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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 (list; graph; refs; listen; history; text; internal format)
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

Charles R Greathouse IV, Table of n, a(n) for n = 1..5000

T. D. Browning and K. Van Valckenborgh, Sums of three squareful numbers (2011).

FORMULA

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

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

Subsequence of A001694 and of A076871.

Cf. A001694, A007532, A005934, A005188, A003321, A014576, A023052, A046074, A013929, A076871, A143813. - Jonathan Vos Post, Jul 10 2011

Sequence in context: A169817 A227649 A227648 * A265731 A227646 A145820

Adjacent sequences:  A192633 A192634 A192635 * A192637 A192638 A192639

KEYWORD

nonn

AUTHOR

Charles R Greathouse IV, Jul 06 2011

EXTENSIONS

Corrected (on the advice of Donovan Johnson) by Charles R Greathouse IV, Sep 25 2012

STATUS

approved

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Last modified November 20 12:19 EST 2017. Contains 294967 sequences.