Conjectured Periods (or Starting Places where it becomes divisible by) of Dan Romik's Theta3 sequence By Shalosh B. Ekhad Here are the conjectured periods of the sequence d(n), modulo prime powers \ when p mod 4=1, and starting places where it 0 when p mod 4=3 . The sequence d(n) is defined and described in Dan Romik's nice article "The Taylor coefficients of the Jacobi theta constant theta_3" Modulo , 3, to the power, 1, it starta being zero at: , 3 --------------------------------------- Modulo , 3, to the power, 2, it starta being zero at: , 5 --------------------------------------- Modulo , 3, to the power, 3, it starta being zero at: , 7 --------------------------------------- Modulo , 3, to the power, 4, it starta being zero at: , 7 --------------------------------------- Modulo , 3, to the power, 5, it starta being zero at: , 12 --------------------------------------- Modulo , 3, to the power, 6, it starta being zero at: , 14 --------------------------------------- Modulo , 3, to the power, 7, it starta being zero at: , 14 --------------------------------------- Modulo , 3, to the power, 8, it starta being zero at: , 16 --------------------------------------- Modulo , 3, to the power, 9, it starta being zero at: , 21 --------------------------------------- Modulo , 3, to the power, 10, it starta being zero at: , 21 --------------------------------------- Modulo , 3, to the power, 11, it starta being zero at: , 21 --------------------------------------- Modulo , 3, to the power, 12, it starta being zero at: , 23 --------------------------------------- Modulo , 3, to the power, 13, it starta being zero at: , 25 --------------------------------------- Modulo , 3, to the power, 14, it starta being zero at: , 30 --------------------------------------- Modulo , 3, to the power, 15, it starta being zero at: , 32 --------------------------------------- Modulo , 3, to the power, 16, it starta being zero at: , 34 --------------------------------------- Modulo , 3, to the power, 17, it starta being zero at: , 34 --------------------------------------- Modulo , 3, to the power, 18, it starta being zero at: , 39 --------------------------------------- Modulo , 3, to the power, 19, it starta being zero at: , 41 --------------------------------------- Modulo , 3, to the power, 20, it starta being zero at: , 41 --------------------------------------- Modulo , 3, to the power, 21, it starta being zero at: , 41 --------------------------------------- Modulo , 3, to the power, 22, it starta being zero at: , 43 --------------------------------------- Modulo , 3, to the power, 23, it starta being zero at: , 48 --------------------------------------- Modulo , 3, to the power, 24, it starta being zero at: , 48 --------------------------------------- Modulo , 3, to the power, 25, it starta being zero at: , 50 --------------------------------------- Modulo , 3, to the power, 26, it starta being zero at: , 52 --------------------------------------- Modulo , 3, to the power, 27, it starta being zero at: , 57 --------------------------------------- Modulo , 3, to the power, 28, it starta being zero at: , 59 --------------------------------------- Modulo , 3, to the power, 29, it starta being zero at: , 61 --------------------------------------- Modulo , 3, to the power, 30, it starta being zero at: , 61 --------------------------------------- Modulo , 3, to the power, 31, it starta being zero at: , 61 --------------------------------------- Modulo , 3, to the power, 32, it starta being zero at: , 61 --------------------------------------- Modulo , 3, to the power, 33, it starta being zero at: , 66 --------------------------------------- Modulo , 3, to the power, 34, it starta being zero at: , 68 --------------------------------------- Modulo , 3, to the power, 35, it starta being zero at: , 68 --------------------------------------- Modulo , 3, to the power, 36, it starta being zero at: , 70 --------------------------------------- Modulo , 3, to the power, 37, it starta being zero at: , 75 --------------------------------------- Modulo , 3, to the power, 38, it starta being zero at: , 75 --------------------------------------- Modulo , 3, to the power, 39, it starta being zero at: , 77 --------------------------------------- Modulo , 3, to the power, 40, it starta being zero at: , 79 --------------------------------------- Modulo , 3, to the power, 41, it starta being zero at: , 84 --------------------------------------- Modulo , 3, to the power, 42, it starta being zero at: , 86 --------------------------------------- Modulo , 3, to the power, 43, it starta being zero at: , 88 --------------------------------------- Modulo , 3, to the power, 44, it starta being zero at: , 88 --------------------------------------- Modulo , 3, to the power, 45, it starta being zero at: , 93 --------------------------------------- Modulo , 3, to the power, 46, it starta being zero at: , 95 --------------------------------------- Modulo , 3, to the power, 47, it starta being zero at: , 95 --------------------------------------- Modulo , 3, to the power, 48, it starta being zero at: , 97 --------------------------------------- Modulo , 3, to the power, 49, it starta being zero at: , 102 --------------------------------------- Modulo , 3, to the power, 50, it starta being zero at: , 102 --------------------------------------- Modulo , 3, to the power, 51, it starta being zero at: , 102 --------------------------------------- Modulo , 3, to the power, 52, it starta being zero at: , 104 --------------------------------------- Modulo , 3, to the power, 53, it starta being zero at: , 106 --------------------------------------- Modulo , 3, to the power, 54, it starta being zero at: , 111 --------------------------------------- Modulo , 3, to the power, 55, it starta being zero at: , 113 --------------------------------------- Modulo , 3, to the power, 56, it starta being zero at: , 115 --------------------------------------- Modulo , 3, to the power, 57, it starta being zero at: , 115 --------------------------------------- Modulo , 3, to the power, 58, it starta being zero at: , 120 --------------------------------------- Modulo , 3, to the power, 59, it starta being zero at: , 122 --------------------------------------- Modulo , 3, to the power, 60, it starta being zero at: , 122 --------------------------------------- Modulo , 3, to the power, 61, it starta being zero at: , 122 --------------------------------------- Modulo , 3, to the power, 62, it starta being zero at: , 122 --------------------------------------- Modulo , 3, to the power, 63, it starta being zero at: , 124 --------------------------------------- Modulo , 3, to the power, 64, it starta being zero at: , 129 --------------------------------------- Modulo , 3, to the power, 65, it starta being zero at: , 129 --------------------------------------- Modulo , 3, to the power, 66, it starta being zero at: , 131 --------------------------------------- Modulo , 3, to the power, 67, it starta being zero at: , 133 --------------------------------------- Modulo , 3, to the power, 68, it starta being zero at: , 138 --------------------------------------- Modulo , 3, to the power, 69, it starta being zero at: , 140 --------------------------------------- Modulo , 3, to the power, 70, it starta being zero at: , 142 --------------------------------------- Modulo , 3, to the power, 71, it starta being zero at: , 142 --------------------------------------- Modulo , 3, to the power, 72, it starta being zero at: , 142 --------------------------------------- Modulo , 3, to the power, 73, it starta being zero at: , 147 --------------------------------------- Modulo , 3, to the power, 74, it starta being zero at: , 149 --------------------------------------- Modulo , 3, to the power, 75, it starta being zero at: , 149 --------------------------------------- Modulo , 3, to the power, 76, it starta being zero at: , 151 --------------------------------------- Modulo , 3, to the power, 77, it starta being zero at: , 156 --------------------------------------- Modulo , 3, to the power, 78, it starta being zero at: , 156 --------------------------------------- Modulo , 3, to the power, 79, it starta being zero at: , 158 --------------------------------------- Modulo , 3, to the power, 80, it starta being zero at: , 160 --------------------------------------- Modulo , 3, to the power, 81, it starta being zero at: , 165 --------------------------------------- Modulo , 3, to the power, 82, it starta being zero at: , 167 --------------------------------------- Modulo , 3, to the power, 83, it starta being zero at: , 169 --------------------------------------- Modulo , 3, to the power, 84, it starta being zero at: , 169 --------------------------------------- Modulo , 3, to the power, 85, it starta being zero at: , 174 --------------------------------------- Modulo , 3, to the power, 86, it starta being zero at: , 176 --------------------------------------- Modulo , 3, to the power, 87, it starta being zero at: , 176 --------------------------------------- Modulo , 3, to the power, 88, it starta being zero at: , 178 --------------------------------------- Modulo , 3, to the power, 89, it starta being zero at: , 183 --------------------------------------- Modulo , 3, to the power, 90, it starta being zero at: , 183 --------------------------------------- Modulo , 3, to the power, 91, it starta being zero at: , 183 --------------------------------------- Modulo , 3, to the power, 92, it starta being zero at: , 183 --------------------------------------- Modulo , 3, to the power, 93, it starta being zero at: , 183 --------------------------------------- Modulo , 3, to the power, 94, it starta being zero at: , 185 --------------------------------------- Modulo , 3, to the power, 95, it starta being zero at: , 187 --------------------------------------- --------------------------------------- --------------------------------------- Modulo , 5, to the power, 1, the period is [1, 4] of length, 2 --------------------------------------- Modulo , 5, to the power, 2, the period is [1, 24] of length, 2 --------------------------------------- Modulo , 5, to the power, 3, the period is [1, 124, 51, 99, 51, 124, 1, 74, 26, 74] of length, 10 --------------------------------------- Modulo , 5, to the power, 4, the period is [1, 624, 51, 224, 51, 124, 126, 199, 401, 324, 376, 374, 176, 224, 551, 374, 376, 74, 401, 449, 126, 124, 301, 224, 426, 624, 1, 574, 401, 574, 501, 499, 426, 224, 301, 249, 251, 449, 401, 74, 251, 249, 551, 224, 176, 499, 501, 324, 401, 199] of length, 50 --------------------------------------- --------------------------------------- --------------------------------------- Modulo , 7, to the power, 1, it starta being zero at: , 13 --------------------------------------- Modulo , 7, to the power, 2, it starta being zero at: , 25 --------------------------------------- Modulo , 7, to the power, 3, it starta being zero at: , 37 --------------------------------------- Modulo , 7, to the power, 4, it starta being zero at: , 49 --------------------------------------- Modulo , 7, to the power, 5, it starta being zero at: , 62 --------------------------------------- Modulo , 7, to the power, 6, it starta being zero at: , 74 --------------------------------------- Modulo , 7, to the power, 7, it starta being zero at: , 86 --------------------------------------- Modulo , 7, to the power, 8, it starta being zero at: , 86 --------------------------------------- Modulo , 7, to the power, 9, it starta being zero at: , 111 --------------------------------------- Modulo , 7, to the power, 10, it starta being zero at: , 123 --------------------------------------- Modulo , 7, to the power, 11, it starta being zero at: , 135 --------------------------------------- Modulo , 7, to the power, 12, it starta being zero at: , 147 --------------------------------------- Modulo , 7, to the power, 13, it starta being zero at: , 160 --------------------------------------- Modulo , 7, to the power, 14, it starta being zero at: , 172 --------------------------------------- Modulo , 7, to the power, 15, it starta being zero at: , 172 --------------------------------------- Modulo , 7, to the power, 16, it starta being zero at: , 184 --------------------------------------- --------------------------------------- --------------------------------------- Modulo , 11, to the power, 1, it starta being zero at: , 31 --------------------------------------- Modulo , 11, to the power, 2, it starta being zero at: , 61 --------------------------------------- Modulo , 11, to the power, 3, it starta being zero at: , 91 --------------------------------------- Modulo , 11, to the power, 4, it starta being zero at: , 121 --------------------------------------- Modulo , 11, to the power, 5, it starta being zero at: , 152 --------------------------------------- Modulo , 11, to the power, 6, it starta being zero at: , 182 --------------------------------------- --------------------------------------- --------------------------------------- Modulo , 13, to the power, 1, the period is [1, 12, 12, 4, 9, 9, 3, 10, 10, 12, 1, 1, 9, 4, 4, 10, 3, 3] of length, 18 --------------------------------------- --------------------------------------- --------------------------------------- Modulo , 17, to the power, 1, the period is [1, 16, 0, 16, 15, 2, 0, 2, 4, 13, 0, 13, 9, 8, 0, 8, 16, 1, 0, 1, 2, 15, 0, 15, 13, 4, 0, 4, 8, 9, 0, 9] of length, 32 --------------------------------------- --------------------------------------- --------------------------------------- Modulo , 19, to the power, 1, it starta being zero at: , 91 --------------------------------------- Modulo , 19, to the power, 2, it starta being zero at: , 181 --------------------------------------- --------------------------------------- ---------------------------------------