Smarandache–Wellin number
In mathematics, a Smarandache–Wellin number is an integer that in a given base is the concatenation of the first n prime numbers written in that base. Smarandache–Wellin numbers are named after Florentin Smarandache and Paul R. Wellin.
The first decimal Smarandache–Wellin numbers are:
- 2, 23, 235, 2357, 235711, 23571113, 2357111317, 235711131719, 23571113171923, 2357111317192329, ... (sequence A019518 in OEIS).
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Smarandache–Wellin prime
A Smarandache–Wellin number that is also prime is called a Smarandache–Wellin prime. The first three are 2, 23 and 2357 (sequence A069151 in OEIS). The fourth has 355 digits and ends with the digits 719.[1]
The primes at the end of the concatenation in the Smarandache–Wellin primes are
The indices of the Smarandache–Wellin primes in the sequence of Smarandache–Wellin numbers are:
The 1429th Smarandache–Wellin number is a probable prime with 5719 digits ending in 11927, discovered by Eric W. Weisstein in 1998.[2] If it is proven prime, it will be the eighth Smarandache–Wellin prime. In March 2009 Weisstein's search showed the index of the next Smarandache–Wellin prime (if one exists) is at least 22077.[3]
Smarandache number
The Smarandache numbers are the concatenation of the numbers 1 to n. That is:
- 1, 12, 123, 1234, 12345, 123456, 1234567, 12345678, 123456789, 12345678910, 1234567891011, 123456789101112, 12345678910111213, 1234567891011121314, 123456789101112131415, ... (sequence A007908 in OEIS)
Smarandache prime
A Smarandache prime is a Smarandache number that is also prime. However, all of the first 200000 Smarandache numbers are not prime. It is conjectured there are infinitely many Smarandache primes.
Factorization of Smarandache numbers
| 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
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 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 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, ?, ...
See also
References
- ↑ Lua error in package.lua at line 80: module 'strict' not found.
- ↑ Rivera, Carlos, Primes by Listing
- ↑ Weisstein, Eric W., "Integer Sequence Primes", MathWorld. Retrieved 2011-07-28.
- Weisstein, Eric W., "Smarandache Number", MathWorld.
- Weisstein, Eric W., "Smarandache–Wellin Number", MathWorld.
- Smarandache-Wellin number at PlanetMath.org.
- List of first 200 Smarandache numbers with factorisations
- List of first 54 Smarandache–Wellin numbers with factorisations
- Factorization of Smarandache numbers
- Triangle of the Gods
- Smarandache–Wellin primes at The Prime Glossary
- Smith, S. "A Set of Conjectures on Smarandache Sequences." Bull. Pure Appl. Sci. 15E, 101–107, 1996.
