NOMBRES - Curiosités, théorie et usages

 

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DIVISIBILITÉ

 

Débutants

Division

Somme

 

Glossaire

Division

 

 

INDEX

 

Partition

 

Divisibilité

 

Général

Somme de q par q

Consécutifs

Nombres polis

 

Sommaire de cette page

>>> Approche

>>> Formule

>>> Table

 

 

 

 

NOMBRES POLIS

et MULTISOMME de CONSÉCUTIFS

Partition en nombres consécutifs

 

Pratiquement tous les nombres, sauf les puissances de 2, sont somme de deux nombres consécutifs. Ils sont même souvent somme plusieurs fois de k nombres consécutifs

Les nombres polis sont  somme non triviale de nombres consécutifs. Ils sont très nombreux. Ceux qui ne le sont pas sont les nombres impolis. Ceux qui le sont k fois sont k-polis.

Tous les nombres impairs (donc tous les nombres premiers, sauf 2) sont la somme de deux nombres consécutifs. Ils sont tous polis.

 

Exemple: le nombre 2019 est 3-polis, somme trois fois de nombres consécutifs:

Voir Nombres par leur petit nom /  PartitionsIndex

 

Approche

 

Établissement des règles

En prenant un nombre n comme élément central, on calcule la somme de k nombres consécutifs autour de lui.

 

 

 

 

 

 

Si k est impair, la somme est un multiple de k.

 

Si un nombre n est divisible par k, il est somme de k nombres consécutifs*.

Exemple: 15 = 3 x 5 = 4 + 5 + 6

 

* à condition que les termes de la somme soient positifs.

 

Si k est pair, la somme est un multiple e k plus la moitié de k.

 

Si un nombre n diminué de k/2 est divisible par k, il est somme de k nombres consécutifs*.

Exemple: 45 – 3 = 42 = 6 x 7

45 = 5 + 6 + 7 + 8 + 9

 

 

 

Exemples complets

Le nombre 15 est trois fois sommes de nombres consécutifs.

Deux fois du fait de ces deux facteurs 3 et 5, et

Une fois avec k = 2, ce qui toujours le cas pour un nombre impair comme 15.

Avec k = 6, on retrouve la somme précédente  avec un 0 en plus.

 

Le nombre 21 est trois fois somme de nombres consécutifs.

 

Diviseurs de 21:
1, 3, 7, 21

Ils sont quatre un de plus que la quantité de sommes.

Le nombre 45 est cinq fois somme de nombres consécutifs.

 

Diviseurs de 45:
1, 3, 5, 9, 15, 45

Ils sont six, un de plus de la quantité de sommes.

 

 

Formule

 

 

La quantité de k-sommes d'un nombre n est égale à sa quantité de diviseurs impairs moins 1.

 

Voir Diviseurs / Impair

 

Exemples

Div(15) = 1, 3, 5, 15 => 3 sommes

 

Div(45) = 1, 3, 5, 9, 15, 45 => 5 sommes

 

Div(100) =  1, 2, 4, 5, 10, 20, 25, 50, 100 => 2 sommes

100 = 18 + 19 + 20 + 21 + 22 

        = 9 + 10 + 11 + 12 + 13 + 14 + 15 + 16

 

Div(2019) = 1, 3, 673, 2019 => 3 sommes

2019 = 1009 + 1010

          = 672 + 673 + 674

          = 334 + 335 + 336 + 337 + 338 + 339

 

Div(1000) = 1, 2, 4, 5, 8, 10, 20, 25, 40, 50, 100, 125, 200, 250, 500, 1000 = 3 sommes

1000 = 198 + 199 + 200 + 201 + 202

         =  55 + 56 + … + 62 +… + 70

         =  28 + 29 + … + 40 + … + 52

Merci à Jean B. pour sa relecture attentive

 

Tables

 

Polis (nombres pairs en rouge)

3, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, …

 

 

Impolis

 

1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, …

Ce sont les puissances de 2.

