180611

Ruby


A305756(1)

-3456<=ｂ<=3456 のときの
(q*(j(q)+b))^(1/f(b)) に対して、
conjecture が正しいことを確認してみよう。

require 'prime'

def power0(a, n)
  return 1 if n == 0
  k = power0(a, n >> 1)
  k *= k
  return k if n & 1 == 0
  return k * a
end

# x > 0
def sigma(x, i)
  sum = 1
  pq = i.prime_division
  pq.each{|a, n| sum *= (power0(a, (n + 1) * x) - 1) / (power0(a, x) - 1)}
  sum
end

def A000521(n)
  s3 = [0] + (1..n + 1).map{|i| sigma(3, i)}
  s5 = [0] + (1..n + 1).map{|i| sigma(5, i)}
  ary = [1]
  (0..n).each{|i| ary << (1..i + 1).inject(0){|s, j| s + (504 * s5[j] - 240 * (i - j) * s3[j]) * ary[-j]} / (i + 1)}
  ary
end

def A008683(n)
  ary = n.prime_division
  return (-1) ** (ary.size % 2) if ary.all?{|i| i[1] == 1}
  0
end

# ary[0] = 1
def inverse_Euler_transform(ary, n)
  c = [0]
  (1..n).each{|i| c << (1..i - 1).inject(i * ary[i]){|s, j| s - ary[j] * c[-j]}}
  m_ary = [0] + (1..n).map{|i| A008683(i)}
  a = [0]
  (1..n).each{|i|
    s = 0
    (1..i).each{|j|
      s += m_ary[i / j] * c[j] if i % j == 0
    }
    # aryの要素が整数なら、s / iは整数
    a << s / i
  }
  a
end

def A(ary, n)
  a = inverse_Euler_transform(ary, n)
  g = a[0].gcd(a[1])
  (2..n).each{|i| g = g.gcd(a[i])}
  g
end

def A305762(n)
  n = -n if n < 0
  return 24 if n == 0
  s = 1
  s *= 3 if n % 9 == 0
  n.prime_division.each{|i|
    s *= 2 ** [3, (i[1] - 1)].min if i[0] == 2
  }
  s
end

n = 25
N = 3456
a = A000521(n)
ary0 = []
ary1 = []
(-N).upto(N){|i|
  a[1] = 744 + i
  j = A(a, n)
  ary0 << [i, j]
  ary1 << j
}
p ary0
ary2 = (-N).upto(N).map{|i| A305762(i)}
# 予想が正しければ、true
p ary1 == ary2

