Smarandache number

In mathematics, a Smarandache number is an integer that in a given base is the concatenation of the numbers 1 to n written in that base. Smarandache numbers are named after Florentin Smarandache.

The first decimal Smarandache numbers are:

1, 12, 123, 1234, 12345, 123456, 1234567, 12345678, 123456789, 12345678910, 1234567891011, 123456789101112, 12345678910111213, 1234567891011121314, 123456789101112131415, ... (sequence A007908 in the OEIS)

Smarandache prime[edit]

A Smarandache prime is a Smarandache number that is also prime. However, all of the first 344869 Smarandache numbers are not prime. It is conjectured there are infinitely many Smarandache primes, but none are known as of December 2016.^[1]

Factorization of Smarandache numbers[edit]

n	Factorization of Sm(n)	n	Factorization of Sm(n)
1	1	16	2² × 2507191691 × 1231026625769
2	2² × 3	17	3² × 47 × 4993 × 584538396786764503
3	3 × 41	18	2 × 3² × 97 × 88241 × 801309546900123763
4	2 × 617	19	13 × 43 × 79 × 281 × 1193 × 833929457045867563
5	3 × 5 × 823	20	2⁵ × 3 × 5 × 323339 × 3347983 × 2375923237887317
6	2⁶ × 3 × 643	21	3 × 17 × 37 × 43 × 103 × 131 × 140453 × 802851238177109689
7	127 × 9721	22	2 × 7 × 1427 × 3169 × 85829 × 2271991367799686681549
8	2 × 3² × 47 × 14593	23	3 × 41 × 769 × 13052194181136110820214375991629
9	3² × 3607 × 3803	24	2² × 3 × 7 × 978770977394515241 × 1501601205715706321
10	2 × 5 × 1234567891	25	5² × 15461 × 31309647077 × 1020138683879280489689401
11	3 × 7 × 13 × 67 × 107 × 630803	26	2 × 3⁴ × 21347 × 2345807 × 982658598563 × 154870313069150249
12	2³ × 3 × 2437 × 2110805449	27	3³ × 19² × 4547 × 68891 × 40434918154163992944412000742833
13	113 × 125693 × 869211457	28	2³ × 47 × 409 × 416603295903037 × 192699737522238137890605091
14	2 × 3 × 205761315168520219	29	3 × 859 × 24526282862310130729 × 19532994432886141889218213
15	3 × 5 × 8230452606740808761	30	2 × 3 × 5 × 13 × 49269439 × 370677592383442753 × 17333107067824345178861

Generalization[edit]

Since there are no known original Smarandache primes, there are three generalizations of them to find some related primes.

Least k such that concatenating k consecutive natural numbers beginning with n is prime are

?, 1, 1, 4, 1, 2, 1, 2, 179, ?, 1, 2, 1, 4, 5, 28, 1, 3590, 1, 4, ?, ?, 1, ?, 25, 122, ?, 46, 1, ?, 1, ?, 71, 4, 569, 2, 1, 20, 5, ?, 1, 2, 1, 8, ?, ?, 1, ?, 193, 2, ?, ?, 1, ?, ?, 2, 5, 4, 1, ?, 1, 2, ?, 4, ... (sequence A244424 in the OEIS)

Least k such that the number formed by concatenating the decimal numbers 1, 2, 3, ..., k, but omitting n is prime are

2, 3, 7, 9, 11, 7, 11, 1873, 19, 14513, 13, 961, ?, 653, ?, 5109, 493, 757, 29, 1313, ... (sequence A262300 in the OEIS)

Least k such that concatenation of first k numbers in base n is prime are

2, 15, 2, ?, 2, 11, 10, 3, 2, ?, 2, 5, ?, 3, 2, 13, 2, ?, ?, 3, 2, ?, 9, 7, ?, ?, 2, ?, 2, 7, ?, 3, 5, 25, 2, 323, 226, 3, 2, ?, 2, 5, ?, 3, 2, 31, 85, 7, ?, ?, 2, ?, 14, 5, ?, 3, 2, ?, 2, ?, ?, 15, 10, ?, ...

References[edit]

↑ Smarandache Prime

External Links[edit]

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[1] Smarandache Prime

[1]

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Related topics	Probable prime Industrial-grade prime Illegal prime Formula for primes Prime gap
First 60 primes	2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 101 103 107 109 113 127 131 137 139 149 151 157 163 167 173 179 181 191 193 197 199 211 223 227 229 233 239 241 251 257 263 269 271 277 281
List of prime numbers

Smarandache number

Contents

Smarandache prime[edit]

Factorization of Smarandache numbers[edit]

Generalization[edit]

See also[edit]

References[edit]

External Links[edit]

📰 Article(s) of the same category(ies)[edit]