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

1,1

Comments

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

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.

Example

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

Mathematica

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

Prog

(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

Crossrefs

Subsequence of A070003 and A354558.

A354562 is a subsequence.

Cf. A006530, A071178, A354563.

Sequence in context: A173190 A291879 A281450 * A354567 A356083 A278854

Adjacent sequences: A354561 A354562 A354563 * A354565 A354566 A354567

Keyword

nonn

Author

Amiram Eldar, May 30 2022

Status

approved