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.
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
Keywords
Examples
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.
Links
- Daniel Suteu, Table of n, a(n) for n = 1..1908 (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.
Programs
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Mathematica
p[n_] := FactorInteger[n][[-1, 2]]; Select[Range[10^6], p[#] > 1 && p[# + 1] > 3 &]
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Python
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