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Friedman (numeri di)

Rappresentazione dei numeri 

Indice

  1. 1. Pagina principale
  2. 2. I numeri di Friedman in notazione romana

Utilizzando i numeri romani, qualsiasi numero naturale è un numero di Friedman: si tratta solo di inserire i simboli di addizione e sottrazione nei punti opportuni; per esempio, LIX = L – I + X. Robert Happelberg fu il primo a cercare numeri di Friedman in notazione romana rappresentati combinando le cifre in altro modo, con almeno un’operazione diversa da addizione e sottrazione.

In questo caso non si utilizzano i simboli per le cifre superiori a 1000 (v. numeri romani), quindi il massimo numero considerato è 3999 = MMMCMXCIX.

 

I numeri di Friedman in notazione romana noti (M. Fiorentini, 2018, salvo diversa indicazione):

8 = VIII = II • IV (Robert Happelberg),

27 = IX * (X / V + I) (Bryce Herdt),

33 = XI * (X / X + II) (Robert Happelberg),

36 = VI^(XX / X) = IX * XX / V (Robert Happelberg),

37 = IX * IV + X / X,

44 = XLIV = L – V – IX (Robert Happelberg),

46 = XLVI = L – V + IX (Robert Happelberg),

47 = L – X / V – I * I (Bryce Herdt),

48 = L – XII / VI,

49 = L – I^XX = X * (L – I) / X,

75 = L * XV / X = L / X * XV (Bryce Herdt),

78 = (XV – II) * (L / X + I),

81 = IX^(X * X / L) (Robert Happelberg),

84 = LX / X * XIV = L * XV / X + IX (Alan Frank),

86 = L * XV / X + XI (Robert Happelberg),

88 = L / V * X – XI – I^XI =  L / V * XI – XXII,

89 = X * (X – I^L) – X / X (Robert Happelberg),

94 = XCIV = C – V – IX (Robert Happelberg),

96 = XCIV = C – V + IX (Robert Happelberg),

99 = C – I^XX = X * (C – I) / X,

128 = (C / (X * V))^(X – III),

132 = CX / X * XII,

135 = (CX – X) / X * XV,

143 = III^(L / X) – C,

147 = (L – I^C) * (XV + I),

177 = LIX * (X / V + I^C),

180 = C / L * (X * X – X),

183 = (C / X + LI) * (X / X + II),

187 = (X + I^XI) * (XV + C / L),

231 = (C / C + XX) * XI,

234 = IX * (XXV + C / C),

237 = (C / (V * X) + I) * (C – XXI),

243 = (C / C + II^I)^(L / X),

247 = (XI – C / L) * (C / V – I),

255 = (C / C + L) * V,

256 = (CC / L) * (V – I),

267 = (C – XI) * (C / L + I^V),

273 = (L – XI) * (X – II – C / C),

277 = (L – IX) * (VI + C / C) – X,

286 = CCLX / X * (X + I^V),

287 = (L – IX) * (VI + CX / CX),

297 = (C – I^I) * (X / V + C / C),

303 = CI * (C / C + II),

324 = (IX * CC / C)^(X / V),

327 = CIX * (IX – V – C / C),

333 = CXI * (C / C + X / X + I / I),

343 = (IX – II)^(C / C + C / L),

360 = XC * CC / L,

361 = (CC / X – I)^(C / L),

380 = (–C / C + C / L * X) * XX,

385 = CX / X * (CC / V – L / X),

429 = (CD – X) / X * (I + X),

432 = IX * ((D – X) / X – I^C),