出力結果
[[-3456, 24], [-3455, 1], [-3454, 1], [-3453, 1], [-3452, 2], [-3451, 1], [-3450, 1], [-3449, 1], [-3448, 4], [-3447, 3], [-3446, 1], [-3445, 1], [-3444, 2], [-3443, 1], [-3442, 1], [-3441, 1], [-3440, 8], [-3439, 1], [-3438, 3], [-3437, 1], [-3436, 2], [-3435, 1], [-3434, 1], [-3433, 1], [-3432, 4], [-3431, 1], [-3430, 1], [-3429, 3], [-3428, 2], [-3427, 1], [-3426, 1], [-3425, 1], [-3424, 8], [-3423, 1], [-3422, 1], [-3421, 1], [-3420, 6], [-3419, 1], [-3418, 1], [-3417, 1], [-3416, 4], [-3415, 1], [-3414, 1], [-3413, 1], [-3412, 2], [-3411, 3], [-3410, 1], [-3409, 1], [-3408, 8], [-3407, 1], [-3406, 1], [-3405, 1], [-3404, 2], [-3403, 1], [-3402, 3], [-3401, 1], [-3400, 4], [-3399, 1], [-3398, 1], [-3397, 1], [-3396, 2], [-3395, 1], [-3394, 1], [-3393, 3], [-3392, 8], [-3391, 1], [-3390, 1], [-3389, 1], [-3388, 2], [-3387, 1], [-3386, 1], [-3385, 1], [-3384, 12], [-3383, 1], [-3382, 1], [-3381, 1], [-3380, 2], [-3379, 1], [-3378, 1], [-3377, 1], [-3376, 8], [-3375, 3], [-3374, 1], [-3373, 1], [-3372, 2], [-3371, 1], [-3370, 1], [-3369, 1], [-3368, 4], [-3367, 1], [-3366, 3], [-3365, 1], [-3364, 2], [-3363, 1], [-3362, 1], [-3361, 1], [-3360, 8], [-3359, 1], [-3358, 1], [-3357, 3], [-3356, 2], [-3355, 1], [-3354, 1], [-3353, 1], [-3352, 4], [-3351, 1], [-3350, 1], [-3349, 1], [-3348, 6], [-3347, 1], [-3346, 1], [-3345, 1], [-3344, 8], [-3343, 1], [-3342, 1], [-3341, 1], [-3340, 2], [-3339, 3], [-3338, 1], [-3337, 1], [-3336, 4], [-3335, 1], [-3334, 1], [-3333, 1], [-3332, 2], [-3331, 1], [-3330, 3], [-3329, 1], [-3328, 8], [-3327, 1], [-3326, 1], [-3325, 1], [-3324, 2], [-3323, 1], [-3322, 1], [-3321, 3], [-3320, 4], [-3319, 1], [-3318, 1], [-3317, 1], [-3316, 2], [-3315, 1], [-3314, 1], [-3313, 1], [-3312, 24], [-3311, 1], [-3310, 1], [-3309, 1], [-3308, 2], [-3307, 1], [-3306, 1], [-3305, 1], [-3304, 4], [-3303, 3], [-3302, 1], [-3301, 1], [-3300, 2], [-3299, 1], [-3298, 1], [-3297, 1], [-3296, 8], [-3295, 1], [-3294, 3], [-3293, 1], [-3292, 2], [-3291, 1], [-3290, 1], [-3289, 1], [-3288, 4], [-3287, 1], [-3286, 1], [-3285, 3], [-3284, 2], [-3283, 1], [-3282, 1], [-3281, 1], [-3280, 8], [-3279, 1], [-3278, 1], [-3277, 1], [-3276, 6], [-3275, 1], [-3274, 1], [-3273, 1], [-3272, 4], [-3271, 1], [-3270, 1], [-3269, 1], [-3268, 2], [-3267, 3], [-3266, 1], [-3265, 1], [-3264, 8], [-3263, 1], [-3262, 1], [-3261, 1], [-3260, 2], [-3259, 1], [-3258, 3], [-3257, 1], [-3256, 4], [-3255, 1], [-3254, 1], [-3253, 1], [-3252, 2], [-3251, 1], [-3250, 1], [-3249, 3], [-3248, 8], [-3247, 1], [-3246, 1], [-3245, 1], [-3244, 2], [-3243, 1], [-3242, 1], [-3241, 1], [-3240, 12], [-3239, 1], [-3238, 1], [-3237, 1], [-3236, 2], [-3235, 1], [-3234, 1], [-3233, 1], [-3232, 8], [-3231, 3], [-3230, 1], [-3229, 1], [-3228, 2], [-3227, 1], [-3226, 1], [-3225, 1], [-3224, 