Lucky number

Lucky numbers are natural numbers that are generated using a certain sieving principle. The sieve principle is similar to the sieve of Eratosthenes for determining prime numbers . They were first mentioned by mathematicians Gardiner , Lazarus , Metropolis and Ulam in 1956. They call the sieve principle Josephus Flavius sieve because it is very reminiscent of the Josephus problem .

definition

You start with a list of positive natural numbers. Then you go through the numbers in the list, starting with , and cross out every nth number. In contrast to the sieve of Eratosthenes, when counting the numbers to be crossed out, the numbers that have already been crossed out are not counted, but only those that are still in the list. Also when going through the list to get the next x, the crossed out ones are skipped. ${\displaystyle x=2}$

Explanation

This animation shows the sieving principle used to get lucky numbers. The red numbers left over are the lucky numbers.

The first step is to delete every second number and thus all even numbers.

In the second step, the number following two is in the list , and every third is deleted: ${\displaystyle x=3}$

1	3	5	7th	9	11	13	15th	17th	19th
21st	23	25th	27	29	31	33	35	37	39
41	43	45	47	49	51	53	55	57	59
61	63	65	67	69	71	73	75	77	79
81	83	85	87	89	91	93	95	97	99

The third step is the number following three , and every seventh is deleted: ${\displaystyle x=7}$

1	3	5	7th	9	11	13	15th	17th	19th
21st	23	25th	27	29	31	33	35	37	39
41	43	45	47	49	51	53	55	57	59
61	63	65	67	69	71	73	75	77	79
81	83	85	87	89	91	93	95	97	99

The number follows the seven , and every ninth is deleted: ${\displaystyle x=9}$

1	3	5	7th	9	11	13	15th	17th	19th
21st	23	25th	27	29	31	33	35	37	39
41	43	45	47	49	51	53	55	57	59
61	63	65	67	69	71	73	75	77	79
81	83	85	87	89	91	93	95	97	99

Then you delete every 13th, and so on. This gives the sequence of lucky numbers as all the numbers that are never crossed out:

• 1, 3, 7, 9, 13, 15, 21, 25, 31, 33, 37, 43, 49, 51, 63, 67, 69, 73, 75, 79, 87, 93, 99, 105, 111, 115, 127, 129, 133, 135, 141, 151, 159, 163, 169, 171, 189, 193, 195, 201, 205, 211, 219, 223, 231, 235, 237, 241, 259, 261, 267, 273, 283, 285, 289, 297, ... (sequence A000959 in OEIS )

properties

• There are an infinite number of lucky numbers.
• Be the -th lucky number and the -th prime number. Then: ${\displaystyle L_{n}}$ ${\displaystyle n}$ ${\displaystyle p_{n}}$ ${\displaystyle n}$

${\displaystyle L_{n}>p_{n}}$ for sufficiently large ${\displaystyle n}$

In other words: from a certain index on , the -th lucky number is always greater than the -th prime number. ${\displaystyle n}$ ${\displaystyle n}$ ${\displaystyle n}$

• The counting function of the lucky numbers is asymptotically equivalent to (see prime number theorem ). In other words: ${\displaystyle {\frac {n}{\ln n}}}$

Let be the number of lucky numbers that are less than or equal to. Then: ${\displaystyle g(n)}$ ${\displaystyle n}$

${\displaystyle \lim _{n\to \infty }\,{\frac {g(n)}{n/\ln(n)}}\,=\,1}$

Lucky prime numbers

Prime numbers that are happy numbers are called happy prime numbers . The lucky prime numbers that are less than 1000 are: ${\displaystyle p\in \mathbb {P} }$

3, 7, 13, 31, 37, 43, 67, 73, 79, 127, 151, 163, 193, 211, 223, 241, 283, 307, 331, 349, 367, 409, 421, 433, 463, 487, 541, 577, 601, 613, 619, 631, 643, 673, 727, 739, 769, 787, 823, 883, 937, 991, 997, ... (sequence A031157 in OEIS )

It is unknown whether there are infinitely many lucky prime numbers. There is also a conjecture analogous to Goldbach's .

See also

• Happy number
• Narcissistic number
• Munchausen number

Web links

• Eric W. Weisstein : Lucky Numbers . In: MathWorld (English). and in Wolfram Demonstrations Project

Individual evidence

1. ↑ Verna Gardiner, Roger B. Lazarus, Nicholas Metropolis, Stanisław Marcin Ulam: On certain sequences of integers defined by sieves . In: Mathematics Magazine . 29, No. 3, 1956, ISSN  0025-570X , pp. 117-122. doi : 10.2307 / 3029719 .
2. ^ ^{A b} D. Hawkins, William Egbert Briggs: The lucky number theorem . In: Mathematics Magazine . 31, No. 2, 1957, ISSN  0025-570X , pp. 81-84, 277-280. doi : 10.2307 / 3029213 .