A378629 Powerful numbers k such that both k-1 and k+1 are in A126706.

49, 125, 243, 343, 1681, 1849, 3249, 4913, 6724, 6859, 8649, 9801, 11449, 13689, 13924, 17576, 20449, 24389, 24649, 28125, 28224, 29791, 31212, 36125, 37249, 40328, 42849, 45125, 57121, 59049, 63001, 66049, 68921, 79507, 83349, 85849, 94249, 99127, 106929, 110224

Offset: 1

Author

Michael De Vlieger, Dec 03 2024

Keywords

• nonn
• easy

Comments

Contains certain powerful k in A246547 (perfect powers of primes) or in A286708 (powerful numbers that are not prime powers).

Contains certain Achilles numbers (in A052486); a(20) = 28125 = 3^2 * 5^5.

Examples

			Let S = A126706, the sequence of k that are neither squarefree nor prime powers.
{1, 4, 8, 9} are not in the sequence since S(1) = 12.
a(1) = 49 = 7^2 since both 48 = 2^3 * 3 and 50 = 2 * 5^2 are in S.
64 is not in the sequence since 65 is squarefree.
a(2) = 125 = 5^3 since both 124 = 2^2 * 41 and 126 = 2 * 3^2 * 7 are in S.
128 is not in the sequence since 127 is prime.
a(3) = 243 = 3^5 since both 242 = 2 * 11^2 and 244 = 2^2 * 61 are in S.
a(7) = 3249 = 3^2 * 19^2, since both 3248 = 2^4 * 7 * 29 and 3250 = 2 * 5^3 * 13 are in S, etc.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A001694, A013929, A024619, A052486, A126706, A246547, A286708.

Programs

Mathematica

With[{nn = 2^30}, Select[Union@ Flatten@ Table[a^2*b^3, {b, Surd[nn, 3]}, {a, Sqrt[nn/b^3]}], AllTrue[# + {-1, 1}, Nor[SquareFreeQ[#], PrimePowerQ[#] ] &] &] ]