Densités décroissantes (maths avec gomme)

[1] Nombres N > 99 divisibles par le produit des deux nombres P et Q qui apparaissent quand on efface l'un des chiffres de N. Le nombre Q qui apparaît après le chiffre effacé ne peut commencer par  un zéro (sauf si Q = 0).
Exemple de cas favorable : 3417 qui donne P =3 et Q = 17 [P*Q = 51 et 51 divise 3417 (résultat 67)]

Jean-Marc F. a calculé 10000 termes favorables, lesquels deviennent de plus en plus rares au fil du temps : la suite de ces favorables serait-elle finie  ? C'est la question qui fut posée ce jour à SeqFans (en anglais) :

Voici le début de la liste F de ces favorables pour la multiplication :

F = 101,102,104,105,111,112,115,121,122,123,124,125,126,128,131,132,135,141,142,144,145,147,151,152,153,155,156,161,162,164,165,168,171,172,175,181,182,183,184,185,186,189,191,192,195,208,212,216,224,232,252,264,272,276,288,292,306,312,315,321,324,333,342,345,351,357,372,375,381,384,396,432,448,456,464,472,525,575,612,624,672,721,728,735,742,756,784,791,816,832,864,945,972,981,1010,1020,1025,1040,1050,1110,1111,1120,1122,1125,1144,1150,1155,1210,1212,1215,1216,1220,1224,1225,1230,1240,1248,1250,1260,1272,1275,1280,1296,1310,1313,1320,1325,1326,1350,1352,1365,1391,1410,1414,1420,1425,1428,1435,1440,1450,1456,1470,1510,1512,1515,1520,1525,1530,1550,1560,1575,1610,1616,1620,1625,1632,1640,1650,1664,1680,1710,1717,1720,1725,1734,1750,1751,1768,1785,1810,1812,1815,1818,1820,1824,1825,1830,1836,1840,1845,1850,1860,1872,1875,1890,1910,1919,1920,1925,1938,1950,1976,1995,2016,2080,2112,2120,2121,2128,2142,2160,2184,2240,2288,2320,2346,2392,2432,2496,2520,2640,2652,2712,2720,2736,2760,2781,2816,2856,2880,2920,3015,3024,3060,3120,3131,3150,3162,3210,3216,3225,3240,3264,3312,3330,3366,3375,3417,...

Et le graphe associé à F :

On pourrait ajouter P à Q au lieu de les multiplier:
[2] Nombres N > 99 divisibles par l'addition des deux nombres P et Q qui apparaissent quand on efface l'un des chiffres de N. Le nombre Q qui apparaît après le chiffre effacé ne peut commencer par un zéro (sauf si Q = 0).

Voici le début de la suite G des favorables pour l'addition [exemple : 2409 fait partie de G car 24+9 divise 2409 (résultat = 73)] :

G =
100,102,108,110,120,126,130,132,140,150,160,162,170,180,190,192,196,198,200,201,204,207,209,210,212,216,220,230,231,232,234,240,245,250,252,256,260,261,264,270,272,280,290,291,292,294,296,297,300,306,308,330,360,364,390,396,400,402,405,407,408,413,420,424,429,432,440,460,462,464,468,480,483,492,495,500,504,506,510,520,530,532,540,550,560,570,580,590,594,598,600,603,605,630,632,636,637,651,658,660,672,690,693,696,700,702,704,770,792,800,801,803,804,826,840,845,848,864,880,891,896,900,902,918,990,1000,1001,1005,1008,1010,1020,1026,1030,1032,1035,1036,1037,1040,1050,1053,1056,1060,1064,1065,1070,1080,1090,1092,1095,1098,1100,1156,1178,1200,1206,1210,1232,1239,1260,1261,1264,1275,1296,1300,1394,1400,1404,1407,1408,1425,1449,1470,1472,1494,1500,1518,1530,1560,1590,1596,1600,1602,1606,1608,1612,1638,1640,1653,1680,1692,1700,1764,1800,1804,1809,1881,1890,1896,1898,1900,1932,1998,2000,2002,2016,2020,2025,2035,2040,2052,2059,2060,2064,2072,2075,2080,2093,2100,2106,2162,2200,2212,2231,2232,2295,2296,2300,2400,2403,2409,...

Graphe associé à G :

On peut demander à la différence absolue |P-Q| de diviser le nombre N de départ :
[3] Nombres N > 99 divisibles par la différence absolue |P-Q| des deux nombres P et Q visibles quand on efface l'un des chiffres de N. Le nombre Q qui apparaît après le chiffre effacé ne peut commencer par un zéro sauf si Q = 0.

