Smarandache Constants

The first Smarandache constant is defined as

$$S_1\equiv \sum_{n=2}^\infty {1\over [S(n)]!} > 1.093111,$$

where $S(n)$ is the Smarandache Function. Cojocaru and Cojocaru (1996a) prove that $S_1$ exists and is bounded by $0.717<S_1<1.253$. The lower limit given above is obtained by taking 40,000 terms of the sum.

Cojocaru and Cojocaru (1996b) prove that the second Smarandache constant

$$S_2\equiv \sum_{n=2}^\infty {S(n)\over n!}\approx 1.71400629359162$$

is an Irrational Number.

Cojocaru and Cojocaru (1996c) prove that the series

$$S_3\equiv \sum_{n=2}^\infty {1\over\prod_{i=2}^n S(i)}\approx 0.719960700043708$$

converges to a number $0.71<S_3<1.01$, and that

$$S_4(a)\equiv \sum_{n=2}^\infty {n^a\over\prod_{i=2}^n S(i)}$$

converges for a fixed Real Number $a\geq 1$. The values for small $a$ are

$$S_4(1) \approx 1.72875760530223$$

$$S_4(2) \approx 4.50251200619297$$

$$S_4(3) \approx 13.0111441949445$$

$$S_4(4) \approx 42.4818449849626$$

$$S_4(5) \approx 158.105463729329.$$

Sandor (1997) shows that the series

$$S_5\equiv \sum_{n=1}^\infty {(-1)^{n-1}S(n)\over n!}$$

converges to an Irrational. Burton (1995) and Dumitrescu and Seleacu (1996) show that the series

$$S_6\equiv \sum_{n=2}^\infty {S(n)\over(n+1)!}$$

converges. Dumitrescu and Seleacu (1996) show that the series

$$S_7\equiv\sum_{n=r}^\infty {S(n)\over(n+r)!}$$

and

$$S_8\equiv\sum_{n=r}^\infty {S(n)\over(n-r)!}$$

converge for $r$ a natural number (which must be nonzero in the latter case). Dumitrescu and Seleacu (1996) show that

$$S_9\equiv\sum_{n=2}^\infty {1\over\sum_{i=2}^n {S(i)\over i!}}$$

converges. Burton (1995) and Dumitrescu and Seleacu (1996) show that the series

$$S_{10}\equiv\sum_{n=2}^\infty {1\over [S(n)]^\alpha\sqrt{S(n)!}}$$

and

$$S_{11}\equiv\sum_{n=2}^\infty {1\over [S(n)]^\alpha\sqrt{[S(n)+1]!}}$$

converge for $\alpha>1$.

See also Smarandache Function

References

Burton, E. ``On Some Series Involving the Smarandache Function.'' Smarandache Notions J. 6, 13-15, 1995.

Burton, E. ``On Some Convergent Series.'' Smarandache Notions J. 7, 7-9, 1996.

Cojocaru, I. and Cojocaru, S. ``The First Constant of Smarandache.'' Smarandache Notions J. 7, 116-118, 1996a.

Cojocaru, I. and Cojocaru, S. ``The Second Constant of Smarandache.'' Smarandache Notions J. 7, 119-120, 1996b.

Cojocaru, I. and Cojocaru, S. ``The Third and Fourth Constants of Smarandache.'' Smarandache Notions J. 7, 121-126, 1996c.

``Constants Involving the Smarandache Function.'' http://www.gallup.unm.edu/~smarandache/CONSTANT.TXT.

Dumitrescu, C. and Seleacu, V. ``Numerical Series Involving the Function $S$.'' The Smarandache Function. Vail: Erhus University Press, pp. 48-61, 1996.

Ibstedt, H. Surfing on the Ocean of Numbers--A Few Smarandache Notions and Similar Topics. Lupton, AZ: Erhus University Press, pp. 27-30, 1997.

Sandor, J. `On The Irrationality Of Certain Alternative Smarandache Series.'' Smarandache Notions J. 8, 143-144, 1997.

Smarandache, F. Collected Papers, Vol. 1. Bucharest, Romania: Tempus, 1996.

Smarandache, F. Collected Papers, Vol. 2. Kishinev, Moldova: Kishinev University Press, 1997.