 

Les premières valeurs explicitées jusqu'à 50

[nombre n, quantité de sommes,[nombre central, quantité de termes], [idem]]

Exemple: [9, 2, [4, 2], [3, 3]] => 9 est somme 2 fois: 9 = 4 + 5 = 2 + 3 + 4

 

[1, 0]

[2, 0]

[3, 1, [1, 2]]

[4, 0]

[5, 1, [2, 2]]

[6, 1, [2, 3]]

[7, 1, [3, 2]]

[8, 0]

[9, 2, [4, 2], [3, 3]]

[10, 1, [2, 4]]

[11, 1, [5, 2]]

[12, 1, [4, 3]]

[13, 1, [6, 2]]

[14, 1, [3, 4]]

[15, 3, [7, 2], [5, 3], [3, 5]]

[16, 0]

[17, 1, [8, 2]]

[18, 2, [6, 3], [4, 4]]

[19, 1, [9, 2]]

[20, 1, [4, 5]]

[21, 3, [10, 2], [7, 3], [3, 6]]

[22, 1, [5, 4]]

[23, 1, [11, 2]]

[24, 1, [8, 3]]

[25, 2, [12, 2], [5, 5]]

[26, 1, [6, 4]]

[27, 3, [13, 2], [9, 3], [4, 6]]

[28, 1, [4, 7]]

[29, 1, [14, 2]]

[30, 3, [10, 3], [7, 4], [6, 5]]

[31, 1, [15, 2]]

[32, 0]

[33, 3, [16, 2], [11, 3], [5, 6]]

[34, 1, [8, 4]]

[35, 3, [17, 2], [7, 5], [5, 7]]

[36, 2, [12, 3], [4, 8]]

[37, 1, [18, 2]]

[38, 1, [9, 4]]

[39, 3, [19, 2], [13, 3], [6, 6]]

[40, 1, [8, 5]]

[41, 1, [20, 2]]

[42, 3, [14, 3], [10, 4], [6, 7]]

[43, 1, [21, 2]]

[44, 1, [5, 8]]

[45, 5, [22, 2], [15, 3], [9, 5], [7, 6], [5, 9]]

[46, 1, [11, 4]]

[47, 1, [23, 2]]

[48, 1, [16, 3]]

[49, 2, [24, 2], [7, 7]]

[50, 2, [12, 4], [10, 5]]

Voir Table similaire

 

K-Polis

 

Deux sommes jusqu'à 1000

[9, 2, [4, 2], [3, 3]]

[18, 2, [6, 3], [4, 4]]

[25, 2, [12, 2], [5, 5]]

[36, 2, [12, 3], [4, 8]]

[49, 2, [24, 2], [7, 7]]

[50, 2, [12, 4], [10, 5]]

[72, 2, [24, 3], [8, 9]]

[98, 2, [24, 4], [14, 7]]

[100, 2, [20, 5], [12, 8]]

[121, 2, [60, 2], [11, 11]]

[144, 2, [48, 3], [16, 9]]

[169, 2, [84, 2], [13, 13]]

[196, 2, [28, 7], [24, 8]]

[200, 2, [40, 5], [12, 16]]

[242, 2, [60, 4], [22, 11]]

[288, 2, [96, 3], [32, 9]]

[289, 2, [144, 2], [17, 17]]

[338, 2, [84, 4], [26, 13]]

[361, 2, [180, 2], [19, 19]]

[392, 2, [56, 7], [24, 16]]

[400, 2, [80, 5], [16, 25]]

[484, 2, [60, 8], [44, 11]]

[529, 2, [264, 2], [23, 23]]

[576, 2, [192, 3], [64, 9]]

[578, 2, [144, 4], [34, 17]]

[676, 2, [84, 8], [52, 13]]

[722, 2, [180, 4], [38, 19]]

[784, 2, [112, 7], [24, 32]]

[800, 2, [160, 5], [32, 25]]

[841, 2, [420, 2], [29, 29]]

[961, 2, [480, 2], [31, 31]]

[968, 2, [88, 11], [60, 16]]

 

Trois sommes jusqu'à 100

[15, 3, [7, 2], [5, 3], [3, 5]]

[21, 3, [10, 2], [7, 3], [3, 6]]

[27, 3, [13, 2], [9, 3], [4, 6]]

[30, 3, [10, 3], [7, 4], [6, 5]]

[33, 3, [16, 2], [11, 3], [5, 6]]

