Partion des nombres en sommes de palindromes

Édition du: 26/04/2022

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INDEX

Partition

Calculs

PARTITIONS

Approche

En bref et Index

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Quantité de partitions

Palindromiques

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Partitions palindromiques

Quelles sont les sommes (partitions) d'un nombre qui peuvent s'écrire en palindrome ?

Illustration avec 4 partitions palindromiques du nombre 37 Les deux premières lignes sont palindromiques en nombres, les deux suivantes en chiffres.

Il existe 21 637 partitions du nombre 37 dont 297 sont palindromiques en nombres et 68 en chiffres sans 1 ni 2.

Sommaire de cette page

>>> Approche

>>> Partitions palindromiques de n de 2 à 10

>>> Partitions palindromiques de n SANS 1

>>> Partitions de 37

Débutants

Addition

Glossaire

Addition

Voir Brève 883 / Nombre 37

Approche				haut
Exemple Prenons les partitions du nombre 10. Cette liste montre la suite des 42 partitions. En rouge les sommes non palindromes. En noir les 19 sommes palindromes. La somme des nombres entre crochet est 10.	[1, 1, 1, 1, 1, 1, 1, 1, 1, 1] [1, 1, 1, 1, 2, 1, 1, 1, 1] [1, 1, 1, 2, 2,1, 1, 1] [1, 1, 2, 2, 2, 1, 1] [1, 2, 2, 2, 2, 1] [2, 2, 2, 2, 2] [1, 1, 1, 1, 1, 1, 1, 3] [1, 1, 1, 1, 1, 2, 3] [1, 1, 1, 2, 2, 3] [1, 2, 2, 2, 3] [1, 1, 3, 3, 1, 1] [1, 3, 2, 3, 1] [2, 3, 3, 2] [1, 3, 3, 3]	[1, 1, 1, 4, 1, 1, 1] [1, 1, 1, 1, 2, 4] [1, 2, 4, 2, 1] [2, 2, 2, 4] [1, 1, 1, 3, 4] [1, 2, 3, 4] [3, 4, 3] [1, 4, 4, 1] [4, 2, 4] [1, 1, 1, 1, 1, 5] [1, 1, 1, 2, 5] [1, 2, 2, 5] [1, 1, 3, 5] [2, 3, 5]	[1, 4, 5] [5, 5] [1, 1, 6, 1, 1] [1, 1, 2, 6] [2, 6, 2] [1, 3, 6] [4, 6] [1, 1, 1, 7] [1, 2, 7] [3, 7] [1, 8, 1] [2, 8] [1, 9] [10]
Observations	Pour former une somme palindromique: si la quantité de termes est paire, il faut pouvoir arranger les nombres par couples; et si la quantité est impaire, il faut former des couples et disposer d'un nombre isolé, lequel prendra la position centrale.

