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%I #24 Jun 05 2022 11:47:20
%S 8,49,242,288,675,1444,1681,2400,2645,6727,6859,9408,9800,10647,12167,
%T 13689,18490,23762,24299,26010,36517,47915,48734,57121,58080,59535,
%U 75809,85697,101250,103246,113568,118579,131043,142884,158949,182182,201019,212194,235224
%N Numbers k such that k and k+1 are both divisible by the square of their largest prime factor.
%C Numbers k such that P(k)^2 | k and P(k+1)^2 | (k+1), where P(k) = A006530(k).
%H Michael S. Branicky, <a href="/A354558/b354558.txt">Table of n, a(n) for n = 1..12072</a> (all terms <= 10^11)
%H Jean-Marie De Koninck, Nicolas Doyon, and Florian Luca, <a href="https://www.jeanmariedekoninck.mat.ulaval.ca/fileadmin/jmdk/Documents/Publications/2013_consecutive_integers_divisible_by_the_square_of_their_largest_prime_factors.pdf">Consecutive integers divisible by the square of their largest prime factors</a>, Journal of Combinatorics and Number Theory, Vol. 5, No. 2 (2013), pp. 81-93; <a href="https://www.researchgate.net/publication/268171058_Consecutive_integers_divisible_by_the_square_of_their_largest_prime_factors">Researchgate link</a>.
%H Jean-Marie De Koninck and Matthieu Moineau, <a href="http://emis.muni.cz/journals/JIS/VOL21/DeKoninck/dek22.html">Consecutive Integers Divisible by a Power of their Largest Prime Factor</a>, J. Integer Seq., Vol. 21 (2018), Article 18.9.3.
%H Régis de la Bretèche and Sary Drappeau, <a href="https://doi.org/10.4171/jems/951">Niveau de répartition des polynômes quadratiques et crible majorant pour les entiers friables</a>, Journal of the European Mathematical Society, Vol. 22, No. 5 (2020), pp. 1577-1624; <a href="https://arxiv.org/abs/1703.03197">arXiv preprint</a>, arXiv:1703.03197 [math.NT], 2017-2019.
%H Daniel Suteu, <a href="/A354558/a354558.txt">Table of n, a(n) for n = 1..112589</a> (all terms <= 10^13)
%F x^(1/4)/log(x) << N(x) << x*exp(-c*sqrt(2*log(x)*log(log(x)))), where N(x) is the number of terms <= x, c = 25/24 (De Koninck et al., 2013), or 4/sqrt(5) (de la Bretèche and Drappeau, 2020).
%e 8 = 2^3 is a term since P(8) = 2 and 2^2 | 8, 9 = 3^2, P(9) = 3 and 3^2 | 9.
%e 675 = 3^3 * 5^2 is a term since P(675) = 5, 5^2 | 675, 676 = 2^2 * 13^2, P(676) = 13 and 13^2 | 676.
%t q[n_] := FactorInteger[n][[-1, 2]] > 1; Select[Range[250000], q[#] && q[# + 1] &]
%o (Python)
%o from sympy import factorint
%o def c(n): f = factorint(n); return f[max(f)] >= 2
%o def ok(n): return n > 1 and c(n) and c(n+1)
%o print(list(filter(ok, range(235225)))) # _Michael S. Branicky_, May 30 2022
%Y Subsequence of A070003.
%Y Subsequences: A060355, A354560, A354562, A354563, A354564, A354565, A354566.
%Y Cf. A006530, A071178, A354559.
%K nonn
%O 1,1
%A _Amiram Eldar_, May 30 2022
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