[35, 3, [17, 2], [7, 5], [5, 7]]

[39, 3, [19, 2], [13, 3], [6, 6]]

[42, 3, [14, 3], [10, 4], [6, 7]]

[51, 3, [25, 2], [17, 3], [8, 6]]

[54, 3, [18, 3], [13, 4], [6, 9]]

[55, 3, [27, 2], [11, 5], [5, 10]]

[57, 3, [28, 2], [19, 3], [9, 6]]

[60, 3, [20, 3], [12, 5], [7, 8]]

[65, 3, [32, 2], [13, 5], [6, 10]]

[66, 3, [22, 3], [16, 4], [6, 11]]

[69, 3, [34, 2], [23, 3], [11, 6]]

[70, 3, [17, 4], [14, 5], [10, 7]]

[77, 3, [38, 2], [11, 7], [7, 11]]

[78, 3, [26, 3], [19, 4], [6, 12]]

[84, 3, [28, 3], [12, 7], [10, 8]]

[85, 3, [42, 2], [17, 5], [8, 10]]

[87, 3, [43, 2], [29, 3], [14, 6]]

[91, 3, [45, 2], [13, 7], [7, 13]]

[93, 3, [46, 2], [31, 3], [15, 6]]

[95, 3, [47, 2], [19, 5], [9, 10]]

[102, 3, [34, 3], [25, 4], [8, 12]]

 

 

Quatre sommes jusqu'à 1000

[81, 4, [40, 2], [27, 3], [13, 6], [9, 9]]

[162, 4, [54, 3], [40, 4], [18, 9], [13, 12]]

[324, 4, [108, 3], [40, 8], [36, 9], [13, 24]]

[625, 4, [312, 2], [125, 5], [62, 10], [25, 25]]

[648, 4, [216, 3], [72, 9], [40, 16], [24, 27]]

 

Cinq sommes jusqu'à 300

[45, 5, [22, 2], [15, 3], [9, 5], [7, 6], [5, 9]]

[63, 5, [31, 2], [21, 3], [10, 6], [9, 7], [7, 9]]

[75, 5, [37, 2], [25, 3], [15, 5], [12, 6], [7, 10]]

[90, 5, [30, 3], [22, 4], [18, 5], [10, 9], [7, 12]]

[99, 5, [49, 2], [33, 3], [16, 6], [11, 9], [9, 11]]

[117, 5, [58, 2], [39, 3], [19, 6], [13, 9], [9, 13]]

[126, 5, [42, 3], [31, 4], [18, 7], [14, 9], [10, 12]]

[147, 5, [73, 2], [49, 3], [24, 6], [21, 7], [10, 14]]

[150, 5, [50, 3], [37, 4], [30, 5], [12, 12], [10, 15]]

[153, 5, [76, 2], [51, 3], [25, 6], [17, 9], [9, 17]]

[171, 5, [85, 2], [57, 3], [28, 6], [19, 9], [9, 18]]

[175, 5, [87, 2], [35, 5], [25, 7], [17, 10], [12, 14]]

[180, 5, [60, 3], [36, 5], [22, 8], [20, 9], [12, 15]]

[198, 5, [66, 3], [49, 4], [22, 9], [18, 11], [16, 12]]

[207, 5, [103, 2], [69, 3], [34, 6], [23, 9], [11, 18]]

[234, 5, [78, 3], [58, 4], [26, 9], [19, 12], [18, 13]]

[243, 5, [121, 2], [81, 3], [40, 6], [27, 9], [13, 18]]

[245, 5, [122, 2], [49, 5], [35, 7], [24, 10], [17, 14]]

[252, 5, [84, 3], [36, 7], [31, 8], [28, 9], [12, 21]]

[261, 5, [130, 2], [87, 3], [43, 6], [29, 9], [14, 18]]

[275, 5, [137, 2], [55, 5], [27, 10], [25, 11], [12, 22]]

[279, 5, [139, 2], [93, 3], [46, 6], [31, 9], [15, 18]]

[294, 5, [98, 3], [73, 4], [42, 7], [24, 12], [14, 21]]

[300, 5, [100, 3], [60, 5], [37, 8], [20, 15], [12, 24]]

