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.

8, 6859, 12167, 101250, 328509, 453962, 482447, 536238, 598950, 5619712, 7170366, 11449008, 11667159, 11859210, 13428095, 15054335, 16541965, 18085704, 18253460, 19450850, 22173969, 23049600, 24039994, 29911714, 30959144, 32580250, 33229625, 44126385, 44321375

Offset: 1

Author

Amiram Eldar, May 30 2022

Keywords

• nonn

Comments

De Koninck and Moineau (2018) proved that this sequence is infinite assuming the Bunyakovsky conjecture.

Examples

			8 = 2^3 is a term since P(8) = 2 and 2^3 | 8, 9 = 3^2, P(9) = 3, and 3^2 | 9.
		

References

• Jean-Marie De Koninck and Nicolas Doyon, The Life of Primes in 37 Episodes, American Mathematical Society, 2021, p. 232.

Links

• Daniel Suteu, Table of n, a(n) for n = 1..9686 (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.
• Eric Weisstein's World of Mathematics, Bouniakowsky Conjecture.
• Wikipedia, Bunyakovsky conjecture.

Crossrefs

Subsequence of A070003 and A354558.

A354562 is a subsequence.

Cf. A006530, A071178, A354563.

Programs

Mathematica

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

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, 3) and c(n+1, 2)
print([k for k in range(10**6) if ok(k)]) # Michael S. Branicky, May 30 2022