Voici le début de la suite H des favorables pour la différence absolue [exemple : 694 fait partie de H car 6-4 divise 694 (résultat = 347)] :

H =
100,102,110,112,114,120,122,130,132,140,142,144,150,152,160,162,168,170,172,174,180,182,190,192,200,201,203,204,210,211,213,214,216,220,221,223,224,225,228,230,231,233,234,236,240,241,243,244,250,251,253,254,255,256,258,259,260,261,263,264,270,271,273,274,276,280,281,283,284,285,288,290,291,293,294,296,300,302,304,306,312,314,322,324,330,332,334,336,342,344,352,354,360,362,364,366,372,374,382,384,390,392,394,396,400,402,403,405,406,408,411,412,413,415,416,417,420,422,423,425,426,428,432,433,435,436,440,441,442,443,445,446,447,448,452,453,455,456,460,462,463,465,466,468,471,472,473,475,476,477,480,482,483,485,486,488,492,493,495,496,500,504,506,510,514,516,520,522,524,526,528,530,534,536,540,544,546,550,552,554,556,558,560,564,566,570,574,576,580,582,584,586,588,590,594,596,600,603,604,605,607,608,609,612,614,615,617,618,624,625,627,628,630,632,633,634,635,637,638,639,644,645,647,648,652,654,655,657,658,660,663,664,665,667,668,669,672,674,675,677,678,684,685,687,688,690,692,693,694,695,697,698,699,700,706,708

Graphe associé à H :

On pourrait aussi demander à P/Q de diviser N :
[4] Nombres N > 99 divisibles par P/Q, P et Q apparaissant quand on efface l'un des chiffres de N. Le nombre Q qui apparaît après le chiffre effacé ne peut commencer par un zéro.

Voici le début de la suite I des favorables pour la division par P/Q [exemple : 2157 fait partie de I car 21/7 divise 2157 (résultat = 719)] :


I =
101,111,121,131,141,151,161,171,181,191,202,212,222,232,242,252,262,272,282,292,303,313,321,323,333,343,351,353,363,373,381,383,393,402,404,412,414,422,424,432,434,442,444,452,454,462,464,472,474,482,484,492,494,505,515,525,535,545,555,565,575,585,595,606,612,616,626,636,642,646,656,666,672,676,686,696,707,717,721,727,737,747,757,767,777,787,791,797,804,808,812,814,818,824,828,832,834,838,844,848,852,854,858,864,868,872,874,878,884,888,892,894,898,903,909,919,929,933,939,949,959,963,969,979,981,989,993,999,1111,1206,1212,1216,1224,1226,1236,1242,1246,1254,1256,1266,1272,1276,1284,1286,1296,1391,1442,1515,1545,1575,1604,1608,1618,1624,1628,1632,1638,1644,1648,1658,1664,1668,1672,1678,1684,1688,1698,1751,1806,1836,1866,1872,1896,2121,2127,2157,2163,2187,2222,2412,2416,2418,2424,2436,2448,2454,2456,2472,2476,2478,2484,2496,2505,2515,2525,2535,2545,2555,2565,2575,2585,2595,2652,2709,2739,2763,2769,2781,2799,2814,2842,2884,3131,3208,3224,3228,3232,3248,3264,3268,3288,3333,3535,3606,3636,3654,3666,3672,3696,3913...

Graphe associé à la suite I :

De même, symétriquement, pourrait-on demander à Q/P de diviser N :
[5] Nombres N > 99 divisibles par Q/P, P et Q apparaissant quand on efface l'un des chiffres de N. Le nombre Q qui apparaît après le chiffre effacé ne peut commencer par un zéro.

Voici le début de la suite J des favorables pour la division par Q/P [exemple : 2016 fait partie de J car 16/2 divise 2016 (résultat = 252)] :

J =
101,102,104,105,111,112,115,121,122,123,124,125,126,128,131,132,135,141,142,144,145,147,151,152,153,155,156,161,162,164,165,168,171,172,175,181,182,183,184,185,186,189,191,192,195,202,204,208,212,214,216,222,224,228,232,234,242,244,246,248,252,254,262,264,268,272,274,276,282,284,288,292,294,303,306,309,313,316,323,326,333,336,339,343,346,353,356,363,366,369,373,376,383,386,393,396,399,404,408,414,418,424,428,434,438,444,448,454,458,464,468,474,478,484,488,494,498,505,515,525,535,545,555,565,575,585,595,606,616,626,636,646,656,666,676,686,696,707,717,727,737,747,757,767,777,787,797,808,818,828,838,848,858,868,878,888,898,909,919,929,939,949,959,969,979,989,999,1001,1002,1004,1005,1008,1010,1020,1025,1040,1050,1101,1102,1104,1105,1110,1111,1120,1122,1125,1144,1150,1155,1201,1202,1203,1204,1205,1206,1208,1210,1212,1215,1216,1220,1224,1225,1230,1240,1248,1250,1260,1275,1280,1301,1302,1304,1305,1310,1313,1320,1325,1326,1350,1352,1365,1401,1402,1404,1405,1407,1408,1410,1414,1420,1425,1428,1435,1440,1450,1456,1470,1501,1502,1503,1504,1505,1506,1510,1512,1515,1520,1525,1530,1550,1560,1575,1601,1602,1604,1605,1608,1610,1616,1620,1625,1632,1640,1650,1664,1680,1701,1702,1704,1705,1710,1717,1720,1725,1734,1750,1768,1785,1801,1802,1803,1804,1805,1806,1808,1809,1810,1812,1815,1818,1820,1824,1825,1830,1836,1840,1845,1850,1860,1872,1875,1890,1901,1902,1904,1905,1910,1919,1920,1925,1938,1950,1976,1995,2002,2004,2008,2010,2016,...