 

 

 

 

Six sommes jusqu'à 5000

[729, 6, [364, 2], [243, 3], [121, 6], [81, 9], [40, 18], [27, 27]]

[1458, 6, [486, 3], [364, 4], [162, 9], [121, 12], [54, 27], [40, 36]]

[2916, 6, [972, 3], [364, 8], [324, 9], [121, 24], [108, 27], [40, 72]]

 

Sept sommes jusqu'à 300

[105, 7, [52, 2], [35, 3], [21, 5], [17, 6], [15, 7], [10, 10], [7, 14]]

[135, 7, [67, 2], [45, 3], [27, 5], [22, 6], [15, 9], [13, 10], [9, 15]]

[165, 7, [82, 2], [55, 3], [33, 5], [27, 6], [16, 10], [15, 11], [11, 15]]

[189, 7, [94, 2], [63, 3], [31, 6], [27, 7], [21, 9], [13, 14], [10, 18]]

[195, 7, [97, 2], [65, 3], [39, 5], [32, 6], [19, 10], [15, 13], [13, 15]]

[210, 7, [70, 3], [52, 4], [42, 5], [30, 7], [17, 12], [14, 15], [10, 20]]

[231, 7, [115, 2], [77, 3], [38, 6], [33, 7], [21, 11], [16, 14], [11, 21]]

[255, 7, [127, 2], [85, 3], [51, 5], [42, 6], [25, 10], [17, 15], [15, 17]]

[270, 7, [90, 3], [67, 4], [54, 5], [30, 9], [22, 12], [18, 15], [13, 20]]

[273, 7, [136, 2], [91, 3], [45, 6], [39, 7], [21, 13], [19, 14], [13, 21]]

[285, 7, [142, 2], [95, 3], [57, 5], [47, 6], [28, 10], [19, 15], [15, 19]]

[297, 7, [148, 2], [99, 3], [49, 6], [33, 9], [27, 11], [16, 18], [13, 22]]

[330, 7, [110, 3], [82, 4], [66, 5], [30, 11], [27, 12], [22, 15], [16, 20]]

Voir Records en quantité de sommes de consécutifs

Quantité de partitions en nombres consécutifs

 

Pour les nombres de 1 à 1000.

Les nombres avec quantité 0 ou 1 ne sont pas indiqués.

Par exemple 45 a cinq partitions.

 

