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
Keywords
Examples
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.
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)
Crossrefs
Programs
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Mathematica
q[n_] := FactorInteger[n][[-1, 2]] > 1; Select[Range[250000], q[#] && q[# + 1] &]
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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
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).
Comments