Partitions palindromiques de n de 2 à 10				haut
n, p, pp	Toutes les partitions palindromes (pp) de n parmi les p partitions de n. On donne la liste croissante des nombres; il suffit de les réorganiser pour constituer la somme palindrome. Ainsi, par exemple [1, 1, 2] deviendra [1, 2, 1]. Exemple de lecture: 5, 7, 4 veut dire que le nombre 5 a 7 partitions dont 4 sont palindromes.
2, 2, 2	[1, 1]	[2]
3, 3, 2	[1, 1, 1]	[3]
4, 5, 4	[1, 1, 1, 1] [1, 1, 2]	[2, 2]	[4]
5, 7, 4	[1, 1, 1, 1, 1] [1, 2, 2]	[1, 1, 3]	[5]
6, 11, 7	[1, 1, 1, 1, 1, 1] [1, 1, 1, 1, 2] [1, 1, 2, 2]	[2, 2, 2] [3, 3]	[1, 1, 4] [6
7, 15, 7	[1, 1, 1, 1, 1, 1, 1] [1, 1, 1, 2, 2] [1, 1, 1, 1, 3]	[2, 2, 3] [1, 3, 3]	[1, 1, 5] [7]
8, 22, 12	[1, 1, 1, 1, 1, 1, 1, 1] [1, 1, 1, 1, 1, 1, 2] [1, 1, 1, 1, 2, 2] [1, 1, 2, 2, 2]	[2, 2, 2, 2] [1, 1, 3, 3] [2, 3, 3] [1, 1, 1, 1, 4]	[2, 2, 4] [4, 4] [1, 1, 6] [8]
9, 30, 12	[1, 1, 1, 1, 1, 1, 1, 1, 1] [1, 1, 1, 1, 1, 2, 2] [1, 2, 2, 2, 2] [1, 1, 1, 1, 1, 1, 3]	[1, 1, 2, 2, 3] [1, 1, 1, 3, 3] [3, 3, 3] [1, 4, 4]	[1, 1, 1, 1, 5] [2, 2, 5] [1, 1, 7] [9]
10, 42, 19	[1, 1, 1, 1, 1, 1, 1, 1, 1, 1] [1, 1, 1, 1, 1, 1, 1, 1, 2] [1, 1, 1, 1, 1, 1, 2, 2] [1, 1, 1, 1, 2, 2, 2] [1, 1, 2, 2, 2, 2] [2, 2, 2, 2, 2] [1, 1, 1, 1, 3, 3]	[1, 1, 2, 3, 3] [2, 2, 3, 3] [1, 1, 1, 1, 1, 1, 4] [1, 1, 2, 2, 4] [3, 3, 4] [1, 1, 4, 4]	[2, 4, 4] [5, 5] [1, 1, 1, 1, 6] [2, 2, 6] [1, 1, 8] [10]
Liste des triplets [n, p, pp] pour n de 2 à 69	[2, 2, 2], [3, 3, 2], [4, 5, 4], [5, 7, 4], [6, 11, 7], [7, 15, 7], [8, 22, 12], [9, 30, 12], [10, 42, 19], [11, 56, 19], [12, 77, 30], [13, 101, 30], [14, 135, 45], [15, 176, 45], [16, 231, 67], [17, 297, 67], [18, 385, 97], [19, 490, 97], [20, 627, 139], [21, 792, 139], [22, 1002, 195], [23, 1255, 195], [24, 1575, 272], [25, 1958, 272], [26, 2436, 373], [27, 3010, 373], [28, 3718, 508], [29, 4565, 508], [30, 5604, 684], [31, 6842, 684], [32, 8349, 915], [33, 10143, 915], [34, 12310, 1212], [35, 14883, 1212], [36, 17977, 1597], [37, 21637, 1597], [38, 26015, 2087], [39, 31185, 2087], [40, 37338, 2714], [41, 44583, 2714], [42, 53174, 3506], [43, 63261, 3506], [44, 75175, 4508], [45, 89134, 4508], [46, 105558, 5763], [47, 124754, 5763], [48, 147273, 7338], [49, 173525, 7338], [50, 204226, 9296], [51, 239943, 9296], [52, 281589, 11732], [53, 329931, 11732], [54, 386155, 14742], [55, 451276, 14742], [56, 526823, 18460], [57, 614154, 18460], [58, 715220, 23025], [59, 831820, 23025], [60, 966467, 28629], [61, 1121505, 28629], [62, 1300156, 35471], [63, 1505499, 35471], [64, 1741630, 43820], [65, 2012558, 43820], [66, 2323520, 53963], [67, 2679689, 53963], [68, 3087735, 66273], [69, 3554345, 66273] …

Partitions palindromiques de n SANS 1

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10, 42, 7

[2, 2, 2, 2, 2]

[2, 2, 3, 3]

[3, 3, 4]

[2, 4, 4]

[5, 5]

[2, 2, 6]

[10]

11, 56, 5

[2, 2, 2, 2, 3]

[3, 4, 4]

[3, 3, 5]

[2, 2, 7]

[11]

12, 77, 11

[2, 2, 2, 2, 2, 2]

[2, 2, 2, 3, 3]

[3, 3, 3, 3]

[2, 2, 2, 2, 4]

[2, 2, 4, 4]

[4, 4, 4]

[2, 5, 5]

[3, 3, 6]

[6, 6]

[2, 2, 8]

[12]

13, 101, 7

[2, 2, 3, 3, 3]

[2, 2, 2, 2, 5]

[4, 4, 5]

[3, 5, 5]

[3, 3, 7]

[2, 2, 9]

[13]

14, 135, 15

[2, 2, 2, 2, 2, 2, 2]

[2, 2, 2, 2, 3, 3]

[2, 3, 3, 3, 3]

[2, 2, 3, 3, 4]

[2, 2, 2, 4, 4]

[3, 3, 4, 4]

[2, 2, 5, 5]

[4, 5, 5]

[2, 2, 2, 2, 6]

[4, 4, 6]

[2, 6, 6]

[7, 7]

[3, 3, 8]

[2, 2, 10]

