A354559
The number of terms of A354558 that are <= 10^n.
Original entry on oeis.org
1, 2, 5, 13, 28, 79, 204, 549, 1509, 4231, 12072, 36426, 112589
Offset: 1
There is one term <= 10 in A354558, 8, therefore a(1) = 1.
There are 2 terms <= 10^2 in A354558, 8 and 49, therefore a(2) = 2.
- 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.
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q[n_] := FactorInteger[n][[-1, 2]] > 1; c[s_, n_] := Count[s, _?(# <= n &)]; m = 6; c[Select[Range[10^m], q[#] && q[# + 1] &], #] & /@ (10^Range[m])
A354562
Numbers k such that k and k+1 are both divisible by the cube of their largest prime factor.
Original entry on oeis.org
6859, 11859210, 18253460, 38331320423, 41807225999, 49335445119, 50788425848, 67479324240, 203534609200, 245934780371, 250355343420, 581146348824, 779369813871, 1378677994836, 2152196307260, 2730426690524, 3616995855087, 5473549133744, 6213312123347, 6371699408179, 8817143116903
Offset: 1
6859 = 19^3 is a term since P(6859) = 19 and 19^3 | 6859, 6860 = 2^2 * 5 * 7^3, P(6860) = 7 and 7^3 | 6860.
- David A. Corneth, Table of n, a(n) for n = 1..81 (terms <= 10^15)
- 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.
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q[n_] := FactorInteger[n][[-1, 2]] > 2; Select[Range[2*10^7], q[#] && q[# + 1] &]
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from sympy import factorint
def c(n): f = factorint(n); return f[max(f)] >= 3
def ok(n): return n > 1 and c(n) and c(n+1)
print([k for k in range(10**5) if ok(k)]) # Michael S. Branicky, May 30 2022
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
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.
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p[n_] := FactorInteger[n][[-1, 2]]; Select[Range[10^6], p[#] > 1 && p[# + 1] > 2 &]
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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
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.
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p[n_] := FactorInteger[n][[-1, 2]]; Select[Range[10^6], p[#] > 2 && p[# + 1] > 1 &]
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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
A354565
Numbers k such that P(k)^2 | k and P(k+1)^4 | (k+1), where P(k) = A006530(k) is the largest prime dividing k.
Original entry on oeis.org
242, 2400, 57121, 499999, 1012499, 2825760, 2829123, 11859210, 18279039, 21093749, 37218852, 38740085, 70799772, 96393374, 413428949, 642837222, 656356767, 675975026, 1065352364, 1333564323, 1418528255, 2654744949, 5547008142, 8576868299, 9515377949, 10022519999
Offset: 1
242 = 2 * 11^2 is a term since P(242) = 11 and 11^2 | 242, 243 = 3^5, P(243) = 3, and 3^4 | 243.
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p[n_] := FactorInteger[n][[-1, 2]]; Select[Range[10^6], p[#] > 1 && p[# + 1] > 3 &]
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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, 4)
print([k for k in range(10**6) if ok(k)]) # Michael S. Branicky, May 30 2022
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
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.
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p[n_] := FactorInteger[n][[-1, 2]]; Select[Range[3*10^7], p[#] > 3 && p[# + 1] > 1 &]
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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
A354560
Numbers k such that k, k+1 and k+2 are all divisible by the square of their largest prime factor.
Original entry on oeis.org
1294298, 9841094, 158385500, 1947793550, 5833093013, 11587121710, 20944167840, 22979821310, 24604784814, 267631935500, 290672026412, 956544588350, 987988937343, 2399283556900, 2816075601855, 4174608151758, 4322550249043, 6789218799999, 10617595679778, 16036630184409
Offset: 1
1294298 = 2 * 61 * 103^2 is a term since P(1294298) = 103 and 103^2 | 1294298, 1294299 = 3^4 * 19 * 29^2, P(1294299) = 29 and 29^2 | 1294299, 1294300 = 2^2 * 5^2 * 7 * 43^2, P(1294300) = 43 and 43^2 | 1294300.
- Amiram Eldar, Table of n, a(n) for n = 1..60
- Jean-Marie De Koninck, Those Fascinating Numbers, American Mathematical Society, 2009, p. 277, entry 1294298.
- 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; Known members of E_{3,2} with at most 21 digits, addendum, 2025.
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
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.
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