Graphe associé à J :

On pourrait aussi demander à P puissance Q de diviser N :
[6] Nombres N > 99 divisibles par P^Q, P et Q apparaissant quand on efface l'un des chiffres de N. Le nombre Q qui apparaît après le chiffre effacé ne peut commencer par un zéro (sauf si Q = 0).

Voici le début de la suite K des favorables pour l'exponentiation P^Q [exemple : 256 fait partie de K car 2^6 divise 256 (résultat = 4)] :

K =
100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,210,212,220,224,230,232,240,250,252,256,260,270,272,280,290,292,300,310,320,321,324,330,340,342,350,351,360,370,380,381,390,400,410,420,430,432,440,450,460,470,480,490,500,510,520,530,540,550,560,570,580,590,600,610,612,620,630,640,650,660,670,680,690,700,710,720,721,730,740,750,760,770,780,790,791,800,810,820,830,832,840,850,860,870,880,890,900,910,920,930,940,950,960,970,972,980,981,990,1000,1010,1011,1012,1013,1014,1015,1016,1017,1018,1019,1020,1021,1022,1023,1024,1025,1026,1027,1028,1029,1030,1031,1032,1033,1034,1035,1036,1037,1038,1039,1040,1041,1042,1043,1044,1045,1046,1047,1048,1049,1050,1051,1052,1053,1054,

Graphe associé à K :

On pourrait demander de même à Q puissance P de diviser N :
[7] Nombres N > 99 divisibles par Q^P, P et Q apparaissant quand on efface l'un des chiffres de N. Le nombre Q qui apparaît après le chiffre effacé ne peut commencer par un zéro (sauf si Q = 0).

Voici le début de la suite L des favorables pour l'exponentiation Q^P [exemple : 1951 fait partie de L car 1^19 divise 1951 (résultat 1951) ] :

L =
101,102,104,105,111,112,115,121,122,123,124,125,126,128,131,132,135,141,142,144,145,147,151,152,153,155,156,161,162,164,165,168,171,172,175,181,182,183,184,185,186,189,191,192,195,201,211,212,216,221,224,225,231,232,241,243,251,252,261,271,272,275,281,291,292,301,311,312,321,331,341,351,352,361,371,375,381,384,391,392,401,411,421,431,432,441,451,461,471,481,491,501,511,512,521,531,541,551,561,571,581,591,601,611,621,631,641,651,661,671,681,691,701,711,721,731,741,751,761,771,781,791,801,811,821,831,841,851,861,871,881,891,901,911,921,931,941,951,961,971,981,991,1001,1010,1011,1020,1021,1025,1031,1040,1041,1050,1051,1061,1071,1081,1091,1101,1110,1111,1120,1121,1122,1125,1131,1141,1144,1150,1151,1155,1161,1171,1181,1191,1201,1210,1211,1212,1215,1216,1220,1221,1224,1225,1230,1231,1240,1241,1248,1250,1251,1260,1261,1271,1275,1280,1281,1291,1301,1310,1311,1313,1320,1321,1325,1326,1331,1341,1350,1351,1352,1361,1365,1371,1381,1391,1401,1410,1411,1414,1420,1421,1425,1428,1431,1435,1440,1441,1450,1451,1456,1461,1470,1471,1481,1491,1501,1510,1511,1512,1515,1520,1521,1525,1530,1531,1541,1550,1551,1560,1561,1571,1575,1581,1591,1601,1610,1611,1616,1620,1621,1625,1631,1632,1640,1641,1650,1651,1661,1664,1671,1680,1681,1691,1701,1710,1711,1717,1720,1721,1725,1731,1734,1741,1750,1751,1761,1768,1771,1781,1785,1791,1801,1810,1811,1812,1815,1818,1820,1821,1824,1825,1830,1831,1836,1840,1841,1845,1850,1851,1860,1861,1871,1872,1875,1881,1890,1891,1901,1910,1911,1919,1920,1921,1925,1931,1938,1941,1950,1951,1961,1971,1976,1981,1991,1995,2001,2011,2021,2031,2041,2051,2061,2071,2081,2091,2101,2111,2121,2131,2141,2151,2161,2171,2181,2191,2201,2211,2221,2231,2241,2251,2261,2271,2281,2291,2301,2311,2321,2331,2341,2351,2361,2371,2381,2391,2401,2411,2421,2431,2441,2451,2461,2471,2481,2491,2501,2511,2521,2531,2541,2551,2561,2571,2581,2591,2601,2611,2621,2631,2641,2651,2661,2671,2681,2691,2701,2711,2721,2731,2741,2751,2761,2771,2781,2791,2801,2811,2816,2821,2831,2841,2851,2861,2871,2881,2891,2901,2911,2921,2931,2941,2951,2961,2971,2981,2991,3001,...

Graphe associé à L :

G O M M E                                      
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