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

3,1

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.

References

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

Links

Table of n, a(n) for n=3..37.

Eric Weisstein's World of Mathematics, Smarandache Function

Example

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.

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}]

Crossrefs

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

Sequence in context: A262411 A280488 A250072 * A199620 A375829 A016553

Adjacent sequences: A099117 A099118 A099119 * A099121 A099122 A099123

Keyword

nonn

Author

T. D. Noe, Sep 28 2004

Status

approved