A354566 Numbers k such that P(k)^4 | k and P(k+1)^2 | (k+1), where P(k) = A006530(k) is the largest prime dividing k.

101250, 11859210, 23049600, 32580250, 131545575, 162364824, 969697050, 1176565754, 1271688417, 1612089680, 1862719859, 2409451520, 2441023914, 3182903731, 3697778084, 4010283270, 4329214629, 6666661950, 6932744126, 7739389944, 9188994752, 11717364285, 17306002674

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..1897 (terms <= 10^17)

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

101250 = 2 * 3^4 * 5^4 is a term since P(101250) = 5 and 5^4 | 101250, 101251 = 19 * 73^2, P(101251) = 73, and 73^2 | 101251.

Mathematica

p[n_] := FactorInteger[n][[-1, 2]]; Select[Range[3*10^7], p[#] > 3 && 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, 4) 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, A354558 and A354564.

Cf. A006530, A071178, A354565.

Sequence in context: A161796 A096929 A031678 * A251959 A250631 A229684

Adjacent sequences: A354563 A354564 A354565 * A354567 A354568 A354569

Keyword

nonn

Author

Amiram Eldar, May 30 2022

Status

approved