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Number Buffet
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500 số hạnh phúc đầu tiên

1, 7, 10, 13, 19, 23, 28, 31, 32, 44, 49, 68, 70, 79, 82, 86, 91, 94, 97, 100, 103, 109, 129, 130, 133, 139, 167, 176, 188, 190, 192, 193, 203, 208, 219, 226, 230, 236, 239, 262, 263, 280, 291, 293, 301, 302, 310, 313, 319, 320, 326, 329, 331, 338, 356, 362, 365, 367, 368, 376, 379, 383, 386, 391, 392, 397, 404, 409, 440, 446, 464, 469, 478, 487, 490, 496, 536, 556, 563, 565, 566, 608, 617, 622, 623, 632, 635, 637, 638, 644, 649, 653, 655, 656, 665, 671, 673, 680, 683, 694, 700, 709, 716, 736, 739, 748, 761, 763, 784, 790, 793, 802, 806, 818, 820, 833, 836, 847, 860, 863, 874, 881, 888, 899, 901, 904, 907, 910, 912, 913, 921, 923, 931, 932, 937, 940, 946, 964, 970, 973, 989, 998, 1000, 1003, 1009, 1029, 1030, 1033, 1039, 1067, 1076, 1088, 1090, 1092, 1093, 1112, 1114, 1115, 1121, 1122, 1125, 1128, 1141, 1148, 1151, 1152, 1158, 1177, 1182, 1184, 1185, 1188, 1209, 1211, 1212, 1215, 1218, 1221, 1222, 1233, 1247, 1251, 1257, 1258, 1274, 1275, 1277, 1281, 1285, 1288, 1290, 1299, 1300, 1303, 1309, 1323, 1330, 1332, 1333, 1335, 1337, 1339, 1353, 1366, 1373, 1390, 1393, 1411, 1418, 1427, 1444, 1447, 1448, 1457, 1472, 1474, 1475, 1478, 1481, 1484, 1487, 1511, 1512, 1518, 1521, 1527, 1528, 1533, 1547, 1557, 1572, 1574, 1575, 1578, 1581, 1582, 1587, 1599, 1607, 1636, 1663, 1666, 1670, 1679, 1697, 1706, 1717, 1724, 1725, 1727, 1733, 1742, 1744, 1745, 1748, 1752, 1754, 1755, 1758, 1760, 1769, 1771, 1772, 1784, 1785, 1796, 1808, 1812, 1814, 1815, 1818, 1821, 1825, 1828, 1841, 1844, 1847, 1851, 1852, 1857, 1874, 1875, 1880, 1881, 1882, 1888, 1900, 1902, 1903, 1920, 1929, 1930, 1933, 1959, 1967, 1976, 1992, 1995, 2003, 2008, 2019, 2026, 2030, 2036, 2039, 2062, 2063, 2080, 2091, 2093, 2109, 2111, 2112, 2115, 2118, 2121, 2122, 2133, 2147, 2151, 2157, 2158, 2174, 2175, 2177, 2181, 2185, 2188, 2190, 2199, 2206, 2211, 2212, 2221, 2224, 2242, 2245, 2254, 2257, 2258, 2260, 2275, 2285, 2300, 2306, 2309, 2313, 2331, 2333, 2338, 2339, 2360, 2369, 2383, 2390, 2393, 2396, 2417, 2422, 2425, 2448, 2452, 2455, 2457, 2458, 2471, 2475, 2478, 2484, 2485, 2487, 2511, 2517, 2518, 2524, 2527, 2528, 2542, 2545, 2547, 2548, 2554, 2555, 2557, 2568, 2571, 2572, 2574, 2575, 2581, 2582, 2584, 2586, 2602, 2603, 2620, 2630, 2639, 2658, 2685, 2693, 2714, 2715, 2717, 2725, 2741, 2745, 2748, 2751, 2752, 2754, 2755, 2771, 2784, 2800, 2811, 2815, 2818, 2825, 2833, 2844, 2845, 2847, 2851, 2852, 2854, 2856, 2865, 2874, 2881, 2899, 2901, 2903, 2910, 2919, 2930, 2933, 2936, 2963, 2989, 2991, 2998, 3001, 3002, 3010, 3013, 3019, 3020, 3026, 3029, 3031, 3038, 3056, 3062, 3065, 3067, 3068, 3076, 3079, 3083, 3086, 3091, 3092, 3097, 3100, 3103, 3109, 3123, 3130, 3132, 3133, 3135, 3137, 3139, 3153, 3166, 3173, 3190, 3193, 3200, 3206, 3209, 3213, 3231, 3233, 3238, 3239, 3260, 3269, 3283, 3290, 3293, 3296, 3301, 3308, 3310, 3312, 3313, 3315, 3317

Thiết lập

Thiết lập nhanh

Sad numbers are the ones whose chain falls into the 4 → 16 → 37 → 58 → 89 → 145 → 42 → 20 loop.

