cp's OEIS Frontend

A361936 Indices of the squares in the sequence of powerful numbers (A001694).

%I A361936 #15 Sep 10 2024 14:21:14
%S A361936 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,
%T A361936 44,45,46,48,50,51,55,56,59,60,61,65,67,68,70,71,73,75,76,79,81,84,85,
%U A361936 87,88,90,92,94,96,97,100,102,104,107,109,110,111,114,116,117,119,120
%N A361936 Indices of the squares in the sequence of powerful numbers (A001694).
%C A361936 Equivalently, the number of powerful numbers that do not exceed n^2.
%C A361936 The asymptotic density of this sequence is zeta(3)/zeta(3/2) = 1/A090699 = 0.460139... .
%C A361936 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.
%H A361936 Amiram Eldar, <a href="/A361936/b361936.txt">Table of n, a(n) for n = 1..10000</a>
%H A361936 Paul Erdős and George Szekeres, <a href="http://pub.acta.hu/acta/showCustomerArticle.action?id=5508&amp;dataObjectType=article">Über die Anzahl der Abelschen Gruppen gegebener Ordnung und über ein verwandtes zahlentheoretisches Problem</a>, Acta Sci. Math. (Szeged), Vol. 7, No. 2 (1935), pp. 95-102.
%H A361936 Solomon W. Golomb, <a href="http://www.jstor.org/stable/2317020">Powerful numbers</a>, Amer. Math. Monthly, Vol. 77, No. 8 (1970), pp. 848-852.
%F A361936 a(n) = A217038(n^2).
%F A361936 a(n+1) - a(n) = A119241(n) + 1.
%F A361936 a(n) = (zeta(3/2)/zeta(3)) * n + O(n^(2/3)).
%t A361936 Position[Select[Range[5000], # == 1 || Min[FactorInteger[#][[;; , 2]]] > 1 &], _?(IntegerQ[Sqrt[#]] &)] // Flatten
%o A361936 (PARI) lista(kmax) = {my(c = 0); for(k = 1, kmax, if(ispowerful(k), c++); if(issquare(k), print1(c, ", "))); }
%o A361936 (Python)
%o A361936 from math import isqrt
%o A361936 from sympy import integer_nthroot, factorint
%o A361936 def A361936(n):
%o A361936     m = n**2
%o A361936     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
%Y A361936 Cf. A000290, A001694, A090699, A119241, A217038, A336175.
%K A361936 nonn
%O A361936 1,2
%A A361936 _Amiram Eldar_, Mar 31 2023