Smarandache Constants

"The" Smarandache constant is the smallest solution to the generalized Andrica's conjecture, 👁 x approx 0.567148
(OEIS A038458).

The first Smarandache constant is defined as

(OEIS A048799), where 👁 mu(n)
is the Smarandache function. Cojocaru and Cojocaru (1996a) prove that 👁 S_1
exists and is bounded by 👁 0.717<S_1<1.253
.

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

(OEIS A048834) is an irrational number.

Cojocaru and Cojocaru (1996c) prove that the series

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

converges for a fixed real number 👁 a>=1
. The values for small 👁 a
are

(OEIS A048836, A048837, and A048838).

Sandor (1997) shows that the series

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

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

and

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

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

and

converge for 👁 alpha>1
.

See also

Andrica's Conjecture, Smarandache Function

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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 in Number Theory. 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.Sloane, N. J. A. Sequences A038458, A048799, A048834, A048835, A048836, A048837, A048838, and A071120 in "The On-Line Encyclopedia of Integer Sequences."Smarandache, F. Collected Papers, Vol. 1. Bucharest, Romania: Tempus, 1996.Smarandache, F. Collected Papers, Vol. 2. Kishinev, Moldova: Kishinev University Press, 1997.

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Smarandache Constants

Cite this as:

Weisstein, Eric W. "Smarandache Constants." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/SmarandacheConstants.html

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