Terms are returned in increasing order from here. Happy primes start from 2 whatever you set.

Prints the whole trajectory, e.g. 19 → 82 → 68 → 100 → 1.

Tinh chỉnh dáng vẻ

Hãy chọn một thiết lập cạnh ảnh trước — các điều khiển này điều chỉnh nó.

Frame

A border drawn inside the edge of the image.

Nâng cao

Kết quả

500 giá trị

1, 7, 10, 13, 19, 23, 28, 31, 32, 44, 49, 68, 70, 79, 82, 86, 91, 94, 97, 100, 103, 109, 129, 130, 133, 139, 167, 176, 188, 190, 192, 193, 203, 208, 219, 226, 230, 236, 239, 262, 263, 280, 291, 293, 301, 302, 310, 313, 319, 320, 326, 329, 331, 338, 356, 362, 365, 367, 368, 376, 379, 383, 386, 391, 392, 397, 404, 409, 440, 446, 464, 469, 478, 487, 490, 496, 536, 556, 563, 565, 566, 608, 617, 622, 623, 632, 635, 637, 638, 644, 649, 653, 655, 656, 665, 671, 673, 680, 683, 694, 700, 709, 716, 736, 739, 748, 761, 763, 784, 790, 793, 802, 806, 818, 820, 833, 836, 847, 860, 863, 874, 881, 888, 899, 901, 904, 907, 910, 912, 913, 921, 923, 931, 932, 937, 940, 946, 964, 970, 973, 989, 998, 1000, 1003, 1009, 1029, 1030, 1033, 1039, 1067, 1076, 1088, 1090, 1092, 1093, 1112, 1114, 1115, 1121, 1122, 1125, 1128, 1141, 1148, 1151, 1152, 1158, 1177, 1182, 1184, 1185, 1188, 1209, 1211, 1212, 1215, 1218, 1221, 1222, 1233, 1247, 1251, 1257, 1258, 1274, 1275, 1277, 1281, 1285, 1288, 1290, 1299, 1300, 1303, 1309, 1323, 1330, 1332, 1333, 1335, 1337, 1339, 1353, 1366, 1373, 1390, 1393, 1411, 1418, 1427, 1444, 1447, 1448, 1457, 1472, 1474, 1475, 1478, 1481, 1484, 1487, 1511, 1512, 1518, 1521, 1527, 1528, 1533, 1547, 1557, 1572, 1574, 1575, 1578, 1581, 1582, 1587, 1599, 1607, 1636, 1663, 1666, 1670, 1679, 1697, 1706, 1717, 1724, 1725, 1727, 1733, 1742, 1744, 1745, 1748, 1752, 1754, 1755, 1758, 1760, 1769, 1771, 1772, 1784, 1785, 1796, 1808, 1812, 1814, 1815, 1818, 1821, 1825, 1828, 1841, 1844, 1847, 1851, 1852, 1857, 1874, 1875, 1880, 1881, 1882, 1888, 1900, 1902, 1903, 1920, 1929, 1930, 1933, 1959, 1967, 1976, 1992, 1995, 2003, 2008, 2019, 2026, 2030, 2036, 2039, 2062, 2063, 2080, 2091, 2093, 2109, 2111, 2112, 2115, 2118, 2121, 2122, 2133, 2147, 2151, 2157, 2158, 2174, 2175, 2177, 2181, 2185, 2188, 2190, 2199, 2206, 2211, 2212, 2221, 2224, 2242, 2245, 2254, 2257, 2258, 2260, 2275, 2285, 2300, 2306, 2309, 2313, 2331, 2333, 2338, 2339, 2360, 2369, 2383, 2390, 2393, 2396, 2417, 2422, 2425, 2448, 2452, 2455, 2457, 2458, 2471, 2475, 2478, 2484, 2485, 2487, 2511, 2517, 2518, 2524, 2527, 2528, 2542, 2545, 2547, 2548, 2554, 2555, 2557, 2568, 2571, 2572, 2574, 2575, 2581, 2582, 2584, 2586, 2602, 2603, 2620, 2630, 2639, 2658, 2685, 2693, 2714, 2715, 2717, 2725, 2741, 2745, 2748, 2751, 2752, 2754, 2755, 2771, 2784, 2800, 2811, 2815, 2818, 2825, 2833, 2844, 2845, 2847, 2851, 2852, 2854, 2856, 2865, 2874, 2881, 2899, 2901, 2903, 2910, 2919, 2930, 2933, 2936, 2963, 2989, 2991, 2998, 3001, 3002, 3010, 3013, 3019, 3020, 3026, 3029, 3031, 3038, 3056, 3062, 3065, 3067, 3068, 3076, 3079, 3083, 3086, 3091, 3092, 3097, 3100, 3103, 3109, 3123, 3130, 3132, 3133, 3135, 3137, 3139, 3153, 3166, 3173, 3190, 3193, 3200, 3206, 3209, 3213, 3231, 3233, 3238, 3239, 3260, 3269, 3283, 3290, 3293, 3296, 3301, 3308, 3310, 3312, 3313, 3315, 3317

