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%I #25 Feb 12 2023 03:15:12
%S 8,9,16,25,32,36,64,72,81,100,108,121,125,128,144,169,196,200,216,225,
%T 243,256,288,289,324,343,361,392,400,432,441,484,500,512,576,625,648,
%U 675,676,729,784,800,841,864,900,961,968,972,1000,1024,1089,1125,1152,1156,1225
%N Powerful sums of two powerful numbers.
%C Browning & Valckenborgh conjecture that a(n) ~ kn^2 with k approximately 0.139485255. See their Conjecture 1 and equation (14). Their Theorems 1 and 2 establish upper and lower asymptotic bounds.
%H Amiram Eldar, <a href="/A192636/b192636.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..5000 from Charles R Greathouse IV)
%H Tim D. Browning and K. Van Valckenborgh, <a href="https://doi.org/10.1080/10586458.2011.605733">Sums of three squareful numbers</a>, Experimental Mathematics, Vol. 21, No. 2 (2012), pp. 204-211; <a href="https://arxiv.org/abs/1106.4472">arXiv preprint</a>, arXiv:1106.4472 [math.NT], 2011.
%F Numbers k such that there exists some a, b, c with A001694(a) + A001694(b) = k = A001694(c).
%t With[{m = 1225}, pow = Select[Range[m], # == 1 || Min[FactorInteger[#][[;; , 2]]] > 1 &]; Intersection[pow, Plus @@@ Tuples[pow, {2}]]] (* _Amiram Eldar_, Feb 12 2023 *)
%o (PARI) isPowerful(n)=if(n>3,vecmin(factor(n)[,2])>1,n==1)
%o sumset(a,b)={
%o my(c=vectorsmall(#a*#b));
%o for(i=1,#a,
%o for(j=1,#b,
%o c[(i-1)*#b+j]=a[i]+b[j]
%o )
%o );
%o vecsort(c,,8)
%o }; selfsum(a)={
%o my(c=vectorsmall(binomial(#a+1,2)),k);
%o for(i=1,#a,
%o for(j=i,#a,
%o c[k++]=a[i]+a[j]
%o )
%o );
%o vecsort(c,,8)
%o };
%o list(lim)={
%o my(v=select(isPowerful, vector(floor(lim),i,i)));
%o select(n->n<=lim && isPowerful(n), Vec(selfsum(v)))
%o };
%Y Subsequence of A001694 and of A076871.
%Y Cf. A001694, A007532, A005934, A005188, A003321, A014576, A023052, A046074, A013929, A076871, A143813. - _Jonathan Vos Post_, Jul 10 2011
%K nonn
%O 1,1
%A _Charles R Greathouse IV_, Jul 06 2011
%E Corrected (on the advice of _Donovan Johnson_) by _Charles R Greathouse IV_, Sep 25 2012
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