4], [-3223, 1], [-3222, 3], [-3221, 1], [-3220, 2], [-3219, 1], [-3218, 1], [-3217, 1], [-3216, 8], [-3215, 1], [-3214, 1], [-3213, 3], [-3212, 2], [-3211, 1], [-3210, 1], [-3209, 1], [-3208, 4], [-3207, 1], [-3206, 1], [-3205, 1], [-3204, 6], [-3203, 1], [-3202, 1], [-3201, 1], [-3200, 8], [-3199, 1], [-3198, 1], [-3197, 1], [-3196, 2], [-3195, 3], [-3194, 1], [-3193, 1], [-3192, 4], [-3191, 1], [-3190, 1], [-3189, 1], [-3188, 2], [-3187, 1], [-3186, 3], [-3185, 1], [-3184, 8], [-3183, 1], [-3182, 1], [-3181, 1], [-3180, 2], [-3179, 1], [-3178, 1], [-3177, 3], [-3176, 4], [-3175, 1], [-3174, 1], [-3173, 1], [-3172, 2], [-3171, 1], [-3170, 1], [-3169, 1], [-3168, 24], [-3167, 1], [-3166, 1], [-3165, 1], [-3164, 2], [-3163, 1], [-3162, 1], [-3161, 1], [-3160, 4], [-3159, 3], [-3158, 1], [-3157, 1], [-3156, 2], [-3155, 1], [-3154, 1], [-3153, 1], [-3152, 8], [-3151, 1], [-3150, 3], [-3149, 1], [-3148, 2], [-3147, 1], [-3146, 1], [-3145, 1], [-3144, 4], [-3143, 1], [-3142, 1], [-3141, 3], [-3140, 2], [-3139, 1], [-3138, 1], [-3137, 1], [-3136, 8], [-3135, 1], [-3134, 1], [-3133, 1], [-3132, 6], [-3131, 1], [-3130, 1], [-3129, 1], [-3128, 4], [-3127, 1], [-3126, 1], [-3125, 1], [-3124, 2], [-3123, 3], [-3122, 1], [-3121, 1], [-3120, 8], [-3119, 1], [-3118, 1], [-3117, 1], [-3116, 2], [-3115, 1], [-3114, 3], [-3113, 1], [-3112, 4], [-3111, 1], [-3110, 1], [-3109, 1], [-3108, 2], [-3107, 1], [-3106, 1], [-3105, 3], [-3104, 8], [-3103, 1], [-3102, 1], [-3101, 1], [-3100, 2], [-3099, 1], [-3098, 1], [-3097, 1], [-3096, 12], [-3095, 1], [-3094, 1], [-3093, 1], [-3092, 2], [-3091, 1], [-3090, 1], [-3089, 1], [-3088, 8], [-3087, 3], [-3086, 1], [-3085, 1], [-3084, 2], [-3083, 1], [-3082, 1], [-3081, 1], [-3080, 4], [-3079, 1], [-3078, 3], [-3077, 1], [-3076, 2], [-3075, 1], [-3074, 1], [-3073, 1], [-3072, 8], [-3071, 1], [-3070, 1], [-3069, 3], [-3068, 2], [-3067, 1], [-3066, 1], [-3065, 1], [-3064, 4], [-3063, 1], [-3062, 1], [-3061, 1], [-3060, 6], [-3059, 1], [-3058, 1], [-3057, 1], [-3056, 8], [-3055, 1], [-3054, 1], [-3053, 1], [-3052, 2], [-3051, 3], [-3050, 1], [-3049, 1], [-3048, 4], [-3047, 1], [-3046, 1], [-3045, 1], [-3044, 2], [-3043, 1], [-3042, 3], [-3041, 1], [-3040, 8], [-3039, 1], [-3038, 1], [-3037, 1], [-3036, 2], [-3035, 1], [-3034, 1], [-3033, 3], [-3032, 4], [-3031, 1], [-3030, 1], [-3029, 1], [-3028, 2], [-3027, 1], [-3026, 1], [-3025, 1], [-3024, 24], [-3023, 1], [-3022, 1], [-3021, 1], [-3020, 2], [-3019, 1], [-3018, 1], [-3017, 1], [-3016, 4], [-3015, 3], [-3014, 1], [-3013, 1], [-3012, 2], [-3011, 1], [-3010, 1], [-3009, 1], [-3008, 8], [-3007, 1], [-3006, 3], [-3005, 1], [-3004, 2], [-3003, 1], [-3002, 1], [-3001, 1], [-3000, 4], [-2999, 1], [-2998, 1], [-2997, 3], [-2996, 2], [-2995, 1], [-2994, 1], [-2993, 1], [-2992, 8], [-2991, 1], [-2990, 