440 = D – L – C / X,

441 = CDXLI = D – LX + IC,

451 = CDLI = D – X + IC,

454 = CDLIV = DV – L – IC,

456 = CDLVI = DV – L + IC,

459 = CDLIX = DX – L – IC,

460 = D – L + C / X,

461 = CDLXI = DX – L + IC,

464 = D – L + CLX / V – I^I,

470 = D – X * C / L – X,

473 = XI * (CDL / X – II),

477 = IX * (D / X + C / I + I^V),

480 = X * D / X – X * C / L,

484 = (CDL – X) / X * (X + I^V),

490 = C * (D – X) / C,

494 = CDIV = D – V – IC,

495 = D – V * (C / C)^X,

496 = CDVI = D – V + IC,

499 = CDXCIX = D – ICCXX,

499 = D – (CX / CX)^I,

523 = DXXIII = IIX – D – IX,

524 = (X / V)^X – D^I,

528 = (D / X – II) * (X + I^V),

539 = (D / X – X / X) * (I + X),

583 = XI^I * (DXL / X – I),

625 = (D / C)^(–X / X + V),

637 = (D / X – I) * (C / X + X / V + I),

682 = XI * (DCX / X + I^L),

693 = (C – I^X) * (D / C + II),

729 = (XX / X + I)^(DC / C),

737 = XI * (D / X + C / C + XVI),

750 = D / C * CL,

781 = (X + I^L) * DCCX / X,

782 = IX * (D / L + XX – C / C) – I,

783 = IX^I * (D / L + C / X + C / X – I),

784 = (XL – C / C – I^DX)^(X / V),

802 = CDI * (C / C + I),

819 = (X – I^D) * (XC + C / C),

836 = (X – I^C) * (DCCX / X + V),

841 = DC / C * CXL,

841 = ((D – CC) / X – I)^(C / L),

869 = XI * (XC – D / L – C / C),

873 = (D / L – I) * (CC – CII – X / X),

880 = D / L * CX – CCXX,

890 = D * CC / C – X – C,

891 = X * (C – C / C) – C + I^D,

894 = DCCCXCIV = X • C – CV – IDCC,

896 = DCCCXCVV = X • C – CV + IDCC,

927 = (X – I^M) * (CI + X / V),

940 = M – L – C / X,

941 = CMXLI = M – LX + IC,

951 = CMLI = M – L + IC,

954 = CMLIV = MV – L – IC,

956 = CMLVI = MV – L + IC,

959 = CMLIX = MX – L – IC,

940 = M – L – C / X,

961 = CMLXI = MX – L + IC,

960 = M – L + C / X,

964 = MIV + C / X – L,

966 = MVI + C / X – L,

967 = M – CLX / V – I^I,

970 = M – X * C / L – X,

979 = (X + I^L) * (M – CX) / X,

980 = M * X / X – X * C / L,

987 = XX^I * (MC / L + X / V),

990 = C * (M – X) / C,

994 = CMXCIV = M – V – ICCX,

995 = M – V * (C / C)^X,

996 = CMXCIV = M – V + ICCX,

999 = CMXCIX = M – ICCXX,

999 = M – (CX / CX)^I,

1023 = MXXIII = IIX – IMX,

1024 = (X / V)^X * I^M,

1178 = LXII * (CM / (V * X) + I),

1187 = –M + (C / L + (X / X)^X)^VII,

1188 = CM / (V * X) * (LIII + X / X) = XI * (CVI + X / X + I^ML)

1286 = XXXVI^(C / L) – M / C,

1313 = (XII + C / C) * (C + I^M),

1320 = MC / C * CXX,

1331 = MC / C * CX / X * XI,

1341 = (X – I^M) * (CL – C / C),

1342 = (X – I^CI) * (C + MC / L),

1350 = (M – C) / C * CL,

1393 = (M / CC + II) * (CC – I^X),

1425 = XV * (C – X * D / M),

1431 = XI * (MDC – X) / X,

1663 = XLII^(M / D) – C,

1672 = (X + I^XI) * (CL + M / D),

1683 = CIX * (MD / (X * X) + I + I^L)

