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A217038 Number of powerful numbers < n. 1
0, 1, 1, 1, 2, 2, 2, 2, 3, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 7, 7, 7, 7, 7, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,5

COMMENTS

Powerful numbers are given by A001694.

LINKS

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

EXAMPLE

a(10)=4 since there are exactly 4 powerful numbers (1,4,8,9) less than 10.

MATHEMATICA

PowQ[n_] := Cases[FactorInteger[n], {p_, 1} -> p] == {}; Q[n_] := Length[Join[{1}, Select[Range[n - 1], PowQ[#] &]]] ; Join[{0}, Table[Q[n], {n, 2, 100}]]

PROG

(PARI) g(n, fe=factor(n)[, 2])=prod(i=1, #fe, (fe[i]+2)\2 - (fe[i]+2)\3)

a(n)=my(v=List(), t); n--; for(m=2, sqrtnint(n, 6), for(y=1, sqrtnint(n\m^6, 3), t=(m^2*y)^3; for(x=1, sqrtint(n\t), listput(v, t*x^2)))); v=Set(v); sum(y=1, sqrtnint(n, 3), sqrtint(n\y^3))-sum(i=1, #v, g(v[i])-1) \\ Charles R Greathouse IV, Jul 31 2017

(PARI) first(n)=my(v=vector(n), s=1); if(n>1, v[2]=1); forfactored(k=2, n-1, if(vecmin(k[2][, 2])>1, s++); v[k[1]+1]=s); v \\ Charles R Greathouse IV, Jul 31 2017

CROSSREFS

Cf. A001597, A001694, A076411.

Sequence in context: A085501 A069623 A076411 * A309196 A243283 A243284

Adjacent sequences:  A217035 A217036 A217037 * A217039 A217040 A217041

KEYWORD

nonn

AUTHOR

Jayanta Basu, Apr 07 2013

STATUS

approved

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Last modified August 19 18:33 EDT 2019. Contains 326133 sequences. (Running on oeis4.)