1], [-2989, 1], [-2988, 6], [-2987, 1], [-2986, 1], [-2985, 1], [-2984, 4], [-2983, 1], [-2982, 1], [-2981, 1], [-2980, 2], [-2979, 3], [-2978, 1], [-2977, 1], [-2976, 8], [-2975, 1], [-2974, 1], [-2973, 1], [-2972, 2], [-2971, 1], [-2970, 3], [-2969, 1], [-2968, 4], [-2967, 1], [-2966, 1], [-2965, 1], [-2964, 2], [-2963, 1], [-2962, 1], [-2961, 3], [-2960, 8], [-2959, 1], [-2958, 1], [-2957, 1], [-2956, 2], [-2955, 1], [-2954, 1], [-2953, 1], [-2952, 12], [-2951, 1], [-2950, 1], [-2949, 1], [-2948, 2], [-2947, 1], [-2946, 1], [-2945, 1], [-2944, 8], [-2943, 3], [-2942, 1], [-2941, 1], [-2940, 2], [-2939, 1], [-2938, 1], [-2937, 1], [-2936, 4], [-2935, 1], [-2934, 3], [-2933, 1], [-2932, 2], [-2931, 1], [-2930, 1], [-2929, 1], [-2928, 8], [-2927, 1], [-2926, 1], [-2925, 3], [-2924, 2], [-2923, 1], [-2922, 1], [-2921, 1], [-2920, 4], [-2919, 1], [-2918, 1], [-2917, 1], [-2916, 6], [-2915, 1], [-2914, 1], [-2913, 1], [-2912, 8], [-2911, 1], [-2910, 1], [-2909, 1], [-2908, 2], [-2907, 3], [-2906, 1], [-2905, 1], [-2904, 4], [-2903, 1], [-2902, 1], [-2901, 1], [-2900, 2], [-2899, 1], [-2898, 3], [-2897, 1], [-2896, 8], [-2895, 1], [-2894, 1], [-2893, 1], [-2892, 2], [-2891, 1], [-2890, 1], [-2889, 3], [-2888, 4], [-2887, 1], [-2886, 1], [-2885, 1], [-2884, 2], [-2883, 1], [-2882, 1], [-2881, 1], [-2880, 24], [-2879, 1], [-2878, 1], [-2877, 1], [-2876, 2], [-2875, 1], [-2874, 1], [-2873, 1], [-2872, 4], [-2871, 3], [-2870, 1], [-2869, 1], [-2868, 2], [-2867, 1], [-2866, 1], [-2865, 1], [-2864, 8], [-2863, 1], [-2862, 3], [-2861, 1], [-2860, 2], [-2859, 1], [-2858, 1], [-2857, 1], [-2856, 4], [-2855, 1], [-2854, 1], [-2853, 3], [-2852, 2], [-2851, 1], [-2850, 1], [-2849, 1], [-2848, 8], [-2847, 1], [-2846, 1], [-2845, 1], [-2844, 6], [-2843, 1], [-2842, 1], [-2841, 1], [-2840, 4], [-2839, 1], [-2838, 1], [-2837, 1], [-2836, 2], [-2835, 3], [-2834, 1], [-2833, 1], [-2832, 8], [-2831, 1], [-2830, 1], [-2829, 1], [-2828, 2], [-2827, 1], [-2826, 3], [-2825, 1], [-2824, 4], [-2823, 1], [-2822, 1], [-2821, 1], [-2820, 2], [-2819, 1], [-2818, 1], [-2817, 3], [-2816, 8], [-2815, 1], [-2814, 1], [-2813, 1], [-2812, 2], [-2811, 1], [-2810, 1], [-2809, 1], [-2808, 12], [-2807, 1], [-2806, 1], [-2805, 1], [-2804, 2], [-2803, 1], [-2802, 1], [-2801, 1], [-2800, 8], [-2799, 3], [-2798, 1], [-2797, 1], [-2796, 2], [-2795, 1], [-2794, 1], [-2793, 1], [-2792, 4], [-2791, 1], [-2790, 3], [-2789, 1], [-2788, 2], [-2787, 1], [-2786, 1], [-2785, 1], [-2784, 8], [-2783, 1], [-2782, 1], [-2781, 3], [-2780, 2], [-2779, 1], [-2778, 1], [-2777, 1], [-2776, 4], [-2775, 1], [-2774, 1], [-2773, 1], [-2772, 6], [-2771, 1], [-2770, 1], [-2769, 1], [-2768, 8], [-2767, 1], [-2766, 1], [-2765, 1], [-2764, 2], [-2763, 3], [-2762, 1], [-2761, 1], [-2760, 4], [-2759, 1], [-2758, 1], [-2757, 