A188672 - OEIS

%I #39 Aug 29 2013 14:06:12

%S 0,0,0,0,0,0,2,4,0,2,3,0,0,0,6,2,2,3,2,7,4,2,4,0,2,7,2,2,4,3,2,2,4,2,

%T 4,4,2,2,3,5,5,2,2,3,2,2,2,3,2,4,3,2,3,4,2,0,2,2,2,5,2,3,4,2,5,2,2,3,

%U 3,2,2,2,2,4,4,2,2,3,2,2,3,2,4,3,2,3,2

%N a(n) is the least r > 1 for which the interval (r*n, r*(n+1)) contains no prime powers (p^k, k >= 1), or a(n) = 0 if no such r exists.

%C Conjecture: a(n) = 0, iff n = 1, 2, 3, 4, 5, 6, 9, 12, 13, 14, 24, 56.

%C A proof that the interval(r*n, r*(n+1)) for r > 1 always contains a term from A000961 for n = 1, 2, 3, 4, 5, 6, 9, 12, 13, 14, 24, 56 uses methods based on the corresponding analog of Ramanujan numbers (cf. A228592) and their generalization.

%F If a(n)*A228518(n) is not 0, then a(n) >= A228518(n).

%Y Cf. A000961, A218831, A228518, A228592.

%K nonn

%O 1,7

%A _Vladimir Shevelev_ and _Peter J. C. Moses_, Aug 26 2013