1717 = CI^I * (C / X * V + M / D),

1728 = (MDC / C – IV)^(X / X + II),

1764 = (XL + I^V + C / C)^(M / D),

1773 = (D / L – (X / X)^M) * (CC – III),

1791 = IX * (M / D * C – C / C),

1795 = CM * C / (X * V) – D / C,

1818 = CI * (X / V + MDC / C + I – I),

1833 = (C / X + C / C + II) * (MD / X – IX),

1843 = (M / L – I) * (XC + D / C + I * I),

1849 = (XL + I^X + CC/ C)^(M / D),

1863 = (CM / C)^II * (DC / L +XI),

1870 = CX * (MC / L + D / C – X),

1883 = DCC / C * (M / X * II + LXIX),

1886 = (LXIX – X) * (CC / C)^V – M / D,

1887 = LIX * (I^MDC + C / C)^V – X / X = (L + M / C – I) * (XX / X)^V – (C / C)^DI,

1888 = II^V * (LX – C / C) – (X – X) * MDCI = LIX * II^V + (C – C) * MDCXX,

1919 = CI * (XX – M / M),

1932 = XCII * (XX + M / M),

1963 = XIII * (M / M + CL),

1964 = (M – CM / L) * X / V,

1994 = MM – X / V – (C / C)^I,

2018 = MMXVIII = M + (I + IM)X – VI,

2048 = (I / I + M / M)^(L / X * VI),

2079 = (M / L + I) * (M / X – X / X),

2121 = CI * (XX + M / M),

2134 = XI * ((MCM – X) / X + V),

2177 = (M / M + C / L)^(VI + I^X) – X,

2187 = (M / M + CXX / LX)^VII = (M / M + C / L * (X / X)^X)^VII)^VII,

2239 = MCXX * C * XX / M – I,

2268 = (CX – II) * (M / M + L / V + I^C),

2310 = MMC / C * CX,

2352 = (CI + MC / C) * (I + M / L),

2424 = (C + I^V) * (XX + MM / D),

2450 = (C * D – M) / M * L,

2471 = LIX^(M / D) – M – C / X,

2505 = (M / M + D) * V,

2670 = (M – CX) * (L / X – M / D),

2860 = MC / C * (M / D * C + LX),

2929 = (C + I^M) * (XXX – M / M),

3003 = MI * (M / M + II),

3033 = MXI * (M / M + X / X + I /I),

3125 = (M / MM * C / X)^(X – V),

3131 = (C + I^M) * (XXX + M / M),

3136 = (C / XX * X + VI)^(MM / M),

3179 = (X + I^L) * (MMM – CX) / X,

3249 = (MC / XX + M / M + I)^C / L,

3270 = MXC * (C / L + (M / M)^X),

3303 = MCI * (M / M + C / C + I / I),

3330 = MCX * (M / M + C / C + X / X),

3364 = (LX – CC / C)^(MM / M) * I^V,

3432 = (X + I^D) * (MMMCX / X + I),

3434 = (X + I^MV) * (XXX + MMM / D),

3600 = CM * MM / D,

3663 = (X + I) * (CI + D / L) * (MM / M + I),

3939 = CI * (XXX + X – MM / MM),

3943 = CIX * (M * L  / MM + I + I^M),

3956 = IV * (M – M / C – (M / M)^L).

 

Da molti di questi se ne possono ricavare vari altri, aggiungendo o sottraendo cifre; per esempio, da 8 = VIII = II • IV si ricavano 18 = XVIII = X + II • IV, 138 = CXXXVIII = CXXX + II • IV ecc. e da 75 = L * XV / X si ricavano 175 = C + L * XV / X74 = L * XV / X – I ecc..

In questo modo, grazie anche al fatto che la rappresentazione romana utilizza quasi sempre più simboli di quella decimale per rappresentare i numeri, mettendo quindi a disposizione più combinazioni, la maggior parte degli interi è un numero di Friedman in notazione romana.

