161, -331/323, -83/81, -333/325, -167/163, -335/327, -42/41, -337/329, -169/165, -339/331, -85/83, -341/333, -171/167, -343/335, -43/42, -345/337, -173/169, -347/339, -87/85, -349/341, -175/171, -351/343, -44/43, -353/345, -177/173, -355/347, -89/87, -357/349, -179/175, -359/351, -45/44, -361/353, -181/177, -363/355, -91/89, -365/357, -183/179, -367/359, -46/45, -369/361, -185/181, -371/363, -93/91, -373/365, -187/183, -375/367, -47/46, -377/369, -189/185, -379/371, -95/93, -381/373, -191/187, -383/375, -48/47, -385/377, -193/189, -387/379, -97/95, -389/381, -195/191, -391/383, -49/48, -393/385, -197/193, -395/387, -99/97, -397/389, -199/195, -399/391, -50/49, -401/393, -201/197, -403/395, -101/99, -405/397, -203/199, -407/399, -51/50, -409/401, -205/201, -411/403, -103/101, -413/405, -207/203, -415/407, -52/51, -417/409, -209/205, -419/411, -105/103, -421/413, -211/207, -423/415, -53/52, -425/417, -213/209, -427/419, -107/105, -429/421, -215/211, -431/423, -54/53, -433/425, -217/213, -435/427, -109/107, -437/429, -219/215, -439/431, -55/54, -441/433, -221/217, -443/435, -111/109, -445/437, -223/219, -447/439, -56/55, -449/441, -225/221, -451/443, -113/111, -453/445, -227/223, -455/447, -57/56, -457/449, -229/225, -459/451, -115/113, -461/453, -231/227, -463/455, -58/57, -465/457, -233/229, -467/459, -117/115, -469/461, -235/231, -471/463, -59/58, -473/465, -237/233, -475/467, -119/117, -477/469, -239/235, -479/471, -60/59, -481/473, -241/237, -483/475, -121/119, -485/477, -243/239, -487/479, -61/60, -489/481, -245/241, -491/483, -123/121, -493/485, -247/243, -495/487, -62/61, -497/489, -249/245, -499/491, -125/123, -501/493, -251/247, -503/495, -63/62, -505/497}
There are infinitely more, and their sum diverges.