A260198 Numbers k such that core(k+1) = core(k)+1 and k is not squarefree, where core(k) = A007913(k).

49, 242, 1681, 2645, 4374, 18490, 23762, 39325, 48734, 57121, 158949, 182182, 410669, 453870, 601425, 656914, 843637, 912950, 1143449, 1228122, 1275125, 1608574, 1940449, 2328482, 2948406, 3041577, 3203050, 3721549, 4495985, 4705074, 5371029, 5606230, 6352825

Offset: 1

Author

Giovanni Resta, Jul 19 2015

Keywords

• nonn

Comments

Each term is the first of a consecutive pair of nonsquarefree integers whose cores are consecutive too.

If c and c+1 are squarefree, then the sequence contains all the numbers c*x^2 where x is a solution to the Diophantine equation c*x^2+1 = (c+1)*y^2, with x,y > 1, hence the sequence is infinite.

Examples

			182182 is a term, because 182182 = 2*7^2*11*13^2 and 182183 = 23*89^2, so core(182182) = 22 and core(182183) = 23.
		

Links

• Giovanni Resta, Table of n, a(n) for n = 1..1328 (terms < 10^12)

Crossrefs

Cf. A007913, A007674.

Programs

Mathematica

core[n_] := Times@@ ((#[[1]] ^ Mod[#[[2]], 2]) & /@ FactorInteger@n); Select[ Range[10^5],# != core[#] == core[#+1]-1 &]