A360543 a(n) = number of numbers k < n, gcd(k, n) > 1, such that omega(k) > omega(n) and rad(n) | rad(k), where omega(n) = A001221(n) and rad(n) = A007947(n).
0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 6, 0, 0, 0, 0, 11, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 2, 5, 1, 0, 0, 0, 2, 0, 1, 0, 0, 0, 0, 0, 0, 1, 26, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 2, 0, 0, 0, 0, 3, 23, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 7, 0, 3, 1, 4
Offset: 1
Keywords
Examples
a(4) = 0 since k = 1..3 are prime powers.
a(8) = 1 since only k = 6 is such that p = 3, q = 5, but gcd(6, 10) = 2.
a(9) = 1 since the following satisfies definition: {6},
a(16) = 4, i.e., {6, 10, 12, 14},
a(25) = 3, i.e., {10, 15, 20},
a(27) = 6, i.e., {6, 12, 15, 18, 21, 24},
a(32) = 11, i.e., {6, 10, 12, 14, 18, 20, 22, 24, 26, 28, 30},
a(36) = 1, i.e., {30},
a(40) = 1, i.e., {30},
a(45) = 1, i.e., {30}, etc.
Links
- Michael De Vlieger, Table of n, a(n) for n = 1..16384
- Michael De Vlieger, Diagram showing k <= n, n = 1..36, where a(n) is the number of numbers k in row n shown in blue. Numbers k in green in row n are counted by A360480(n). Together, blue and green numbers are counted by A243823(n) and appear in row n of A272619. Dots at (k, n) in red are divisors, and in yellow and magenta in row n are counted by A243822(n).
- Michael De Vlieger, Plot k < n at (x, y) = (k, -n) for n = 1..2^10, where black represents k such that gcd(k, n) > 1, such that omega(k) > omega(n) and rad(n) | rad(k).
Crossrefs
Programs
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Mathematica
nn = 120; rad[n_] := rad[n] = Times @@ FactorInteger[n][[All, 1]]; c = Select[Range[4, nn], CompositeQ]; Table[Function[{q, r}, Count[TakeWhile[c, # <= n &], _?(And[PrimeNu[#] > q, Divisible[rad[#], r]] &)]] @@ {PrimeNu[n], rad[n]}, {n, nn}]
Formula
For prime power n = p^e, n > 1, a(n) = p^(e-1) - e.
For n in A360765, a(n) > 0.