Tạo ảnh

Hãy bật JavaScript để tạo kiểu cho những số này và tải về dưới dạng ảnh. Bản thân các giá trị đã được liệt kê ở trên.

Text on the image

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500 số hạnh phúc đầu tiên là những số nào?

500 số hạnh phúc đầu tiên là:

1, 7, 10, 13, 19, 23, 28, 31, 32, 44, 49, 68, 70, 79, 82, 86, 91, 94, 97, 100, 103, 109, 129, 130, 133, 139, 167, 176, 188, 190, 192, 193, 203, 208, 219, 226, 230, 236, 239, 262, 263, 280, 291, 293, 301, 302, 310, 313, 319, 320, 326, 329, 331, 338, 356, 362, 365, 367, 368, 376, 379, 383, 386, 391, 392, 397, 404, 409, 440, 446, 464, 469, 478, 487, 490, 496, 536, 556, 563, 565, 566, 608, 617, 622, 623, 632, 635, 637, 638, 644, 649, 653, 655, 656, 665, 671, 673, 680, 683, 694, 700, 709, 716, 736, 739, 748, 761, 763, 784, 790, 793, 802, 806, 818, 820, 833, 836, 847, 860, 863, 874, 881, 888, 899, 901, 904, 907, 910, 912, 913, 921, 923, 931, 932, 937, 940, 946, 964, 970, 973, 989, 998, 1000, 1003, 1009, 1029, 1030, 1033, 1039, 1067, 1076, 1088, 1090, 1092, 1093, 1112, 1114, 1115, 1121, 1122, 1125, 1128, 1141, 1148, 1151, 1152, 1158, 1177, 1182, 1184, 1185, 1188, 1209, 1211, 1212, 1215, 1218, 1221, 1222, 1233, 1247, 1251, 1257, 1258, 1274, 1275, 1277, 1281, 1285, 1288, 1290, 1299, 1300, 1303, 1309, 1323, 1330, 1332, 1333, 1335, 1337, 1339, 1353, 1366, 1373, 1390, 1393, 1411, 1418, 1427, 1444, 1447, 1448, 1457, 1472, 1474, 1475, 1478, 1481, 1484, 1487, 1511, 1512, 1518, 1521, 1527, 1528, 1533, 1547, 1557, 1572, 1574, 1575, 1578, 1581, 1582, 1587, 1599, 1607, 1636, 1663, 1666, 1670, 1679, 1697, 1706, 1717, 1724, 1725, 1727, 1733, 1742, 1744, 1745, 1748, 1752, 1754, 1755, 1758, 1760, 1769, 1771, 1772, 1784, 1785, 1796, 1808, 1812, 1814, 1815, 1818, 1821, 1825, 1828, 1841, 1844, 1847, 1851, 1852, 1857, 1874, 1875, 1880, 1881, 1882, 1888, 1900, 1902, 1903, 1920, 1929, 1930, 1933, 1959, 1967, 1976, 1992, 1995, 2003, 2008, 2019, 2026, 2030, 2036, 2039, 2062, 2063, 2080, 2091, 2093, 2109, 2111, 2112, 2115, 2118, 2121, 2122, 2133, 2147, 2151, 2157, 2158, 2174, 2175, 2177, 2181, 2185, 2188, 2190, 2199, 2206, 2211, 2212, 2221, 2224, 2242, 2245, 2254, 2257, 2258, 2260, 2275, 2285, 2300, 2306, 2309, 2313, 2331, 2333, 2338, 2339, 2360, 2369, 2383, 2390, 2393, 2396, 2417, 2422, 2425, 2448, 2452, 2455, 2457, 2458, 2471, 2475, 2478, 2484, 2485, 2487, 2511, 2517, 2518, 2524, 2527, 2528, 2542, 2545, 2547, 2548, 2554, 2555, 2557, 2568, 2571, 2572, 2574, 2575, 2581, 2582, 2584, 2586, 2602, 2603, 2620, 2630, 2639, 2658, 2685, 2693, 2714, 2715, 2717, 2725, 2741, 2745, 2748, 2751, 2752, 2754, 2755, 2771, 2784, 2800, 2811, 2815, 2818, 2825, 2833, 2844, 2845, 2847, 2851, 2852, 2854, 2856, 2865, 2874, 2881, 2899, 2901, 2903, 2910, 2919, 2930, 2933, 2936, 2963, 2989, 2991, 2998, 3001, 3002, 3010, 3013, 3019, 3020, 3026, 3029, 3031, 3038, 3056, 3062, 3065, 3067, 3068, 3076, 3079, 3083, 3086, 3091, 3092, 3097, 3100, 3103, 3109, 3123, 3130, 3132, 3133, 3135, 3137, 3139, 3153, 3166, 3173, 3190, 3193, 3200, 3206, 3209, 3213, 3231, 3233, 3238, 3239, 3260, 3269, 3283, 3290, 3293, 3296, 3301, 3308, 3310, 3312, 3313, 3315, 3317