[14]

15, 176, 11

[2, 2, 2, 2, 2, 2, 3]

[3, 3, 3, 3, 3]

[2, 2, 3, 4, 4]

[2, 2, 3, 3, 5]

[5, 5, 5]

[3, 6, 6]

[2, 2, 2, 2, 7]

[4, 4, 7]

[3, 3, 9]

[2, 2, 11]

[15]

16, 231, 22

[2, 2, 2, 2, 2, 2, 2, 2]

[2, 2, 2, 2, 2, 3, 3]

[2, 2, 3, 3, 3, 3]

[2, 2, 2, 2, 2, 2, 4]

[3, 3, 3, 3, 4]

[2, 2, 2, 2, 4, 4]

[2, 3, 3, 4, 4]

[2, 2, 4, 4, 4]

[4, 4, 4, 4]

[2, 2, 2, 5, 5]

[3, 3, 5, 5]

[2, 2, 3, 3, 6]

[5, 5, 6]

[2, 2, 6, 6]

[4, 6, 6]

[2, 7, 7]

[2, 2, 2, 2, 8]

[4, 4, 8]

[8, 8]

[3, 3, 10]

[2, 2, 12]

[16]

17, 297, 15

[2, 2, 2, 2, 3, 3, 3]

[3, 3, 3, 4, 4]

[2, 2, 2, 2, 2, 2, 5]

[3, 3, 3, 3, 5]

[2, 2, 4, 4, 5]

[2, 2, 3, 5, 5]

[5, 6, 6]

[2, 2, 3, 3, 7]

[5, 5, 7]

[3, 7, 7]

[2, 2, 2, 2, 9]

[4, 4, 9]

[3, 3, 11]

[2, 2, 13]

[17]

18, 385, 30

[2, 2, 2, 2, 2, 2, 2, 2, 2]

[2, 2, 2, 2, 2, 2, 3, 3]

[2, 2, 2, 3, 3, 3, 3]

[3, 3, 3, 3, 3, 3]

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

[2, 2, 2, 2, 2, 4, 4]

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

[3, 3, 4, 4, 4]

[2, 4, 4, 4, 4]

[2, 2, 2, 2, 5, 5]

[2, 3, 3, 5, 5]

[2, 2, 4, 5, 5]

[4, 4, 5, 5]

[2, 2, 2, 2, 2, 2, 6]

[3, 3, 3, 3, 6]

[2, 2, 4, 4, 6]

[2, 2, 2, 6, 6]

[3, 3, 6, 6]

[6, 6, 6]

[2, 2, 7, 7]

[4, 7, 7]

[2, 2, 3, 3, 8]

[5, 5, 8]

[2, 8, 8]

[9, 9]

[2, 2, 2, 2, 10]

[4, 4, 10]

[3, 3, 12]

[2, 2, 14]

[18]

19, 490, 22

[2, 2, 2, 2, 2, 2, 2, 2, 3]

[2, 2, 3, 3, 3, 3, 3]

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

[3, 4, 4, 4, 4]

[2, 2, 2, 2, 3, 3, 5]

[3, 3, 4, 4, 5]

[3, 3, 3, 5, 5]

[2, 2, 5, 5, 5]

[2, 2, 3, 6, 6]

[2, 2, 2, 2, 2, 2, 7]

[3, 3, 3, 3, 7]

[2, 2, 4, 4, 7]

[6, 6, 7]

[5, 7, 7]

[3, 8, 8]

[2, 2, 3, 3, 9]

[5, 5, 9]

[2, 2, 2, 2, 11]

[4, 4, 11]

[3, 3, 13]

[2, 2, 15]

[19]

20, 627, 42

[2, 2, 2, 2, 2, 2, 2, 2, 2, 2]

[2, 2, 2, 2, 2, 2, 2, 3, 3]

[2, 2, 2, 2, 3, 3, 3, 3]

[2, 3, 3, 3, 3, 3, 3]

[2, 2, 2, 2, 2, 2, 2, 2, 4]

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

[2, 2, 2, 2, 2, 2, 4, 4]

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

[3, 3, 3, 3, 4, 4]

[2, 2, 2, 2, 4, 4, 4]

[2, 2, 4, 4, 4, 4]

[4, 4, 4, 4, 4]

[2, 2, 2, 2, 2, 5, 5]

[2, 2, 3, 3, 5, 5]

[3, 3, 4, 5, 5]

[2, 4, 4, 5, 5]

[5, 5, 5, 5]

