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A354564 Numbers k such that P(k)^3 | k and P(k+1)^2 | (k+1), where P(k) = A006530(k) is the largest prime dividing k.

%I A354564 #13 Jun 04 2022 02:01:49
%S A354564 8,6859,12167,101250,328509,453962,482447,536238,598950,5619712,
%T A354564 7170366,11449008,11667159,11859210,13428095,15054335,16541965,
%U A354564 18085704,18253460,19450850,22173969,23049600,24039994,29911714,30959144,32580250,33229625,44126385,44321375
%N A354564 Numbers k such that P(k)^3 | k and P(k+1)^2 | (k+1), where P(k) = A006530(k) is the largest prime dividing k.
%C A354564 De Koninck and Moineau (2018) proved that this sequence is infinite assuming the Bunyakovsky conjecture.
%D A354564 Jean-Marie De Koninck and Nicolas Doyon, The Life of Primes in 37 Episodes, American Mathematical Society, 2021, p. 232.
%H A354564 Daniel Suteu, <a href="/A354564/b354564.txt">Table of n, a(n) for n = 1..9686</a> (terms <= 10^15)
%H A354564 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 A354564 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/BouniakowskyConjecture.html">Bouniakowsky Conjecture</a>.
%H A354564 Wikipedia, <a href="https://en.wikipedia.org/wiki/Bunyakovsky_conjecture">Bunyakovsky conjecture</a>.
%e A354564 8 = 2^3 is a term since P(8) = 2 and 2^3 | 8, 9 = 3^2, P(9) = 3, and 3^2 | 9.
%t A354564 p[n_] := FactorInteger[n][[-1, 2]]; Select[Range[10^6], p[#] > 2 && p[# + 1] > 1 &]
%o A354564 (Python)
%o A354564 from sympy import factorint
%o A354564 def c(n, e): f = factorint(n); return f[max(f)] >= e
%o A354564 def ok(n): return n > 1 and c(n, 3) and c(n+1, 2)
%o A354564 print([k for k in range(10**6) if ok(k)]) # _Michael S. Branicky_, May 30 2022
%Y A354564 Subsequence of A070003 and A354558.
%Y A354564 A354562 is a subsequence.
%Y A354564 Cf. A006530, A071178, A354563.
%K A354564 nonn
%O A354564 1,1
%A A354564 _Amiram Eldar_, May 30 2022