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.

242, 2400, 6859, 10647, 47915, 57121, 344604, 499999, 830465, 1012499, 1431125, 2098853, 2825760, 2829123, 3930399, 5560691, 11859210, 12323584, 13137830, 18253460, 18279039, 21093749, 30664296, 32279841, 33999932, 37218852, 38640401, 38740085, 41485688, 45222737

Offset: 1

Author

Amiram Eldar, May 30 2022

Keywords

• nonn

Examples

			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.
		

Links

• Daniel Suteu, Table of n, a(n) for n = 1..9503 (terms <= 10^15)
• Jean-Marie De Koninck and Matthieu Moineau, Consecutive Integers Divisible by a Power of their Largest Prime Factor, J. Integer Seq., Vol. 21 (2018), Article 18.9.3.

Crossrefs

Subsequence of A070003 and A354558.

A354562 is a subsequence.

Cf. A006530, A071178, A354564.

Programs

Mathematica

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

Python

from sympy import factorint
def c(n, e): f = factorint(n); return f[max(f)] >= e
def ok(n): return n > 1 and c(n, 2) and c(n+1, 3)
print([k for k in range(10**6) if ok(k)]) # Michael S. Branicky, May 30 2022