A099120 Least m such that n = S(k) = S(k+m) for some k, where S is the Kempner function A002034.

3, 4, 5, 2, 7, 32, 27, 8, 11, 26, 13, 48, 19, 4096, 17, 74, 19, 447, 27, 121, 23, 4005, 3125, 169, 177147, 2401, 29, 1203, 31, 134217728, 459, 289, 551, 2684163, 37

Offset: 3

Author

T. D. Noe, Sep 28 2004

Keywords

• nonn

Comments

Consider the set Sn of d(n!)-d((n-1)!) positive integers k with S(k) = n, where d is the divisor counting function A000005. For each n, a(n) gives the least difference of integers in the set Sn. For prime n, a(n) = n. For n a power of a prime, a(n) = A046021(n), the least k in Sn. The Tutescu conjecture, which states that the equation S(k) = S(k+1) has no solutions, is equivalent to a(n) > 1 for all n.

Examples

			a(6) = 2 because S(k) = 6 for k = 9, 16, 18, 36, 45, 48, 72, 80, 90, 144, 180, 240, 360, 720 and the least difference is 2, between 16 and 18.
		

References

• L. Tutescu, "On a Conjecture Concerning the Smarandache Function." Abstracts of Papers Presented to the Amer. Math. Soc. 17, 583, 1996.

Links

• Eric Weisstein's World of Mathematics, Smarandache Function

Crossrefs

Cf. A099118 (number of times S(k+n) = S(k)), A099119 (greatest k such that S(k) = S(k-n)).

Programs

Mathematica

(*See A002034 for the Kempner function*) a=Table[Kempner[n], {n, 10!}]; Table[lst=Flatten[Position[a, n]]; mn=Infinity; Do[mn=Min[mn, lst[[i+1]]-lst[[i]]], {i, Length[lst]-1}]; mn, {n, 10}]