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%I #95 Sep 08 2022 08:46:26
%S 4,8,9,16,18,24,25,27,32,36,48,49,54,64,72,80,81,96,100,108,112,121,
%T 125,128,135,144,160,162,169,192,196,200,216,224,225,243,250,256,288,
%U 289,320,324,343,352,360,361,375,384,392,400,405,416,432,441,448,450,480,484,486,500
%N 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).
%C Prime powers p^e where p is a prime and e >= 2 (A246547) form a subsequence.
%C 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.
%C Table with percentage of terms <= 10^k for k = 1, 2, ..., 8, 9 (first rows coming from b-file):
%C +-------+------------------------+----------------------------+
%C | k |number of terms <= 10^k |percentage of terms <= 10^k |
%C | | | % |
%C +-------+------------------------+----------------------------+
%C | 1 | 3 | 30 |
%C | 2 | 19 | 19 |
%C | 3 | 95 | 9.5 |
%C | 4 | 435 | 4.35 |
%C | 5 | 1853 | 1.85 |
%C | 6 | 7793 | 0.78 |
%C | 7 | 32365 | 0.32 |
%C | 8 | 131200 | 0.13 |
%C | 9 | 527161 | 0.05 |
%C | | | |
%C +-------+------------------------+----------------------------+
%C The percentage of terms decreases as 10^k increases, and a plausible conjecture is that the asymptotic density of this sequence is 0.
%D Richard K. Guy, Unsolved Problems in Theory of Numbers, Springer-Verlag, Third Edition, 2004, B11, p. 102.
%H Marius A. Burtea, <a href="/A339744/b339744.txt">Table of n, a(n) for n = 1..10000</a>
%e rad(18)^2 - sigma(18) = (2*3)^2 - (1+2+3+6+9+18) = 36 - 39 = -3 and 18 is a term.
%e rad(25)^2 - sigma(25) = 5^2 - (1+5+25) = 25 - 31 = -6 and 25 is a term.
%e 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.
%p Rad := n -> convert(NumberTheory:-PrimeFactors(n), `*`):
%p Sigma := n -> NumberTheory:-SumOfDivisors(n):
%p Is_a := n -> Rad(n)^2 < Sigma(n):
%p select(Is_a, [`$`(1..500)]); # _Peter Luschny_, Dec 16 2020
%t 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 *)
%o (Magma) s:=func<n|&*PrimeDivisors(n)>; [k:k in [2..500]|s(k)^2 lt DivisorSigma(1,k)]; // _Marius A. Burtea_, Dec 15 2020
%o (PARI) isok(k) = factorback(factorint(k)[, 1])^2 < sigma(k); \\ _Michel Marcus_, Dec 15 2020
%Y Cf. A000203, A007947, A078615, A338790, A339794.
%Y Subsequence: A246547.
%K nonn
%O 1,1
%A _Bernard Schott_, Dec 15 2020
%E More terms from _Marius A. Burtea_, Dec 15 2020
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