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A361936
Indices of the squares in the sequence of powerful numbers (A001694).
4
1, 2, 4, 5, 6, 9, 10, 11, 13, 14, 16, 19, 20, 21, 24, 26, 28, 29, 31, 33, 35, 36, 39, 40, 41, 44, 45, 46, 48, 50, 51, 55, 56, 59, 60, 61, 65, 67, 68, 70, 71, 73, 75, 76, 79, 81, 84, 85, 87, 88, 90, 92, 94, 96, 97, 100, 102, 104, 107, 109, 110, 111, 114, 116, 117, 119, 120
OFFSET
1,2
COMMENTS
Equivalently, the number of powerful numbers that do not exceed n^2.
The asymptotic density of this sequence is zeta(3)/zeta(3/2) = 1/A090699 = 0.460139... .
If k is a term of A336175 then a(k) and a(k+1) are consecutive integers, i.e., a(k+1) = a(k) + 1.
LINKS
Paul Erdős and George Szekeres, Über die Anzahl der Abelschen Gruppen gegebener Ordnung und über ein verwandtes zahlentheoretisches Problem, Acta Sci. Math. (Szeged), Vol. 7, No. 2 (1935), pp. 95-102.
Solomon W. Golomb, Powerful numbers, Amer. Math. Monthly, Vol. 77, No. 8 (1970), pp. 848-852.
FORMULA
a(n) = A217038(n^2).
a(n+1) - a(n) = A119241(n) + 1.
a(n) = (zeta(3/2)/zeta(3)) * n + O(n^(2/3)).
MATHEMATICA
Position[Select[Range[5000], # == 1 || Min[FactorInteger[#][[;; , 2]]] > 1 &], _?(IntegerQ[Sqrt[#]] &)] // Flatten
PROG
(PARI) lista(kmax) = {my(c = 0); for(k = 1, kmax, if(ispowerful(k), c++); if(issquare(k), print1(c, ", "))); }
(Python)
from math import isqrt
from sympy import integer_nthroot, factorint
def A361936(n):
m = n**2
return int(sum(isqrt(m//k**3) for k in range(1, integer_nthroot(m, 3)[0]+1) if all(d<=1 for d in factorint(k).values()))) # Chai Wah Wu, Sep 10 2024
CROSSREFS
KEYWORD
nonn
AUTHOR
Amiram Eldar, Mar 31 2023
STATUS
approved