I numeri fino a 3999 per i quali non si conosce una rappresentazione come numeri di Friedman in notazione romana sono 1291: 1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 16, 17, 19, 20, 21, 22, 23, 24, 25, 26, 29, 30, 31, 32, 34, 35, 39, 40, 41, 42, 43, 45, 50, 51, 52, 53, 54, 55, 56, 57, 59, 60, 61, 62, 63, 64, 65, 66, 67, 69, 70, 71, 72, 73, 79, 80, 90, 91, 92, 93, 95, 100, 101, 102, 103, 104, 105, 106, 107, 109, 110, 111, 112, 113, 114, 115, 116, 117, 119, 120, 121, 122, 123, 124, 125, 126, 129, 130, 131, 139, 140, 141, 142, 145, 150, 151, 152, 153, 154, 155, 156, 157, 159, 160, 161, 162, 163, 164, 165, 166, 167, 169, 170, 171, 172, 173, 190, 191, 192, 193, 195, 200, 201, 202, 203, 204, 205, 206, 207, 209, 210, 211, 212, 213, 214, 215, 216, 217, 219, 220, 221, 222, 223, 224, 225, 226, 229, 230, 239, 240, 241, 242, 250, 251, 252, 253, 259, 260, 261, 262, 263, 269, 270, 271, 272, 290, 291, 292, 295, 300, 301, 302, 304, 305, 306, 307, 309, 310, 311, 312, 314, 315, 316, 317, 319, 320, 321, 322, 325, 326, 329, 330, 339, 340, 341, 342, 350, 351, 352, 390, 391, 392, 395, 400, 401, 402, 403, 404, 405, 406, 407, 409, 410, 411, 412, 413, 414, 415, 416, 417, 419, 420, 421, 422, 425, 426, 430, 431, 435, 450, 455, 500, 501, 502, 503, 504, 505, 506, 507, 509, 510, 511, 512, 513, 514, 515, 516, 517, 519, 520, 521, 522, 525, 526, 529, 530, 531, 532, 535, 540, 541, 542, 543, 545, 550, 551, 552, 553, 554, 555, 556, 557, 559, 560, 561, 562, 563, 564, 565, 566, 567, 569, 570, 571, 572, 579, 580, 590, 591, 592, 593, 595, 600, 601, 602, 603, 604, 605, 606, 607, 609, 610, 611, 612, 613, 614, 615, 616, 617, 619, 620, 621, 622, 629, 630, 631, 640, 641, 642, 645, 650, 651, 652, 653, 654, 655, 656, 657, 659, 660, 661, 662, 663, 664, 665, 666, 667, 669, 670, 671, 672, 690, 691, 692, 695, 700, 701, 702, 703, 704, 705, 706, 707, 709, 710, 711, 712, 713, 714, 715, 716, 717, 719, 720, 721, 722, 730, 790, 791, 795, 800, 801, 804, 805, 806, 809, 810, 811, 814, 815, 816, 820, 821, 830, 900, 901, 902, 903, 904, 905, 906, 907, 909, 910, 911, 912, 913, 914, 915, 916, 917, 919, 920, 921, 922, 925, 926, 929, 930, 931, 932, 935, 939, 950, 955, 1000, 1001, 1002, 1003, 1004, 1005, 1006, 1007, 1009, 1010, 1011, 1012, 1013, 1014, 1015, 1016, 1017, 1019, 1020, 1021, 1022, 1025, 1026, 1029, 1030, 1031, 1032, 1035, 1039, 1040, 1041, 1042, 1043, 1045, 1050, 1051, 1052, 1053, 1054, 1055, 1056, 1057, 1059, 1060, 1061, 1062, 1063, 1064, 1065, 1066, 1067, 1069, 1070, 1071, 1072, 1079, 1080, 1090, 