A076871 -id:A076871 - cp's OEIS frontend

A192636 Powerful sums of two powerful numbers.

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

Charles R Greathouse IV, Jul 06 2011

nonn

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.

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.

Subsequence of A001694 and of A076871.

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

With[{m = 1225}, pow = Select[Range[m], # == 1 || Min[FactorInteger[#][[;; , 2]]] > 1 &]; Intersection[pow, Plus @@@ Tuples[pow, {2}]]] (* Amiram Eldar, Feb 12 2023 *)

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)))
};

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

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

A331801 Integers that are sum of two nonsquarefree numbers.

Original entry on oeis.org

8, 12, 13, 16, 17, 18, 20, 21, 22, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85

Offset: 1

Bernard Schott, Jan 26 2020

nonn

Proposition: All integers > 23 are terms of this sequence (see link Prime Curios!).
Proof by exhaustion:
1) For numbers {4*k} with k>=6, then 4*k = 4*(k-1) + 4 is a term as 4*(k-1) and 4 are nonsquarefree;
2) For numbers {4*k+1} with k>=6, then 4*k+1 = 4*(k-2) + 9 is a term as 4*(k-2) and 9 are nonsquarefree;
3) For numbers {4*k+2} with k>=6, then 4*k+2 = 4*(k-4) + 18 is a term as 4*(k-4) and 18 are nonsquarefree;
4) For numbers {4*k+3}; with k=6, 27 = 9+18 is a term as 9 and 18 are nonsquarefree, and with k>=7, 4*k+3 = 4*(k-6) + 27 is also a term as 4*(k-6) and 27 are nonsquarefree.
Conclusion: every integer > 23 is sum of two nonsquarefree numbers (QED).

			13 = 4 + 9 and 21 = 9 + 12 are terms of this sequence as 4, 9 and 12 are nonsquarefree numbers.

G. L. Honaker, Jr. and Chris K. Caldwell, Prime Curios! 23 (Rupinski)

Cf. A005117 (squarefree), A013929 (nonsquarefree), A331802 (complement).
Cf. A000404 (sum of 2 nonzero squares), A018825 (not the sum of 2 nonzero squares).
Cf. A001694 (squareful), A052485 (not squareful), A076871 (sum of 2 squareful), A085253 (not the sum of 2 squareful).

max = 85; Union @ Select[Total /@ Tuples[Select[Range[max], !SquareFreeQ[#] &], 2], # <= max &] (* Amiram Eldar, Feb 04 2020 *)
Join[{8,12,13,16,17,18,20,21,22},Range[24,100]] (* or *) Complement[Range[100],{1,2,3,4,5,6,7,9,10,11,14,15,19,23}] (* Harvey P. Dale, Dec 04 2024 *)

isok(m) = {for (i=1, m-1, if (!issquarefree(i) && !issquarefree(m-i), return (1));); return(0);} \\ Michel Marcus, Jan 31 2020

A187054 Numbers that are not the sum of three powerful numbers (A001694).

Original entry on oeis.org

1, 2, 4, 5, 7, 8, 15, 23, 31, 87, 111, 119

Offset: 1

Charles R Greathouse IV, Mar 02 2011

Heath-Brown shows that this sequence is finite, resolving a conjecture of Erdos. Presumably a(12) = 119 is the last term.

D. R. Heath-Brown, "Sums of three square-full numbers". Number theory, Vol. I (Budapest, 1987), pp. 163-171, Colloq. Math. Soc. János Bolyai, 51, North-Holland, Amsterdam, 1990.
D. R. Heath-Brown, "Ternary quadratic forms and sums of three square-full numbers". Séminaire de Théorie des Nombres, Paris 1986-87, pp. 137-163, Progr. Math., 75, Birkhäuser Boston, Boston, MA, 1988.

P. Erdos, Problems and results on number theoretic properties of consecutive integers and related questions, Proceedings of the Fifth Manitoba Conference on Numerical Mathematics (Univ. Manitoba, Winnipeg, Man., 1975), Congress. Numer. XVI (1976), pp. 25-44.

Proper subsequence of A135367.

Cf. A076871, A135693.

powerfulQ[n_] := n == 1 || Min[Last /@ FactorInteger[n]] > 1; nn = 1000; pow = Select[Range[nn], powerfulQ]; Complement[Range[nn], Select[Union[Flatten[Outer[Plus, pow, pow, pow]]], # <= nn &]] (* T. D. Noe, Mar 02 2011 *)

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A192636 Powerful sums of two powerful numbers.

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A331801 Integers that are sum of two nonsquarefree numbers.

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A187054 Numbers that are not the sum of three powerful numbers (A001694).

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