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A354799
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Integers m in A001694 such that 3 | d(m^2), where d(n) = A000005(n).
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1
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16, 81, 128, 144, 324, 400, 432, 625, 648, 784, 1024, 1152, 1296, 1936, 2000, 2025, 2187, 2401, 2500, 2592, 2704, 3200, 3456, 3600, 3888, 3969, 4624, 5000, 5184, 5488, 5625, 5776, 6272, 7056, 8100, 8192, 8464, 8748, 9216, 9604, 9801, 10000, 10125, 10368, 10800
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OFFSET
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1,1
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LINKS
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FORMULA
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Equals { m in A001694 : d(m^2) mod 3 = 0 }.
Sum_{n>=1} 1/a(n) = zeta(2)*zeta(3)/zeta(6) - 5*zeta(3)/(2*zeta(2)) = 0.1166890133... . - Amiram Eldar, Jun 28 2022
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EXAMPLE
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A001694(5) = 16 is a term since d(16^2) = d(256) = 9, and 9 is a multiple of 3.
A001694(13) = 81 is a term since d(81^2) = d(6561) = 9, and 9 is a multiple of 3.
A001694(3) = 8 is not a term since d(8^2) = d(64) = 7, which is not divisible by 3.
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MATHEMATICA
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With[{nn = 10800}, Select[Union@ Flatten@ Table[a^2*b^3, {b, nn^(1/3)}, {a, Sqrt[nn/b^3]}], Mod[DivisorSigma[0, #^2], 3] == 0 &]]
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PROG
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(PARI) isok(m) = ispowerful(m) && !(numdiv(m^2) % 3); \\ Michel Marcus, Jun 27 2022
(Python)
from sympy import divisor_count as d, factorint as f
def ok(k): return k > 1 and min(f(k).values()) > 1 and d(k*k)%3 == 0
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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