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

1,1

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.

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.

Example

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.

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]] ]

Crossrefs

Cf. A001694, A126706, A286708, A332785.

Sequence in context: A075261 A254766 A185201 * A112956 A258995 A335554

Adjacent sequences: A378695 A378697 A378699 * A378701 A378704 A378706

Keyword

nonn,easy,new

Author

Michael De Vlieger, Dec 04 2024

Status

approved