1091, 1092, 1093, 1095, 1100, 1101, 1102, 1103, 1104, 1105, 1106, 1107, 1109, 1110, 1111, 1112, 1113, 1114, 1115, 1116, 1117, 1119, 1120, 1121, 1122, 1125, 1126, 1129, 1130, 1131, 1139, 1140, 1141, 1142, 1145, 1150, 1151, 1152, 1153, 1154, 1155, 1156, 1157, 1159, 1160, 1161, 1162, 1163, 1164, 1165, 1166, 1167, 1169, 1170, 1171, 1172, 1190, 1191, 1192, 1193, 1195, 1200, 1201, 1202, 1203, 1204, 1205, 1206, 1207, 1209, 1210, 1211, 1212, 1213, 1214, 1215, 1216, 1217, 1219, 1220, 1221, 1222, 1225, 1226, 1229, 1230, 1239, 1240, 1241, 1242, 1250, 1251, 1252, 1253, 1259, 1260, 1261, 1262, 1263, 1269, 1270, 1271, 1272, 1290, 1291, 1292, 1295, 1300, 1301, 1302, 1304, 1305, 1306, 1307, 1309, 1310, 1311, 1312, 1314, 1315, 1316, 1317, 1390, 1391, 1392, 1395, 1400, 1401, 1402, 1403, 1404, 1405, 1406, 1407, 1409, 1410, 1411, 1412, 1413, 1414, 1415, 1416, 1417, 1419, 1420, 1421, 1422, 1430, 1450, 1455, 1500, 1501, 1502, 1503, 1504, 1505, 1506, 1507, 1509, 1510, 1511, 1512, 1513, 1514, 1515, 1516, 1517, 1519, 1520, 1521, 1522, 1525, 1526, 1529, 1530, 1531, 1532, 1535, 1540, 1541, 1542, 1543, 1545, 1550, 1551, 1552, 1553, 1554, 1555, 1556, 1557, 1559, 1560, 1561, 1562, 1563, 1564, 1565, 1566, 1567, 1569, 1570, 1571, 1572, 1579, 1580, 1590, 1591, 1592, 1593, 1595, 1600, 1601, 1602, 1603, 1604, 1605, 1606, 1607, 1609, 1610, 1611, 1612, 1613, 1614, 1615, 1616, 1617, 1619, 1620, 1621, 1622, 1629, 1630, 1631, 1640, 1641, 1642, 1645, 1650, 1651, 1652, 1653, 1654, 1655, 1656, 1657, 1659, 1660, 1661, 1662, 1664, 1665, 1666, 1667, 1669, 1670, 1671, 1690, 1691, 1692, 1695, 1700, 1701, 1702, 1703, 1704, 1705, 1706, 1707, 1709, 1710, 1711, 1712, 1713, 1714, 1715, 1716, 1719, 1720, 1721, 1722, 1730, 1790, 1800, 1801, 1804, 1805, 1806, 1809, 1810, 1811, 1814, 1815, 1816, 1900, 1901, 1902, 1903, 1904, 1905, 1906, 1907, 1909, 1910, 1911, 1912, 1913, 1914, 1915, 1916, 1917, 1920, 1921, 1922, 1925, 1926, 1930, 1931, 1935, 1950, 1955, 2000, 2001, 2002, 2003, 2004, 2005, 2006, 2007, 2009, 2010, 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2019, 2020, 2021, 2022, 2025, 2026, 2029, 2030, 2031, 2032, 2035, 2039, 2040, 2041, 2042, 2043, 2045, 2050, 2051, 2052, 2053, 2054, 2055, 2056, 2057, 2059, 2060, 2061, 2062, 2063, 2064, 2065, 2066, 2067, 2069, 2070, 2071, 2072, 2080, 2090, 2091, 2092, 2093, 2095, 2100, 2101, 2102, 2103, 2104, 2105, 2106, 2107, 2109, 2110, 2111, 2112, 2113, 2114, 2115, 2116, 2117, 2119, 2120, 2125, 2129, 2130, 2139, 2140, 2141, 2142, 2145, 2150, 2151, 2152, 2153, 