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A192636 Powerful sums of two powerful numbers. 1


%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 Charles R Greathouse IV, <a href="/A192636/b192636.txt">Table of n, a(n) for n = 1..5000</a>

%H T. D. Browning and K. Van Valckenborgh, <a href="http://arxiv.org/abs/1106.4472">Sums of three squareful numbers</a> (2011).

%F n such that there exists some a, b, c with A001694(a) + A001694(b) = n = A001694(c).

%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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Last modified June 20 09:12 EDT 2019. Contains 324234 sequences. (Running on oeis4.)