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Smarandache Constants

"The" Smarandache constant x approx 0.567148 (OEIS A038458) is the unique positive solution of 127^x-113^x=1. The conjecture that it is the smallest positive solution of q^x-p^x=1 for consecutive primes p<q generalizes Andrica's conjecture (Smarandache 2007).

The first Smarandache constant is defined as

S_1=sum_(n=2)^infty1/([mu(n)]!)=1.09317...
(1)

(OEIS A048799), where mu(n)=S(n) is the Smarandache function and mu(1)=1. 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

S_2=sum_(n=2)^infty(mu(n))/(n!) approx 1.71400629359162
(2)

(OEIS A048834) is an irrational number.

Cojocaru and Cojocaru (1996c) prove that the series

S_3=sum_(n=2)^infty1/(product_(i=2)^(n)S(i)) approx 0.719960700043708
(3)

(OEIS A048835) converges to a number 0.71<S_3<1.01, and that

S_4(a)=sum_(n=2)^infty(n^a)/(product_(i=2)^(n)S(i))
(4)

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

S_4(1) approx 1.72875760530223
(5)
S_4(2) approx 4.50251200619297
(6)
S_4(3) approx 13.0111441949445
(7)
S_4(4) approx 42.4818449849626
(8)
S_4(5) approx 158.105463729329
(9)

(OEIS A048836, A048837, and A048838 for a=1, 2, and 3, respectively).

Sándor (1997) shows that the series

S_5=sum_(n=1)^infty((-1)^(n-1)mu(n))/(n!)
(10)

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

S_6=sum_(n=2)^infty(mu(n))/((n+1)!)
(11)

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

S_7=sum_(n=r)^infty(mu(n))/((n+r)!)
(12)

and

S_8=sum_(n=r)^infty(mu(n))/((n-r)!)
(13)

converge for r a positive integer. Dumitrescu and Seleacu (1996) show that

S_9=sum_(n=2)^infty1/(sum_(i=2)^(n)([S(i)]!)/i)
(14)

converges. Burton (1996) shows that the series

S_(10)=sum_(n=2)^infty1/([mu(n)]^alphasqrt([mu(n)]!))
(15)

converges for alpha>1. For the same values of alpha, the comparison test with S_(10) shows that the series

S_(11)=sum_(n=2)^infty1/([mu(n)]^alphasqrt([mu(n)+1]!))
(16)

also converges.

See also

Andrica's Conjecture, Smarandache Function

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References

--. "Constants Involving the Smarandache Function." https://fs.unm.edu/CONSTANT.TXT.Burton, E. "On Some Series Involving Smarandache Function." Smarandache Function 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.Dumitrescu, C. and Seleacu, V. "Numerical Series Involving the Function S." The Smarandache Function in Number Theory. Vail, AZ: 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.Sándor, 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.Smarandache, F. "Six Conjectures Which Generalize or Are Related to Andrica's Conjecture." 17 Jul 2007. https://arxiv.org/abs/0707.2584.

Referenced on Wolfram|Alpha

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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