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A192636
Powerful sums of two powerful numbers.
1
8, 9, 16, 25, 32, 36, 64, 72, 81, 100, 108, 121, 125, 128, 144, 169, 196, 200, 216, 225, 243, 256, 288, 289, 324, 343, 361, 392, 400, 432, 441, 484, 500, 512, 576, 625, 648, 675, 676, 729, 784, 800, 841, 864, 900, 961, 968, 972, 1000, 1024, 1089, 1125, 1152, 1156, 1225
OFFSET
1,1
COMMENTS
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.
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..5000 from Charles R Greathouse IV)
Tim D. Browning and K. Van Valckenborgh, Sums of three squareful numbers, Experimental Mathematics, Vol. 21, No. 2 (2012), pp. 204-211; arXiv preprint, arXiv:1106.4472 [math.NT], 2011.
FORMULA
Numbers k such that there exists some a, b, c with A001694(a) + A001694(b) = k = A001694(c).
MATHEMATICA
With[{m = 1225}, pow = Select[Range[m], # == 1 || Min[FactorInteger[#][[;; , 2]]] > 1 &]; Intersection[pow, Plus @@@ Tuples[pow, {2}]]] (* Amiram Eldar, Feb 12 2023 *)
PROG
(PARI) isPowerful(n)=if(n>3, vecmin(factor(n)[, 2])>1, n==1)
sumset(a, b)={
my(c=vectorsmall(#a*#b));
for(i=1, #a,
for(j=1, #b,
c[(i-1)*#b+j]=a[i]+b[j]
)
);
vecsort(c, , 8)
}; selfsum(a)={
my(c=vectorsmall(binomial(#a+1, 2)), k);
for(i=1, #a,
for(j=i, #a,
c[k++]=a[i]+a[j]
)
);
vecsort(c, , 8)
};
list(lim)={
my(v=select(isPowerful, vector(floor(lim), i, i)));
select(n->n<=lim && isPowerful(n), Vec(selfsum(v)))
};
CROSSREFS
KEYWORD
nonn
AUTHOR
EXTENSIONS
Corrected (on the advice of Donovan Johnson) by Charles R Greathouse IV, Sep 25 2012
STATUS
approved