A363189 Indices of the odd terms in the sequence of powerful numbers (A001694).

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

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

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

Cf. A001694, A062739, A363190, A363191, A363192.

Sequence in context: A103401 A285091 A286485 * A277572 A107622 A049306

Adjacent sequences: A363186 A363187 A363188 * A363190 A363191 A363192

Keyword

nonn,easy

Author

Amiram Eldar, May 21 2023

Status

approved