A354562 -id:A354562 - 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).

A354563 Numbers k such that P(k)^2 | k and P(k+1)^3 | (k+1), where P(k) = A006530(k) is the largest prime dividing k.

Original entry on oeis.org

242, 2400, 6859, 10647, 47915, 57121, 344604, 499999, 830465, 1012499, 1431125, 2098853, 2825760, 2829123, 3930399, 5560691, 11859210, 12323584, 13137830, 18253460, 18279039, 21093749, 30664296, 32279841, 33999932, 37218852, 38640401, 38740085, 41485688, 45222737

Offset: 1

Amiram Eldar, May 30 2022

nonn

			242 = 2 * 11^2 is a term since P(242) = 11 and 11^2 | 242, 243 = 3^5, P(243) = 3, and 3^3 | 243.

Daniel Suteu, Table of n, a(n) for n = 1..9503 (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.

Subsequence of A070003 and A354558.
A354562 is a subsequence.
Cf. A006530, A071178, A354564.

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

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, 2) and c(n+1, 3)
print([k for k in range(10**6) if ok(k)]) # Michael S. Branicky, May 30 2022

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.

Original entry on oeis.org

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

Amiram Eldar, May 30 2022

nonn

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

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

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..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.

Subsequence of A070003 and A354558.
A354562 is a subsequence.
Cf. A006530, A071178, A354563.

p[n_] := FactorInteger[n][[-1, 2]]; Select[Range[10^6], p[#] > 2 && 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, 3) and c(n+1, 2)
print([k for k in range(10**6) if ok(k)]) # Michael S. Branicky, May 30 2022

A354561 Numbers divisible by the cube of their largest prime factor.

Original entry on oeis.org

8, 16, 27, 32, 54, 64, 81, 108, 125, 128, 162, 216, 243, 250, 256, 324, 343, 375, 432, 486, 500, 512, 625, 648, 686, 729, 750, 864, 972, 1000, 1024, 1029, 1125, 1250, 1296, 1331, 1372, 1458, 1500, 1715, 1728, 1875, 1944, 2000, 2048, 2058, 2187, 2197, 2250, 2401, 2500

Offset: 1

Amiram Eldar, May 30 2022

nonn

Numbers k such that P(k)^3 | k, where P(k) = A006530(k).

Numbers k such that A071178(k) >= 3.

			8 is a term since 2^3|8 and 2 is the largest prime factor of 8.
54 = 2*3^3 is a term since 3^3|8 and 3 is the largest prime factor of 54.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Subsequence of A070003.
Subsequences: A036966 \ {1}, A349306 \ {1}, A354562.
Cf. A006530, A071178.

Select[Range[2500], FactorInteger[#][[-1, 2]] > 2 &]

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)
print([k for k in range(2501) if ok(k)]) # Michael S. Branicky, May 30 2022

A354567 a(n) is the least number k such that P(k)^n | k and P(k+1)^n | (k+1), where P(k) = A006530(k) is the largest prime dividing k, or -1 if no such k exists.

Original entry on oeis.org

1, 8, 6859, 11859210

Offset: 1

Amiram Eldar, May 30 2022

a(1) = 1 since P(1) = 1 by convention. Without this convention we would have a(1) = 2.

a(5) <= 437489361912143559513287483711091603378 (De Koninck, 2009).

			a(2) = 8 since 8 = 2^3, P(8) = 2 and 2^2|8, 9 = 3^2, P(9) = 3 and 3^2 | 9, and 8 is the least number with this property.
a(3) = 6859 since 6859 = 19^3, P(6859) = 19 and 19^3 | 6859, 6860 = 2^2 * 5 * 7^3, P(6860) = 7 and 7^3 | 6860, and 6859 is the least number with this property.

Jean-Marie De Koninck, Those Fascinating Numbers, American Mathematical Society, 2009, p. 173, entry 6859.
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.

Cf. A006530, A071178, A354558, A354562.

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

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A354563 Numbers k such that P(k)^2 | k and P(k+1)^3 | (k+1), where P(k) = A006530(k) is the largest prime dividing k.

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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.

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A354561 Numbers divisible by the cube of their largest prime factor.

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A354567 a(n) is the least number k such that P(k)^n | k and P(k+1)^n | (k+1), where P(k) = A006530(k) is the largest prime dividing k, or -1 if no such k exists.

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