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A339744
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Numbers k such that rad(k)^2 < sigma(k), where rad(k) is the squarefree kernel of k (A007947) and sigma(k) is the sum of divisors of k (A000203).
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2
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4, 8, 9, 16, 18, 24, 25, 27, 32, 36, 48, 49, 54, 64, 72, 80, 81, 96, 100, 108, 112, 121, 125, 128, 135, 144, 160, 162, 169, 192, 196, 200, 216, 224, 225, 243, 250, 256, 288, 289, 320, 324, 343, 352, 360, 361, 375, 384, 392, 400, 405, 416, 432, 441, 448, 450, 480, 484, 486, 500
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OFFSET
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1,1
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COMMENTS
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Prime powers p^e where p is a prime and e >= 2 (A246547) form a subsequence.
For numbers whose prime factors set is {p_1, p_2, ..., p_r}, there exists a minimal element u such that k is a term iff k >= u. This smallest element u satisfies p_1*p_2*...*p_r < u <= (p_1*p_2*...*p_r)^2. These minimal elements are in A339794.
Table with percentage of terms <= 10^k for k = 1, 2, ..., 8, 9 (first rows coming from b-file):
+-------+------------------------+----------------------------+
| k |number of terms <= 10^k |percentage of terms <= 10^k |
| | | % |
+-------+------------------------+----------------------------+
| 1 | 3 | 30 |
| 2 | 19 | 19 |
| 3 | 95 | 9.5 |
| 4 | 435 | 4.35 |
| 5 | 1853 | 1.85 |
| 6 | 7793 | 0.78 |
| 7 | 32365 | 0.32 |
| 8 | 131200 | 0.13 |
| 9 | 527161 | 0.05 |
| | | |
+-------+------------------------+----------------------------+
The percentage of terms decreases as 10^k increases, and a plausible conjecture is that the asymptotic density of this sequence is 0.
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REFERENCES
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Richard K. Guy, Unsolved Problems in Theory of Numbers, Springer-Verlag, Third Edition, 2004, B11, p. 102.
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LINKS
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EXAMPLE
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rad(18)^2 - sigma(18) = (2*3)^2 - (1+2+3+6+9+18) = 36 - 39 = -3 and 18 is a term.
rad(25)^2 - sigma(25) = 5^2 - (1+5+25) = 25 - 31 = -6 and 25 is a term.
rad(40)^2 - sigma(40) = (2*5)^2 - (1+2+4+5+8+10+20+40) = 100 - 90 = 10 and 40 is not a term.
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MAPLE
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Rad := n -> convert(NumberTheory:-PrimeFactors(n), `*`):
Sigma := n -> NumberTheory:-SumOfDivisors(n):
Is_a := n -> Rad(n)^2 < Sigma(n):
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MATHEMATICA
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frad2[p_, e_] := p^2; fsig[p_, e_] := (p^(e + 1) - 1)/(p - 1); Select[Range[2, 500], Times @@ frad2 @@@ (f = FactorInteger[#]) < Times @@ fsig @@@ f &] (* Amiram Eldar, Dec 15 2020 *)
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PROG
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(Magma) s:=func<n|&*PrimeDivisors(n)>; [k:k in [2..500]|s(k)^2 lt DivisorSigma(1, k)]; // Marius A. Burtea, Dec 15 2020
(PARI) isok(k) = factorback(factorint(k)[, 1])^2 < sigma(k); \\ Michel Marcus, Dec 15 2020
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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