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A378629
Powerful numbers k such that both k-1 and k+1 are in A126706.
1
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
1,1
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.
LINKS
EXAMPLE
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.
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[#] ] &] &] ]
KEYWORD
nonn,easy
AUTHOR
Michael De Vlieger, Dec 03 2024
STATUS
approved