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A336175 Numbers k such that there are no powerful numbers between k^2 and (k+1)^2. 5
1, 3, 4, 6, 7, 9, 12, 13, 17, 21, 23, 24, 26, 27, 30, 32, 34, 35, 38, 40, 43, 47, 49, 54, 60, 61, 64, 66, 68, 69, 71, 75, 80, 81, 85, 86, 91, 95, 97, 99, 105, 106, 108, 112, 120, 123, 125, 128, 131, 133, 136, 137, 139, 142, 143, 151, 153, 154, 159, 160, 162, 163 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Positions of 0's in A119241.

Shiu (1980) proved that this sequence has an asymptotic density 0.2759... A more accurate calculation using his formula gives 0.275965511407718981...

REFERENCES

József Sándor, Dragoslav S. Mitrinovic and Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, chapter VI, p. 226.

LINKS

Amiram Eldar, Table of n, a(n) for n = 1..10000

P. Shiu, On the number of square-full integers between successive squares, Mathematika, Vol. 27, No. 2 (1980), pp. 171-178.

EXAMPLE

1 is a term since the two numbers between 1^2 = 1 and (1+1)^2 = 4, 2 and 3, are not powerful.

2 is not a term since there is a powerful number, 8 = 2^3, between 2^2 = 4 and (2+1)^2 = 9.

MATHEMATICA

powQ[n_] := (n == 1) || Min @@ FactorInteger[n][[;; , 2]] > 1; Select[Range[150], ! AnyTrue[Range[#^2 + 1, (# + 1)^2 - 1], powQ] &]

CROSSREFS

Cf. A001694, A119241, A119242, A336176, A336177, A336178.

Sequence in context: A136110 A032725 A089038 * A137673 A226245 A236386

Adjacent sequences:  A336172 A336173 A336174 * A336176 A336177 A336178

KEYWORD

nonn

AUTHOR

Amiram Eldar, Jul 10 2020

STATUS

approved

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Last modified September 25 15:08 EDT 2021. Contains 347657 sequences. (Running on oeis4.)