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 the OEIS).

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 the 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

2, 3, 7, 719, 1033, 2297, 3037, 11927, ... (sequence A046284 in the OEIS).

The indices of the Smarandache–Wellin primes in the sequence of Smarandache–Wellin numbers are:

1, 2, 4, 128, 174, 342, 435, 1429, ... (sequence A046035 in the OEIS).

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 the 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, but none are known as of November 2015.^[4]

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 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

↑ Pomerance, Carl B.; Crandall, Richard E. (2001). Prime Numbers: a computational perspective. Springer. pp. 78 Ex 1.86. ISBN 0-387-25282-7.
↑ Rivera, Carlos, Primes by Listing
↑ Weisstein, Eric W. "Integer Sequence Primes". MathWorld. Retrieved 2011-07-28.
↑ Smarandache Prime

Weisstein, Eric W. "Smarandache Number". MathWorld.

Weisstein, Eric W. "Smarandache–Wellin Number". MathWorld.

"Smarandache-Wellin number". PlanetMath.
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.

Prime number classes

By formula	Fermat (2^2ⁿ + 1) Mersenne (2^p − 1) Double Mersenne (2^{2^p−1} − 1) Wagstaff (2^p + 1)/3 Proth (k·2ⁿ + 1) Factorial (n! ± 1) Primorial (p_n# ± 1) Euclid (p_n# + 1) Pythagorean (4n + 1) Pierpont (2^u·3^v + 1) Quartan (x⁴ + y⁴) Solinas (2^a ± 2^b ± 1) Cullen (n·2ⁿ + 1) Woodall (n·2ⁿ − 1) Cuban (x³ − y³)/(x − y) Carol (2ⁿ − 1)² − 2 Kynea (2ⁿ + 1)² − 2 Leyland (x^y + y^x) Thabit (3·2ⁿ − 1) Mills (floor(A^3ⁿ))

By integer sequence	Fibonacci Lucas Pell Newman–Shanks–Williams Perrin Partitions Bell Motzkin

By property	Wieferich (pair) Wall–Sun–Sun Wolstenholme Wilson Lucky Fortunate Ramanujan Pillai Regular Strong Stern Supersingular (elliptic curve) Supersingular (moonshine theory) Good Super Higgs Highly cototient

Base-dependent	Happy Dihedral Palindromic Emirp Repunit (10ⁿ − 1)/9 Permutable Circular Truncatable Strobogrammatic Minimal Weakly Full reptend Unique Primeval Self Smarandache–Wellin

Patterns	Twin (p, p + 2) Bi-twin chain (n − 1, n + 1, 2n − 1, 2n + 1, …) Triplet (p, p + 2 or p + 4, p + 6) Quadruplet (p, p + 2, p + 6, p + 8) k−Tuple Cousin (p, p + 4) Sexy (p, p + 6) Chen Sophie Germain (p, 2p + 1) Cunningham chain (p, 2p ± 1, …) Safe (p, (p − 1)/2) Arithmetic progression (p + a·n, n = 0, 1, …) Balanced (consecutive p − n, p, p + n)

By size	Titanic (1,000+ digits) Gigantic (10,000+) Mega (1,000,000+) Largest known

Complex numbers	Eisenstein prime Gaussian prime

Composite numbers	Pseudoprime Almost prime Semiprime Interprime

Related topics	Probable prime Industrial-grade prime Illegal prime Formula for primes Prime gap

First 50 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

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