cp's OEIS Frontend

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

%I A099120 #10 Feb 16 2025 08:32:54
%S A099120 3,4,5,2,7,32,27,8,11,26,13,48,19,4096,17,74,19,447,27,121,23,4005,
%T A099120 3125,169,177147,2401,29,1203,31,134217728,459,289,551,2684163,37
%N A099120 Least m such that n = S(k) = S(k+m) for some k, where S is the Kempner function A002034.
%C A099120 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.
%D A099120 L. Tutescu, "On a Conjecture Concerning the Smarandache Function." Abstracts of Papers Presented to the Amer. Math. Soc. 17, 583, 1996.
%H A099120 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SmarandacheFunction.html">Smarandache Function</a>
%e A099120 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.
%t A099120 (*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}]
%Y A099120 Cf. A099118 (number of times S(k+n) = S(k)), A099119 (greatest k such that S(k) = S(k-n)).
%K A099120 nonn
%O A099120 3,1
%A A099120 _T. D. Noe_, Sep 28 2004