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A363189
Indices of the odd terms in the sequence of powerful numbers (A001694).
6
1, 4, 6, 7, 10, 13, 16, 17, 20, 24, 25, 28, 30, 31, 35, 39, 41, 43, 45, 48, 51, 56, 57, 60, 62, 63, 65, 68, 71, 75, 79, 82, 83, 84, 87, 90, 94, 97, 98, 99, 102, 103, 105, 107, 110, 114, 117, 120, 122, 125, 127, 129, 133, 138, 141, 142, 144, 145, 148, 151, 152
OFFSET
1,2
COMMENTS
The asymptotic density of this sequence is (2-sqrt(2))/(3-sqrt(2)) = 0.369398... .
If A001694(k) is a term of A363190 then k and k+1 are consecutive integers in this sequence.
LINKS
Teerapat Srichan, The odd/even dichotomy for the set of square-full numbers, Applied Mathematics E-Notes, Vol. 20 (2020), pp. 528-531.
Wikipedia, Powerful number.
EXAMPLE
The first 6 powerful numbers are 1, 4, 8, 9, 16 and 25. 1, 9 and 25 are odd and their positions in the sequence are 1, 4 and 6, respectively.
MATHEMATICA
Position[Select[Range[7000], # == 1 || Min[FactorInteger[#][[;; , 2]]] > 1 &], _?(OddQ[#] &)] // Flatten
PROG
(PARI) lista(kmax) = {my(c = 0); for(k = 1, kmax, if(ispowerful(k), c++; if(k%2, print1(c, ", ")))); }
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Amiram Eldar, May 21 2023
STATUS
approved