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A339744 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). 2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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.

REFERENCES

Richard K. Guy, Unsolved Problems in Theory of Numbers, Springer-Verlag, Third Edition, 2004, B11, p. 102.

LINKS

Marius A. Burtea, Table of n, a(n) for n = 1..10000

EXAMPLE

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.

MAPLE

Rad := n -> convert(NumberTheory:-PrimeFactors(n), `*`):

Sigma := n -> NumberTheory:-SumOfDivisors(n):

Is_a := n -> Rad(n)^2 < Sigma(n):

select(Is_a, [`$`(1..500)]); # Peter Luschny, Dec 16 2020

MATHEMATICA

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 *)

PROG

(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

CROSSREFS

Cf. A000203, A007947, A078615, A338790, A339794.

Subsequence: A246547.

Sequence in context: A299117 A339462 A140269 * A226385 A070003 A325661

Adjacent sequences:  A339741 A339742 A339743 * A339745 A339746 A339747

KEYWORD

nonn

AUTHOR

Bernard Schott, Dec 15 2020

EXTENSIONS

More terms from Marius A. Burtea, Dec 15 2020

STATUS

approved

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Last modified August 5 04:03 EDT 2021. Contains 346457 sequences. (Running on oeis4.)