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A365517 Numbers k such that the sum of the squarefree part of k and the squarefree kernel of k is a perfect square.

%I A365517 #29 May 25 2025 09:26:25
%S A365517 2,8,9,12,32,48,81,98,108,128,150,192,225,252,363,392,432,512,578,600,
%T A365517 729,768,972,1008,1100,1225,1350,1568,1728,1805,1922,2025,2028,2048,
%U A365517 2268,2312,2366,2400,2940,3072,3174,3267,3750,3888,4032,4400,4802,5400,5625
%N A365517 Numbers k such that the sum of the squarefree part of k and the squarefree kernel of k is a perfect square.
%C A365517 In other words, numbers k such that A007913(k) + A007947(k) is in A000290 (the sum of the core and the kernel (radical) of k is a square).
%C A365517 The corresponding sequence of squares starts: 4,4,4,9,4,9,4,16,9,4,36,9,... and it seems that 25 is the smallest square which is not the sum of the core and kernel of any integer.
%H A365517 Michael De Vlieger, <a href="/A365517/b365517.txt">Table of n, a(n) for n = 1..3140</a> (all terms less than or equal to 2^28.)
%e A365517 108 is a term because A007913(108) = 3, A007947(108) = 6 and 3 + 6 = 9.
%e A365517 Let c, r denote the core and the rad (kernel) respectively, of any number, then for m >= 0, 2^(2*m+1) is a term (c = r = 2)-->4 (2,8,32,128,...).
%e A365517 For m >= 1, h > 0, 2^(2*m)*3^(2*h+1) is a term (c = 6, r = 3)-->9 (12,48,108,...).
%t A365517 squareFreePart[n_Integer?Positive] := squareFreePart[n] = n/Times @@ (First[#]^(2*Floor[Last[#]/2]) & /@ FactorInteger[n]); squareFreeKernel[n_Integer?Positive] := squareFreeKernel[n] = Times @@ (First[#] & /@ FactorInteger[n]); a[max_Integer?Positive] := a[max] = Select[Range[max], IntegerQ@Sqrt[squareFreePart[#] + squareFreeKernel[#]] &]; a[5625] (* _Robert P. P. McKone_, Sep 07 2023 *)
%o A365517 (PARI) isok(s) = issquare(core(s) + factorback(factorint(s)[, 1])); \\ _Michel Marcus_, Sep 08 2023
%o A365517 (Python)
%o A365517 from itertools import count, islice
%o A365517 from sympy.ntheory.primetest import is_square
%o A365517 from sympy import factorint
%o A365517 def A365517_gen(startvalue=1): # generator of terms >= startvalue
%o A365517     for k in count(max(startvalue,1)):
%o A365517         a, b = 1, 1
%o A365517         for p, e in factorint(k).items():
%o A365517             if e&1:
%o A365517                 a *= p
%o A365517             else:
%o A365517                 b *= p
%o A365517         if is_square(a*(b+1)):
%o A365517             yield k
%o A365517 A365517_list = list(islice(A365517_gen(),30)) # _Chai Wah Wu_, Sep 19 2023
%Y A365517 Cf. A000290, A007913, A007947.
%K A365517 nonn
%O A365517 1,1
%A A365517 _David James Sycamore_, Sep 07 2023