Bài viết nền bên dưới chưa được dịch và đang hiển thị bằng tiếng Anh.

Về số hạnh phúc

The iteration came first and the name came later. The earliest known treatment is a three-page paper by Arthur Porges, "A Set of Eight Numbers", in the American Mathematical Monthly volume 52 (1945), pages 379–382. Porges — an American writer who held a bachelor's and a master's degree in mathematics and taught the subject at college level before turning to short fiction full time — proved the fact that makes the whole puzzle well posed: take any positive integer, replace it by the sum of the squares of its digits, repeat, and you must end either at 1 or in the cycle 4, 16, 37, 58, 89, 145, 42, 20. Nothing else can happen in base ten. The reason is a size argument: any number of four or more digits maps to something strictly smaller, so every trajectory falls below 1000 within a few steps and is then trapped in a range small enough to check exhaustively by hand.

Where the cheerful name came from is genuinely unclear, and the usual story is a chain of custody rather than a discovery. The most widely repeated account has the problem reaching Reginald Allenby, a British author and senior lecturer in pure mathematics at the University of Leeds, by way of his daughter, who had met it at school; the same account adds only that the idea may have originated in Russia. Nobody has pinned it down further.

What drew research mathematicians was density. Esam El-Sedy and Samir Siksek proved that the happy numbers contain arbitrarily long runs of consecutive integers. Justin Gilmer later showed that their lower density is below 0.1138 while their upper density is above 0.18577 — the bounds do not meet, so the happy numbers have no natural density at all, even though counting them over any fixed range gives a suspiciously stable answer near 14%.

Tính chất chính

  • In base 10, iterating the sum of the squares of the digits always reaches either 1 or the eight-term cycle 4 → 16 → 37 → 58 → 89 → 145 → 42 → 20 → 4.
  • 7 is the smallest happy number above 1, by way of 7 → 49 → 97 → 130 → 10 → 1.
  • 142 of the integers from 1 to 999 are happy; of the integers from 1 to 2,000,000, exactly 282,005 are, which is about 14.1%.
  • The first pair of consecutive happy numbers is 31 and 32; the first run of three is 1880, 1881, 1882; the first run of four begins at 7839.
  • El-Sedy and Siksek proved that runs of consecutive happy numbers can be made arbitrarily long.
  • Gilmer proved that the lower density of the happy numbers is less than 0.1138 and the upper density greater than 0.18577, so the sequence has no natural density.
  • A number of four or more digits always maps to a strictly smaller value, which is why every trajectory is eventually confined below 1000.
  • Base 2 and base 4 are the only bases below 5×10⁸ known to be "happy", meaning every number in them reaches 1; whether any others exist is an open question.

Số lượng khác

Nguồn