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A309245 Least number k > 0 which is not a divisor of n such that k^2 + n is a nonsquare powerful number (A102834).

%I A309245 #15 Jul 20 2019 12:26:44
%S A309245 682,5,37,11,1879706,463,11,10,2046,341881,31,74,70,5519,793,22,
%T A309245 1952785824219551870,57,559,338,4580728614212333152148,503259461,45,
%U A309245 926,190,109,36,62,436,832836278711,63,88,2451448196948930,7037029,36,33
%N A309245 Least number k > 0 which is not a divisor of n such that k^2 + n is a nonsquare powerful number (A102834).
%C A309245 De Koninck et al. calculated the first 50 terms of this sequence.
%H A309245 Amiram Eldar, <a href="/A309245/b309245.txt">Table of n, a(n) for n = 1..50</a>
%H A309245 Jean-Marie De Koninck, Nicolas Doyon, Florian Luca, and Michoacán Morelia, <a href="http://emis.ams.org/journals/JIS/VOL14/DeKoninck/dek.html">Powerful values of quadratic polynomials</a>, Journal of Integer Sequences, Vol. 14, No. 3 (2011), Article 11.3.3.
%e A309245 a(1) = 682 since 682^2 + 1 = 465125 = 5^3 * 61^2 is a nonsquare powerful number and is the smallest k > 0 such that k^2 + 1 is not a nonsquare powerful number.
%t A309245 powerfulQ[n_] := Min@FactorInteger[n][[All, 2]] > 1; powerfulNonsquare[n_] := !IntegerQ[Sqrt[n]] && powerfulQ[n]; a[n_] := Module[{k=1}, While[Divisible[n, k] || !powerfulNonsquare[k^2 + n], k++]; k]; Table[a[n], {n, 1, 16}]
%o A309245 (PARI) is_a102834(n) = ispowerful(n) && !issquare(n) \\ after _Charles R Greathouse IV_ in A102834
%o A309245 a(n) = for(k=1, oo, if(n%k!=0 && is_a102834(k^2+n), return(k))) \\ _Felix Fröhlich_, Jul 19 2019
%Y A309245 Cf. A001694, A102834, A140746, A309246.
%K A309245 nonn
%O A309245 1,1
%A A309245 _Amiram Eldar_, Jul 18 2019