1], [-2756, 2], [-2755, 1], [-2754, 3], [-2753, 1], [-2752, 8], [-2751, 1], [-2750, 1], [-2749, 1], [-2748, 2], [-2747, 1], [-2746, 1], [-2745, 3], [-2744, 4], [-2743, 1], [-2742, 1], [-2741, 1], [-2740, 2], [-2739, 1], [-2738, 1], [-2737, 1], [-2736, 24], [-2735, 1], [-2734, 1], [-2733, 1], [-2732, 2], [-2731, 1], [-2730, 1], [-2729, 1], [-2728, 4], [-2727, 3], [-2726, 1], [-2725, 1], [-2724, 2], [-2723, 1], [-2722, 1], [-2721, 1], [-2720, 8], [-2719, 1], [-2718, 3], [-2717, 1], [-2716, 2], [-2715, 1], [-2714, 1], [-2713, 1], [-2712, 4], [-2711, 1], [-2710, 1], [-2709, 3], [-2708, 2], [-2707, 1], [-2706, 1], [-2705, 1], [-2704, 8], [-2703, 1], [-2702, 1], [-2701, 1], [-2700, 6], [-2699, 1], [-2698, 1], [-2697, 1], [-2696, 4], [-2695, 1], [-2694, 1], [-2693, 1], [-2692, 2], [-2691, 3], [-2690, 1], [-2689, 1], [-2688, 8], [-2687, 1], [-2686, 1], [-2685, 1], [-2684, 2], [-2683, 1], [-2682, 3], [-2681, 1], [-2680, 4], [-2679, 1], [-2678, 1], [-2677, 1], [-2676, 2], [-2675, 1], [-2674, 1], [-2673, 3], [-2672, 8], [-2671, 1], [-2670, 1], [-2669, 1], [-2668, 2], [-2667, 1], [-2666, 1], [-2665, 1], [-2664, 12], [-2663, 1], [-2662, 1], [-2661, 1], [-2660, 2], [-2659, 1], [-2658, 1], [-2657, 1], [-2656, 8], [-2655, 3], [-2654, 1], [-2653, 1], [-2652, 2], [-2651, 1], [-2650, 1], [-2649, 1], [-2648, 4], [-2647, 1], [-2646, 3], [-2645, 1], [-2644, 2], [-2643, 1], [-2642, 1], [-2641, 1], [-2640, 8], [-2639, 1], [-2638, 1], [-2637, 3], [-2636, 2], [-2635, 1], [-2634, 1], [-2633, 1], [-2632, 4], [-2631, 1], [-2630, 1], [-2629, 1], [-2628, 6], [-2627, 1], [-2626, 1], [-2625, 1], [-2624, 8], [-2623, 1], [-2622, 1], [-2621, 1], [-2620, 2], [-2619, 3], [-2618, 1], [-2617, 1], [-2616, 4], [-2615, 1], [-2614, 1], [-2613, 1], [-2612, 2], [-2611, 1], [-2610, 3], [-2609, 1], [-2608, 8], [-2607, 1], [-2606, 1], [-2605, 1], [-2604, 2], [-2603, 1], [-2602, 1], [-2601, 3], [-2600, 4], [-2599, 1], [-2598, 1], [-2597, 1], [-2596, 2], [-2595, 1], [-2594, 1], [-2593, 1], [-2592, 24], [-2591, 1], [-2590, 1], [-2589, 1], [-2588, 2], [-2587, 1], [-2586, 1], [-2585, 1], [-2584, 4], [-2583, 3], [-2582, 1], [-2581, 1], [-2580, 2], [-2579, 1], [-2578, 1], [-2577, 1], [-2576, 8], [-2575, 1], [-2574, 3], [-2573, 1], [-2572, 2], [-2571, 1], [-2570, 1], [-2569, 1], [-2568, 4], [-2567, 1], [-2566, 1], [-2565, 3], [-2564, 2], [-2563, 1], [-2562, 1], [-2561, 1], [-2560, 8], [-2559, 1], [-2558, 1], [-2557, 1], [-2556, 6], [-2555, 1], [-2554, 1], [-2553, 1], [-2552, 4], [-2551, 1], [-2550, 1], [-2549, 1], [-2548, 2], [-2547, 3], [-2546, 1], [-2545, 1], [-2544, 8], [-2543, 1], [-2542, 1], [-2541, 1], [-2540, 2], [-2539, 1], [-2538, 3], [-2537, 1], [-2536, 4], [-2535, 1], [-2534, 1], [-2533, 1], [-2532, 2], [-2531, 1], [-2530, 1], [-2529, 3], [-2528, 8], [-2527, 1], [-2526, 1], [-2525, 1], [-2524, 