A354565 -id:A354565 - cp's OEIS frontend

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

Amiram Eldar, May 30 2022

nonn

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

			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.

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)

Subsequence of A070003.
Subsequences: A060355, A354560, A354562, A354563, A354564, A354565, A354566.
Cf. A006530, A071178, A354559.

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

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

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).

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.

Original entry on oeis.org

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

Amiram Eldar, May 30 2022

nonn

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

			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.

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

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.

Subsequence of A070003, A354558 and A354564.

Cf. A006530, A071178, A354565.

p[n_] := FactorInteger[n][[-1, 2]]; Select[Range[3*10^7], p[#] > 3 && p[# + 1] > 1 &]

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

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Python

Formula