[[9, 2], [15, 3], [18, 2], [21, 3], [25, 2], [27, 3], [30, 3], [33, 3], [35, 3], [36, 2], [39, 3], [42, 3], [45, 5], [49, 2], [50, 2], [51, 3], [54, 3], [55, 3], [57, 3], [60, 3], [63, 5], [65, 3], [66, 3], [69, 3], [70, 3], [72, 2], [75, 5], [77, 3], [78, 3], [81, 4], [84, 3], [85, 3], [87, 3], [90, 5], [91, 3], [93, 3], [95, 3], [98, 2], [99, 5], [100, 2], [102, 3], [105, 7], [108, 3], [110, 3], [111, 3], [114, 3], [115, 3], [117, 5], [119, 3], [120, 3], [121, 2], [123, 3], [125, 3], [126, 5], [129, 3], [130, 3], [132, 3], [133, 3], [135, 7], [138, 3], [140, 3], [141, 3], [143, 3], [144, 2], [145, 3], [147, 5], [150, 5], [153, 5], [154, 3], [155, 3], [156, 3], [159, 3], [161, 3], [162, 4], [165, 7], [168, 3], [169, 2], [170, 3], [171, 5], [174, 3], [175, 5], [177, 3], [180, 5], [182, 3], [183, 3], [185, 3], [186, 3], [187, 3], [189, 7], [190, 3], [195, 7], [196, 2], [198, 5], [200, 2], [201, 3], [203, 3], [204, 3], [205, 3], [207, 5], [209, 3], [210, 7], [213, 3], [215, 3], [216, 3], [217, 3], [219, 3], [220, 3], [221, 3], [222, 3], [225, 8], [228, 3], [230, 3], [231, 7], [234, 5], [235, 3], [237, 3], [238, 3], [240, 3], [242, 2], [243, 5], [245, 5], [246, 3], [247, 3], [249, 3], [250, 3], [252, 5], [253, 3], [255, 7], [258, 3], [259, 3], [260, 3], [261, 5], [264, 3], [265, 3], [266, 3], [267, 3], [270, 7], [273, 7], [275, 5], [276, 3], [279, 5], [280, 3], [282, 3], [285, 7], [286, 3], [287, 3], [288, 2], [289, 2], [290, 3], [291, 3], [294, 5], [295, 3], [297, 7], [299, 3], [300, 5], [301, 3], [303, 3], [305, 3], [306, 5], [308, 3], [309, 3], [310, 3], [312, 3], [315, 11], [318, 3], [319, 3], [321, 3], [322, 3], [323, 3], [324, 4], [325, 5], [327, 3], [329, 3], [330, 7], [333, 5], [335, 3], [336, 3], [338, 2], [339, 3], [340, 3], [341, 3], [342, 5], [343, 3], [345, 7], [348, 3], [350, 5], [351, 7], [354, 3], [355, 3], [357, 7], [360, 5], [361, 2], [363, 5], [364, 3], [365, 3], [366, 3], [369, 5], [370, 3], [371, 3], [372, 3], [374, 3], [375, 7], [377, 3], [378, 7], [380, 3], [381, 3], [385, 7], [387, 5], [390, 7], [391, 3], [392, 2], [393, 3], [395, 3], [396, 5], [399, 7], [400, 2], [402, 3], [403, 3], [405, 9], [406, 3], [407, 3], [408, 3], [410, 3], [411, 3], [413, 3], [414, 5], [415, 3], [417, 3], [418, 3], [420, 7], [423, 5], [425, 5], [426, 3], [427, 3], [429, 7], [430, 3], [432, 3], [434, 3], [435, 7], [437, 3], [438, 3], [440, 3], [441, 8], [442, 3], [444, 3], [445, 3], [447, 3], [450, 8], [451, 3], [453, 3], [455, 7], [456, 3], [459, 7], [460, 3], [462, 7], [465, 7], [468, 5], [469, 3], [470, 3], [471, 3], [473, 3], [474, 3], [475, 5], [476, 3], [477, 5], [480, 3], [481, 3], [483, 7], [484, 2], [485, 3], [486, 5], [489, 3], [490, 5], [492, 3], [493, 3], [494, 3], [495, 11], [497, 3], [498, 3], [500, 3], [501, 3], [504, 5], [505, 3], [506, 3], [507, 5], [510, 7], [511, 3], [513, 7], [515, 3], [516, 3], [517, 3], [518, 3], [519, 3], [520, 3], [522, 5], [525, 11], [527, 3], [528, 3], [529, 2], [530, 3], [531, 5], [532, 3], [533, 3], [534, 3], [535, 3], [537, 3], [539, 5], [540, 7], [543, 3], [545, 3], [546, 7], [549, 5], [550, 5], [551, 3], [552, 3], [553, 3], [555, 7], [558, 5], [559, 3], [560, 3], [561, 7], [564, 3], [565, 3], [567, 9], [570, 7], [572, 3], [573, 3], [574, 3], [575, 5], [576, 2], [578, 2], [579, 3], [580, 3], [581, 3], [582, 3], [583, 3], [585, 11], [588, 5], [589, 3], [590, 3], [591, 3], [594, 7], [595, 7], [597, 3], [598, 3], [600, 5], [602, 3], [603, 5], [605, 5], [606, 