Son naturales que siguen el siguiente proceso y que deben terminar en 1:
- Se obtiene la suma de los cuadrados de los dígitos que lo componen
- Al número obtenido se le aplica el paso anterior
El proceso finaliza cuando llegamos al número 1 u obtenemos uno que ya se haya identificado como feliz, en estos dos casos tenemos un nº Feliz. En el supuesto de entrar en un proceso infinito lo que obtenemos es un nº infeliz.
Un ejemplo:
Feliz |
Infeliz |
7 -> 72 = 49
49 -> 42 + 92 = 97
97 -> 92 + 72 = 130
130 -> 12 + 32 + 02 = 10
10 -> 12 + 02 = 1 |
4 -> 42 = 16
16 -> 12 + 62 = 37
37 -> 32 + 72 = 58
58 -> 52 + 82 = 89
89 -> 82 +92 = 145
145 -> 12 +42 + 52 = 42
42 -> 42 +22 = 20
20 -> 22 +02 = 4 proceso infinito |
Felices emparentados.
Todo número feliz tiene un conjunto de números formados por los cuadrados de los dígitos que lo componen. En el ejemplo del 7 su conjunto es [49, 97, 130, 10, 1], otros felices pueden tener el mismo conjunto, a esta afinidad la llamamos parentesco. En la tabla siguiente representamos la cantidad de parientes sobre 20 millones de nºs felices, en esta población sólo existen 80 conjuntos de familias felices, las ramas de mayor distribución son:
[97, 130, 10, 1]
[82, 68, 100, 1]
las más pobladas:
[208, 68, 100, 1]
[192, 86, 100, 1]
y todas tienen los mismos ancestros
[10, 1]
[100, 1]
Cantidad de Parientes |
Conjunto Felices |
Cantidad de Parientes |
Conjunto Felices |
9 |
[1] |
147.453 |
[365,70,49,97,130,10,1] |
56 |
[563,70,49,97,130,10,1] |
165.423 |
[356,70,49,97,130,10,1] |
56 |
[566,97,130,10,1] |
176.779 |
[94,97,130,10,1] |
91 |
[536,70,49,97,130,10,1] |
178.816 |
[91,82,68,100,1] |
280 |
[565,86,100,1] |
200.082 |
[338,82,68,100,1] |
484 |
[7,49,97,130,10,1] |
206.712 |
[97,130,10,1] |
876 |
[10,1] |
219.077 |
[331,19,82,68,100,1] |
1.540 |
[13,10,1] |
244.627 |
[100,1] |
1.897 |
[496,133,19,82,68,100,1] |
245.010 |
[103,10,1] |
2.975 |
[490,97,130,10,1] |
257.986 |
[326,49,97,130,10,1] |
3.605 |
[487,129,86,100,1] |
275.828 |
[109,82,68,100,1] |
4.256 |
[19,82,68,100,1] |
286.608 |
[329,94,97,130,10,1] |
5.187 |
[478,129,86,100,1] |
287.515 |
[319,91,82,68,100,1] |
6.105 |
[469,133,19,82,68,100,1] |
325.681 |
[310,10,1] |
6.797 |
[23,13,10,1] |
340.061 |
[320,13,10,1] |
7.771 |
[464,68,100,1] |
360.007 |
[313,19,82,68,100,1] |
10.944 |
[28,68,100,1] |
369.064 |
[130,10,1] |
13.697 |
[31,10,1] |
372.054 |
[302,13,10,1] |
15.015 |
[446,68,100,1] |
405.339 |
[291,86,100,1] |
15.480 |
[32,13,10,1] |
408.987 |
[129,86,100,1] |
20.767 |
[440,32,13,10,1] |
425.001 |
[133,19,82,68,100,1] |
32.969 |
[44,32,13,10,1] |
425.843 |
[301,10,1] |
37.900 |
[49,97,130,10,1] |
457.664 |
[139,91,82,68,100,1] |
47.379 |
[409,97,130,10,1] |
473.883 |
[293,94,97,130,10,1] |
48.190 |
[404,32,13,10,1] |
545.346 |
[262,44,32,13,10,1] |
66.219 |
[391,91,82,68,100,1] |
563.692 |
[280,68,100,1] |
68.466 |
[386,109,82,68,100,1] |
580.587 |
[263,49,97,130,10,1] |
71.798 |
[397,139,91,82,68,100,1] |
629.084 |
[167,86,100,1] |
74.064 |
[392,94,97,130,10,1] |
640.757 |
[226,44,32,13,10,1] |
78.622 |
[379,139,91,82,68,100,1] |
648.013 |
[190,82,68,100,1] |
82.182 |
[383,82,68,100,1] |
659.002 |
[230,13,10,1] |
85.724 |
[70,49,97,130,10,1] |
679.087 |
[239,94,97,130,10,1] |
94.151 |
[68,100,1] |
701.309 |
[219,86,100,1] |
108.773 |
[376,94,97,130,10,1] |
705.660 |
[203,13,10,1] |
116.421 |
[367,94,97,130,10,1] |
725.959 |
[193,91,82,68,100,1] |
123.809 |
[82,68,100,1] |
740.954 |
[176,86,100,1] |
127.961 |
[79,130,10,1] |
760.397 |
[236,49,97,130,10,1] |
130.710 |
[362,49,97,130,10,1] |
768.550 |
[188,129,86,100,1] |
133.087 |
[368,109,82,68,100,1] |
798.373 |
[192,86,100,1] |
143.073 |
[86,100,1] |
808.344 |
[208,68,100,1] |
Infelices emparentados.