[2, 2, 2, 2, 3, 3, 6]

[3, 3, 4, 4, 6]

[2, 2, 5, 5, 6]

[2, 2, 2, 2, 6, 6]

[2, 3, 3, 6, 6]

[2, 2, 4, 6, 6]

[4, 4, 6, 6]

[2, 2, 2, 7, 7]

[3, 3, 7, 7]

[6, 7, 7]

[2, 2, 2, 2, 2, 2, 8]

[3, 3, 3, 3, 8]

[2, 2, 4, 4, 8]

[6, 6, 8]

[2, 2, 8, 8]

[4, 8, 8]

[2, 9, 9]

[2, 2, 3, 3, 10]

[5, 5, 10]

[10, 10]

[2, 2, 2, 2, 12]

[4, 4, 12]

[3, 3, 14]

[2, 2, 16]

[20] 20, 627, 42

Liste des triplets [n, p, pp1] pour n de 2 à 69

[2, 2, 1], [3, 3, 1], [4, 5, 2], [5, 7, 1], [6, 11, 3], [7, 15, 2], [8, 22, 5], [9, 30, 3], [10, 42, 7], [11, 56, 5], [12, 77, 11], [13, 101, 7], [14, 135, 15], [15, 176, 11], [16, 231, 22], [17, 297, 15], [18, 385, 30], [19, 490, 22], [20, 627, 42], [21, 792, 30], [22, 1002, 56], [23, 1255, 42], [24, 1575, 77], [25, 1958, 56], [26, 2436, 101], [27, 3010, 77], [28, 3718, 135], [29, 4565, 101], [30, 5604, 176], [31, 6842, 135], [32, 8349, 231], [33, 10143, 176], [34, 12310, 297], [35, 14883, 231], [36, 17977, 385], [37, 21637, 297], [38, 26015, 490], [39, 31185, 385], [40, 37338, 627], [41, 44583, 490], [42, 53174, 792], [43, 63261, 627], [44, 75175, 1002], [45, 89134, 792], [46, 105558, 1255], [47, 124754, 1002], [48, 147273, 1575], [49, 173525, 1255], [50, 204226, 1958], [51, 239943, 1575], [52, 281589, 2436], [53, 329931, 1958], [54, 386155, 3010], [55, 451276, 2436], [56, 526823, 3718], [57, 614154, 3010], [58, 715220, 4565], [59, 831820, 3718], [60, 966467, 5604], [61, 1121505, 4565], [62, 1300156, 6842], [63, 1505499, 5604], [64, 1741630, 8349], [65, 2012558, 6842], [66, 2323520, 10143], [67, 2679689, 8349], [68, 3087735, 12310], [69, 3554345, 10143]

Partitions de 37

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Les 23 partitions palindromiques de 37 en chiffres sans le 1, 2 et 3

[4, 4, 4, 4, 4, 4, 4, 4, 5]

[5, 5, 5, 5, 5, 6, 6]

[4, 4, 5, 6, 6, 6, 6]

[5, 5, 5, 5, 5, 5, 7]

[4, 4, 5, 5, 6, 6, 7]

[4, 4, 5, 5, 5, 7, 7]

[4, 4, 4, 4, 7, 7, 7]

[4, 4, 4, 4, 5, 8, 8]

[7, 7, 7, 8, 8]

[5, 8, 8, 8, 8]

[4, 4, 5, 5, 5, 5, 9]

[4, 4, 4, 4, 6, 6, 9]

[7, 7, 7, 7, 9]

[6, 6, 8, 8, 9]

[6, 6, 7, 9, 9]

[5, 7, 7, 9, 9]

[5, 5, 9, 9, 9]

[4, 4, 4, 4, 5, 5, 11]

[6, 6, 7, 7, 11]

[5, 5, 8, 8, 11]

[4, 4, 9, 9, 11]

[5, 5, 5, 11, 11]

[4, 4, 7, 11, 11]

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Retour	Partitions – Introduction Itérations palindromiques (Palindromes retard)
Suite	Partitions – En bref et orientation Partitions – Index Autres partitions particulières (Goldbach, Zeckendorf) Partitions distinctes et théorème des nombres pentagonaux
Voir	Partition avec des nombres consécutifs Partition avec des nombres différents Partition et théorème des nombres pentagonaux Partition et montées d'un escalier Compter les marches d'escalier Rectangles magiques à répétitions Répartition de n balles en k boites
Sites	Various palindromic sums – World of Numbers
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