3], [609, 7], [610, 3], [611, 3], [612, 5], [615, 7], [616, 3], [618, 3], [620, 3], [621, 7], [623, 3], [624, 3], [625, 4], [627, 7], [629, 3], [630, 11], [633, 3], [635, 3], [636, 3], [637, 5], [638, 3], [639, 5], [642, 3], [644, 3], [645, 7], [646, 3], [648, 4], [649, 3], [650, 5], [651, 7], [654, 3], [655, 3], [657, 5], [658, 3], [660, 7], [663, 7], [665, 7], [666, 5], [667, 3], [669, 3], [670, 3], [671, 3], [672, 3], [675, 11], [676, 2], [678, 3], [679, 3], [680, 3], [681, 3], [682, 3], [684, 5], [685, 3], [686, 3], [687, 3], [689, 3], [690, 7], [693, 11], [695, 3], [696, 3], [697, 3], [699, 3], [700, 5], [702, 7], [703, 3], [705, 7], [707, 3], [708, 3], [710, 3], [711, 5], [713, 3], [714, 7], [715, 7], [717, 3], [720, 5], [721, 3], [722, 2], [723, 3], [725, 5], [726, 5], [728, 3], [729, 6], [730, 3], [731, 3], [732, 3], [735, 11], [737, 3], [738, 5], [740, 3], [741, 7], [742, 3], [744, 3], [745, 3], [747, 5], [748, 3], [749, 3], [750, 7], [753, 3], [754, 3], [755, 3], [756, 7], [759, 7], [760, 3], [762, 3], [763, 3], [765, 11], [767, 3], [770, 7], [771, 3], [774, 5], [775, 5], [777, 7], [779, 3], [780, 7], [781, 3], [782, 3], [783, 7], [784, 2], [785, 3], [786, 3], [789, 3], [790, 3], [791, 3], [792, 5], [793, 3], [795, 7], [798, 7], [799, 3], [800, 2], [801, 5], [803, 3], [804, 3], [805, 7], [806, 3], [807, 3], [810, 9], [812, 3], [813, 3], [814, 3], [815, 3], [816, 3], [817, 3], [819, 11], [820, 3], [822, 3], [825, 11], [826, 3], [828, 5], [830, 3], [831, 3], [833, 5], [834, 3], [835, 3], [836, 3], [837, 7], [840, 7], [841, 2], [843, 3], [845, 5], [846, 5], [847, 5], [849, 3], [850, 5], [851, 3], [852, 3], [854, 3], [855, 11], [858, 7], [860, 3], [861, 7], [864, 3], [865, 3], [867, 5], [868, 3], [869, 3], [870, 7], [871, 3], [873, 5], [874, 3], [875, 7], [876, 3], [879, 3], [880, 3], [882, 8], [884, 3], [885, 7], [888, 3], [889, 3], [890, 3], [891, 9], [893, 3], [894, 3], [895, 3], [897, 7], [899, 3], [900, 8], [901, 3], [902, 3], [903, 7], [905, 3], [906, 3], [909, 5], [910, 7], [912, 3], [913, 3], [915, 7], [917, 3], [918, 7], [920, 3], [921, 3], [923, 3], [924, 7], [925, 5], [927, 5], [930, 7], [931, 5], [933, 3], [935, 7], [936, 5], [938, 3], [939, 3], [940, 3], [942, 3], [943, 3], [945, 15], [946, 3], [948, 3], [949, 3], [950, 5], [951, 3], [952, 3], [954, 5], [955, 3], [957, 7], [959, 3], [960, 3], [961, 2], [962, 3], [963, 5], [965, 3], [966, 7], [968, 2], [969, 7], [970, 3], [972, 5], [973, 3], [975, 11], [978, 3], [979, 3], [980, 5], [981, 5], [984, 3], [985, 3], [986, 3], [987, 7], [988, 3], [989, 3], [990, 11], [993, 3], [994, 3], [995, 3], [996, 3], [999, 7], [1000, 3]]

 

 

 

Retour

*         Partition en nombres consécutifs

Suite

*         Entiers consécutifs en général

*         Formes diverses

*         Critères généraux

*         DivisibilitéIndex 

Voir

*         Calcul mentalIndex

*         Fermat

*         Produit de consécutifs – Factorielle tronquée

*         Théorie des nombresIndex

DicoNombre

*         Nombre 99

*         Nombre 100

*         Nombre 999

*         Nombre 1000

*         Nombre 2 019 – Propriétés, jeux, humour

Sites

*         Polite number – Wikipedia

*         Polite and impolite numbers – Numbers Aplenty

*         OEIS A138591 – Sums of two or more consecutive nonnegative integers

*         OEIS A069283 – a(n) = -1 + number of odd divisors of n

Cette page

http://villemin.gerard.free.fr/Wwwgvmm/Decompos/aaaDIVIS/NbPoli.htm