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A363084 Numbers k such that sqrt(A007947(k) - A007913(k)) is an integer m > 0.

%I A363084 #36 Sep 19 2023 12:44:13
%S A363084 4,16,18,25,64,72,100,162,180,256,288,289,294,400,507,625,648,676,720,
%T A363084 722,1024,1152,1176,1210,1369,1458,1600,1620,2178,2205,2500,2548,2592,
%U A363084 2646,2704,2880,2888,3150,4096,4225,4500,4563,4608,4704,4840,5202,5832,5887
%N A363084 Numbers k such that sqrt(A007947(k) - A007913(k)) is an integer m > 0.
%C A363084 Let core(k) = A007913(k) and rad(k) = A007947(k).
%C A363084 Squarefree numbers k imply rad(k) - core(k) = k - k = 0.
%C A363084 Perfect squares k^2 such that rad(k) = m^2+1 and k > 1 imply rad(k^2) - core(k^2) = (m^2+1) - 1 = m^2, with integers k, m.
%C A363084 Generally, if there exists a minimal d such that d | k, k/d = m^2, and rad(k) - d = m^2, then k is in the sequence.
%C A363084 Subsets of this sequence include the sets of squares k^2 such that k is in A002496, A003592, and A089653, since A089653 contains both A002496 and A003592.
%H A363084 Michael De Vlieger, <a href="/A363084/b363084.txt">Table of n, a(n) for n = 1..3170</a> (all terms less than 2^28.)
%e A363084 a(1) = 4 since rad(4) = 1+1; rad(4) - core(4) = 2 - 1 = 1, a nonzero square.
%e A363084 a(2) = 18 since 18/2 = 9, and rad(18) - core(18) = 6 - 2 = 4, a nonzero square, etc.
%t A363084 Select[Range[6000], And[IntegerQ[#], # > 0] &[Sqrt[Times @@ FactorInteger[#][[All, 1]] - (Sqrt[#] /. (c_ : 1)*a_^(b_ : 0) :> (c*a^b)^2)] ] &]
%o A363084 (PARI) isok(k) = my(s=factorback(factorint(k)[, 1])-core(k)); (s>0) && issquare(s); \\ _Michel Marcus_, Sep 18 2023
%o A363084 (Python)
%o A363084 from itertools import count, islice
%o A363084 from sympy.ntheory.primetest import is_square
%o A363084 from sympy import factorint
%o A363084 def A363084_gen(startvalue=1): # generator of terms >= startvalue
%o A363084     for k in count(max(startvalue,1)):
%o A363084         a, b = 1, 1
%o A363084         for p, e in factorint(k).items():
%o A363084             if e&1:
%o A363084                 a *= p
%o A363084             else:
%o A363084                 b *= p
%o A363084         if b>1 and is_square(a*(b-1)):
%o A363084             yield k
%o A363084 A363084_list = list(islice(A363084_gen(),30)) # _Chai Wah Wu_, Sep 19 2023
%Y A363084 Cf. A002496, A003592, A007913, A007947, A089653, A365517.
%K A363084 nonn
%O A363084 1,1
%A A363084 _Michael De Vlieger_, Sep 05 2023