A354558 Numbers k such that k and k+1 are both divisible by the square of their largest prime factor.

8, 49, 242, 288, 675, 1444, 1681, 2400, 2645, 6727, 6859, 9408, 9800, 10647, 12167, 13689, 18490, 23762, 24299, 26010, 36517, 47915, 48734, 57121, 58080, 59535, 75809, 85697, 101250, 103246, 113568, 118579, 131043, 142884, 158949, 182182, 201019, 212194, 235224

Offset

1,1

Comments

Numbers k such that P(k)^2 | k and P(k+1)^2 | (k+1), where P(k) = A006530(k).

Links

Michael S. Branicky, Table of n, a(n) for n = 1..12072 (all terms <= 10^11)

Jean-Marie De Koninck, Nicolas Doyon, and Florian Luca, Consecutive integers divisible by the square of their largest prime factors, Journal of Combinatorics and Number Theory, Vol. 5, No. 2 (2013), pp. 81-93; Researchgate link.

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.

Régis de la Bretèche and Sary Drappeau, Niveau de répartition des polynômes quadratiques et crible majorant pour les entiers friables, Journal of the European Mathematical Society, Vol. 22, No. 5 (2020), pp. 1577-1624; arXiv preprint, arXiv:1703.03197 [math.NT], 2017-2019.

Daniel Suteu, Table of n, a(n) for n = 1..112589 (all terms <= 10^13)

Formula

x^(1/4)/log(x) << N(x) << x*exp(-c*sqrt(2*log(x)*log(log(x)))), where N(x) is the number of terms <= x, c = 25/24 (De Koninck et al., 2013), or 4/sqrt(5) (de la Bretèche and Drappeau, 2020).

Example

8 = 2^3 is a term since P(8) = 2 and 2^2 | 8, 9 = 3^2, P(9) = 3 and 3^2 | 9.
675 = 3^3 * 5^2 is a term since P(675) = 5, 5^2 | 675, 676 = 2^2 * 13^2, P(676) = 13 and 13^2 | 676.

Mathematica

q[n_] := FactorInteger[n][[-1, 2]] > 1; Select[Range[250000], q[#] && q[# + 1] &]

Prog

(Python)
from sympy import factorint
def c(n): f = factorint(n); return f[max(f)] >= 2
def ok(n): return n > 1 and c(n) and c(n+1)
print(list(filter(ok, range(235225)))) # Michael S. Branicky, May 30 2022

Crossrefs

Subsequence of A070003.

Subsequences: A060355, A354560, A354562, A354563, A354564, A354565, A354566.

Cf. A006530, A071178, A354559.

Sequence in context: A295777 A319959 A270007 * A344321 A166789 A081901

Adjacent sequences: A354555 A354556 A354557 * A354559 A354560 A354561

Keyword

nonn

Author

Amiram Eldar, May 30 2022

Status

approved