A354563 - OEIS

A354563

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

%I #14 Jun 04 2022 02:01:56

%S 242,2400,6859,10647,47915,57121,344604,499999,830465,1012499,1431125,

%T 2098853,2825760,2829123,3930399,5560691,11859210,12323584,13137830,

%U 18253460,18279039,21093749,30664296,32279841,33999932,37218852,38640401,38740085,41485688,45222737

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

%H Daniel Suteu, <a href="/A354563/b354563.txt">Table of n, a(n) for n = 1..9503</a> (terms <= 10^15)

%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.

%e 242 = 2 * 11^2 is a term since P(242) = 11 and 11^2 | 242, 243 = 3^5, P(243) = 3, and 3^3 | 243.

%t p[n_] := FactorInteger[n][[-1, 2]]; Select[Range[10^6], p[#] > 1 && p[# + 1] > 2 &]

%o (Python)

%o from sympy import factorint

%o def c(n, e): f = factorint(n); return f[max(f)] >= e

%o def ok(n): return n > 1 and c(n, 2) and c(n+1, 3)

%o print([k for k in range(10**6) if ok(k)]) # _Michael S. Branicky_, May 30 2022

%Y Subsequence of A070003 and A354558.

%Y A354562 is a subsequence.

%Y Cf. A006530, A071178, A354564.

%K nonn

%O 1,1

%A _Amiram Eldar_, May 30 2022