A378700 Number of k in A126706 between powerful numbers that are not prime powers.

5, 11, 10, 1, 10, 19, 1, 4, 2, 22, 12, 27, 1, 11, 2, 14, 6, 28, 26, 9, 0, 41, 3, 26, 13, 25, 0, 10, 35, 11, 10, 0, 26, 26, 8, 10, 5, 26, 30, 17, 11, 52, 13, 12, 56, 1, 20, 9, 34, 69, 1, 69, 37, 3, 38, 0, 14, 57, 11, 39, 23, 15, 26, 18, 6, 36, 3, 30, 27, 27, 97

Offset: 1

Author

Michael De Vlieger, Dec 04 2024

Keywords

• nonn
• easy

Comments

Within the sequence S = A126706 of powerful numbers, we have numbers k that are powerful (in A286708) and numbers m that are not powerful (in A332785). This sequence is the number of k between m.

Examples

			We partition S = A126706 by numbers k in A286708 (in brackets) and derive the following irregular table:
    12,   18,  20,  24,  28, [36];                                hence a(1) = 5,
    40,   44,  45,  48,  50,  52,  54,  56,  60,  63,   68, [72];       a(2) = 11,
    75,   76,  80,  84,  88,  90,  92,  96,  98,  99, [100];            a(3) = 10,
   104, [108];                                                          a(4) = 1,
   112,  116, 117, 120, 124, 126, 132, 135, 136, 140, [144];            a(5) = 10, etc.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000
• Michael De Vlieger, Log log scatterplot of a(n), n = 1..11210, rendering a(n) = 0 instead as 1/2 for visibility.

Crossrefs

Cf. A001694, A126706, A286708, A332785.

Programs

Mathematica

s = Select[Range[2^16], Nor[SquareFreeQ[#], PrimePowerQ[#]] &]; -1 + Length /@ TakeList[s, Prepend[Differences[#], First[#]] &@ Position[s, _Integer?(Divisible[#, Apply[Times, FactorInteger[#][[All, 1]] ]^2] &)][[All, 1]] ]