2], [-2523, 1], [-2522, 1], [-2521, 1], [-2520, 12], [-2519, 1], [-2518, 1], [-2517, 1], [-2516, 2], [-2515, 1], [-2514, 1], [-2513, 1], [-2512, 8], [-2511, 3], [-2510, 1], [-2509, 1], [-2508, 2], [-2507, 1], [-2506, 1], [-2505, 1], [-2504, 4], [-2503, 1], [-2502, 3], [-2501, 1], [-2500, 2], [-2499, 1], [-2498, 1], [-2497, 1], [-2496, 8], [-2495, 1], [-2494, 1], [-2493, 3], [-2492, 2], [-2491, 1], [-2490, 1], [-2489, 1], [-2488, 4], [-2487, 1], [-2486, 1], [-2485, 1], [-2484, 6], [-2483, 1], [-2482, 1], [-2481, 1], [-2480, 8], [-2479, 1], [-2478, 1], [-2477, 1], [-2476, 2], [-2475, 3], [-2474, 1], [-2473, 1], [-2472, 4], [-2471, 1], [-2470, 1], [-2469, 1], [-2468, 2], [-2467, 1], [-2466, 3], [-2465, 1], [-2464, 8], [-2463, 1], [-2462, 1], [-2461, 1], [-2460, 2], [-2459, 1], [-2458, 1], [-2457, 3], [-2456, 4], [-2455, 1], [-2454, 1], [-2453, 1], [-2452, 2], [-2451, 1], [-2450, 1], [-2449, 1], [-2448, 24], [-2447, 1], [-2446, 1], [-2445, 1], [-2444, 2], [-2443, 1], [-2442, 1], [-2441, 1], [-2440, 4], [-2439, 3], [-2438, 1], [-2437, 1], [-2436, 2], [-2435, 1], [-2434, 1], [-2433, 1], [-2432, 8], [-2431, 1], [-2430, 3], [-2429, 1], [-2428, 2], [-2427, 1], [-2426, 1], [-2425, 1], [-2424, 4], [-2423, 1], [-2422, 1], [-2421, 3], [-2420, 2], [-2419, 1], [-2418, 1], [-2417, 1], [-2416, 8], [-2415, 1], [-2414, 1], [-2413, 1], [-2412, 6], [-2411, 1], [-2410, 1], [-2409, 1], [-2408, 4], [-2407, 1], [-2406, 1], [-2405, 1], [-2404, 2], [-2403, 3], [-2402, 1], [-2401, 1], [-2400, 8], [-2399, 1], [-2398, 1], [-2397, 1], [-2396, 2], [-2395, 1], [-2394, 3], [-2393, 1], [-2392, 4], [-2391, 1], [-2390, 1], [-2389, 1], [-2388, 2], [-2387, 1], [-2386, 1], [-2385, 3], [-2384, 8], [-2383, 1], [-2382, 1], [-2381, 1], [-2380, 2], [-2379, 1], [-2378, 1], [-2377, 1], [-2376, 12], [-2375, 1], [-2374, 1], [-2373, 1], [-2372, 2], [-2371, 1], [-2370, 1], [-2369, 1], [-2368, 8], [-2367, 3], [-2366, 1], [-2365, 1], [-2364, 2], [-2363, 1], [-2362, 1], [-2361, 1], [-2360, 4], [-2359, 1], [-2358, 3], [-2357, 1], [-2356, 2], [-2355, 1], [-2354, 1], [-2353, 1], [-2352, 8], [-2351, 1], [-2350, 1], [-2349, 3], [-2348, 2], [-2347, 1], [-2346, 1], [-2345, 1], [-2344, 4], [-2343, 1], [-2342, 1], [-2341, 1], [-2340, 6], [-2339, 1], [-2338, 1], [-2337, 1], [-2336, 8], [-2335, 1], [-2334, 1], [-2333, 1], [-2332, 2], [-2331, 3], [-2330, 1], [-2329, 1], [-2328, 4], [-2327, 1], [-2326, 1], [-2325, 1], [-2324, 2], [-2323, 1], [-2322, 3], [-2321, 1], [-2320, 8], [-2319, 1], [-2318, 1], [-2317, 1], [-2316, 2], [-2315, 1], [-2314, 1], [-2313, 3], [-2312, 4], [-2311, 1], [-2310, 1], [-2309, 1], [-2308, 2], [-2307, 1], [-2306, 1], [-2305, 1], [-2304, 24], [-2303, 1], [-2302, 1], [-2301, 1], [-2300, 2], [-2299, 1], [-2298, 1], [-2297, 1], [-2296, 4], [-2295, 3], 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