Sobre 122.940.786 nºs infelices obtenemos 501 familias con la siguiente cantidad de parientes.
Cantidad de Parientes |
Conjunto Infelices |
830.860 |
[216,41,17,50,25,29,85,89,145,42,20,4,16,37,58] |
823.558 |
[200,4,16,37,58,89,145,42,20] |
816.456 |
[224,24,20,4,16,37,58,89,145,42] |
808.539 |
[204,20,4,16,37,58,89,145,42] |
805.057 |
[212,9,81,65,61,37,58,89,145,42,20,4,16] |
795.216 |
[220,8,64,52,29,85,89,145,42,20,4,16,37,58] |
786.500 |
[196,118,66,72,53,34,25,29,85,89,145,42,20,4,16,37,58] |
786.375 |
[232,17,50,25,29,85,89,145,42,20,4,16,37,58] |
782.756 |
[228,72,53,34,25,29,85,89,145,42,20,4,16,37,58] |
775.415 |
[240,20,4,16,37,58,89,145,42] |
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NPN, son el conjunto de todos los números primos que se pueden obtener de las permutaciones de alguno o todos los dígitos de un número natural n.
Un Primitivo NPN, definido por Mike Keith, es un número natural n para el cual la cantidad de números primos que se pueden obtener es mayor que el número de primos obtenidos de la misma manera para cualquier menor a n. Los primeros son :
.
1,2,13,37,107,113,137,1013,1037,1079,1237,1367,1379,10079
Por ejemplo, para el 123.479 los primos que genera son (402):
[ 2, 3, 7, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 71, 73, 79, 97, 127, 137, 139, 149, 173,
179, 193, 197, 239, 241, 271, 293, 317, 347, 349, 379, 397, 419, 421, 431, 439, 479,
491, 719, 739, 743, 937, 941, 947, 971, 1237, 1249, 1279, 1297, 1327, 1423, 1427,
1429, 1439, 1493, 1723, 1973, 2137, 2143, 2179, 2341, 2347, 2371, 2417, 2437,
2473, 2713, 2719, 2731, 2741, 2749, 2791, 2917, 2971, 3217, 3271, 3491, 3719,
3917, 3947, 4127, 4129, 4139, 4217, 4219, 4231, 4271, 4273, 4297, 4327, 4391,
4397, 4721, 4723, 4729, 4793, 4931, 4937, 4973, 7129, 7193, 7213, 7219, 7243,
7321, 7349, 9127, 9137, 9173, 9241, 9341, 9371, 9413, 9421, 9431, 9437, 9473,
9721, 9743, 12347, 12379, 12437, 12473, 12479, 12497, 12739, 12743, 12973,
13249, 13297, 13729, 14293, 14327, 14723, 14923, 17239, 17293, 17923, 19237,
19273, 19423, 19427, 21347, 21379, 21397, 21493, 21739, 21937, 21943, 23197,
23417, 23497, 23719, 23741, 23917, 23971, 24137, 24179, 24197, 24317, 24371,
24379, 24391, 24793, 24917, 24971, 27143, 27431, 27941, 27943, 29137, 29147,
29173, 29347, 29437, 29473, 29741, 31247, 31249, 31729, 32479, 32491, 32497,
32719, 32749, 32917, 32941, 32971, 34127, 34129, 34217, 34297, 34721, 34729,
39217, 39241, 41729, 41927, 42139, 42179, 42193, 42197, 42379, 42391, 42397,
42719, 42793, 42937, 43271, 43291, 43721, 47123, 47129, 47293, 49123, 71249,
71293, 71329, 71429, 72139, 72341, 72431, 72493, 72931, 73291, 73421, 74219,
74231, 74293, 74923, 79231, 79241, 79423, 91237, 91243, 91423, 92143, 92173,
92317, 92347, 92413, 92431, 93241, 93427, 94273, 94321, 94327, 94723, 97213,
97231, 97241, 97423, 123479, 124739, 124793, 127493, 129347, 132749, 132947,
142973, 143729, 147293, 172439, 173249, 173429, 174329, 179243, 192347, 192743, 193247, 194723, 197243, 197423, 213947, 217439, 219437, 231479, 231947, 234197, 234791, 234917, 239147, 239417, 241739, 241793, 241973, 243197, 243917, 247193, 247391, 247913, 249317, 273149, 273941, 274139, 274931, 279143, 279413, 279431, 291437, 291743, 293147, 294317, 294731, 314927, 319427, 321947, 324179, 324791, 327419, 327491, 327941, 329471, 341729, 341927, 342179, 342197, 342791, 342971, 347129, 371249, 372149, 372941, 374219, 374291, 391247, 392741, 394271, 394721, 412397, 412739, 412793, 417239, 417293, 421397, 421739, 421973, 423179, 423791, 427913, 429137, 429731, 431297, 431729, 437219, 439217, 471923, 472139, 472193, 472319, 472391, 473219, 479231, 491273, 491327, 492731, 493127, 493217, 493721, 712493, 721439, 723491, 729143, 729413, 731249, 732491, 734291, 739241, 742193, 742913, 743129, 743921, 792413, 794231, 913247, 914237, 914327, 914723, 917243, 921743, 923147, 923471, 924173, 924713, 924731, 927431, 932417, 932471, 934127, 934721, 937241, 937421, 941723, 942317, 942371, 943127, 972431, 973421, 974123, 974213 ]
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