2154, 2155, 2156, 2157, 2159, 2160, 2161, 2162, 2163, 2164, 2165, 2166, 2167, 2169, 2170, 2190, 2191, 2192, 2193, 2195, 2200, 2201, 2202, 2203, 2204, 2205, 2206, 2207, 2209, 2210, 2211, 2212, 2213, 2214, 2215, 2216, 2217, 2219, 2220, 2225, 2229, 2230, 2240, 2241, 2242, 2250, 2251, 2252, 2253, 2259, 2260, 2261, 2262, 2263, 2269, 2270, 2290, 2291, 2292, 2295, 2300, 2301, 2302, 2304, 2305, 2306, 2307, 2390, 2391, 2392, 2400, 2401, 2402, 2403, 2409, 2410, 2411, 2412, 2413, 2419, 2420, 2421, 2422, 2430, 2500, 2501, 2502, 2503, 2509, 2510, 2511, 2512, 2513, 2519, 2520, 2521, 2522, 2529, 2530, 2531, 2532, 2540, 2541, 2542, 2543, 2550, 2551, 2552, 2553, 2559, 2560, 2561, 2562, 2563, 2569, 2570, 2571, 2572, 2580, 2590, 2591, 2592, 2593, 2600, 2601, 2602, 2603, 2609, 2610, 2611, 2612, 2613, 2619, 2620, 2629, 2630, 2640, 2641, 2642, 2650, 2651, 2652, 2653, 2659, 2660, 2661, 2662, 2690, 2691, 2692, 2700, 2701, 2702, 2703, 2709, 2710, 2711, 2712, 2713, 2719, 2720, 2730, 2790, 2800, 2801, 2900, 2901, 2902, 2904, 2905, 2906, 2907, 2909, 2910, 2911, 2912, 2914, 2915, 2916, 2917, 2920, 2921, 2922, 2925, 2926, 2930, 2931, 2935, 2950, 2955, 3000, 3001, 3002, 3004, 3005, 3006, 3007, 3009, 3010, 3011, 3012, 3014, 3015, 3016, 3017, 3019, 3020, 3021, 3022, 3025, 3026, 3029, 3030, 3031, 3032, 3035, 3039, 3040, 3041, 3042, 3045, 3050, 3051, 3052, 3054, 3055, 3056, 3057, 3059, 3060, 3061, 3062, 3064, 3065, 3066, 3067, 3069, 3070, 3071, 3072, 3080, 3090, 3091, 3092, 3095, 3100, 3101, 3102, 3104, 3105, 3106, 3107, 3109, 3110, 3111, 3112, 3114, 3115, 3116, 3117, 3119, 3120, 3129, 3130, 3139, 3140, 3141, 3142, 3145, 3150, 3151, 3152, 3154, 3155, 3156, 3157, 3159, 3160, 3161, 3162, 3164, 3165, 3166, 3167, 3169, 3170, 3190, 3191, 3192, 3195, 3200, 3201, 3202, 3204, 3205, 3206, 3207, 3209, 3210, 3211, 3212, 3214, 3215, 3216, 3217, 3219, 3220, 3229, 3230, 3240, 3241, 3242, 3250, 3251, 3252, 3259, 3260, 3261, 3262, 3290, 3291, 3292, 3295, 3300, 3301, 3302, 3304, 3305, 3306, 3307, 3390, 3391, 3392, 3400, 3401, 3402, 3409, 3410, 3411, 3412, 3419, 3420, 3421, 3422, 3430, 3500, 3501, 3502, 3509, 3510, 3511, 3512, 3519, 3520, 3521, 3522, 3529, 3530, 3531, 3532, 3540, 3541, 3542, 3550, 3551, 3552, 3559, 3560, 3561, 3562, 3569, 3570, 3571, 3572, 3580, 3900, 3901, 3902, 3904, 3905, 3906, 3907, 3909, 3910, 3911, 3912, 3914, 3915, 3916, 3917, 3920, 3921, 3922, 3925, 3926, 3930, 3931, 3935, 3950, 3955.

Tabelle numeriche

I numeri di Friedman noti.

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