A386257 -id:A386257 - cp's OEIS frontend

A386304 Numbers k such that k - A067666(k) is a square.

1, 16, 27, 75, 128, 343, 475, 600, 663, 715, 759, 1015, 1845, 2679, 3717, 3933, 4440, 5083, 5325, 5467, 6120, 6210, 6325, 6405, 6859, 7029, 8349, 8541, 8664, 9125, 9960, 12045, 12427, 12535, 13509, 15067, 16677, 18693, 18711, 21783, 22797, 23250, 23560, 24605, 25527, 26496, 26967, 27117, 28557

Offset: 1

Will Gosnell and Robert Israel, Jul 17 2025

nonn

Numbers k such that k minus the sum of the squares of its prime factors with multiplicity is a square.
Is there any number other than 1 in both this sequence and A386257?
Contains no semiprimes.

			a(4) = 75 is a term because 75 = 3 * 5^2 and 75 - 3^2 - 2 * 5^2 = 16 = 4^2 is a square.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A067666, A386257.

filter:= proc(n) local t;
  issqr(n - add(t[1]^2*t[2], t=ifactors(n)[2]))
end proc:
select(filter, [$1..10^5]);

spf[{p_,e_}]:=e*p^2;Q[k_]:=IntegerQ[Sqrt[k-Total[spf/@FactorInteger[k]]]];Select[Range[29000],Q[#]&] (* James C. McMahon, Jul 23 2025 *)

isok(k) = my(f=factor(k)); issquare(k - sum(i=1, #f~, f[i, 1]^2*f[i, 2])); \\ Michel Marcus, Jul 20 2025

A386640 Numbers k such that k + A224787(k) is a square.

Original entry on oeis.org

1, 225, 270, 1900, 4988, 5656, 6120, 8704, 11180, 16588, 17710, 19228, 24475, 28449, 29458, 32330, 34606, 38088, 39292, 40221, 41181, 42476, 48545, 48640, 53795, 56832, 57288, 64975, 78793, 84925, 86242, 117116, 124135, 128478, 129673, 134044, 136224, 136896, 147149, 150528, 168055, 183141

Offset: 1

Will Gosnell and Robert Israel, Jul 27 2025

nonn

Numbers k such that the sum of k and the cubes of the prime factors of k, counted with multiplicity, is a square.

			a(3) = 270 = 2 * 3^3 * 5 is a term because 270 + 2^3 + 3 * 3^3 + 5^3 =  484 = 22^2 is a square.

Robert Israel, Table of n, a(n) for n = 1..3000

Cf. A224787, A385238, A386246, A386257, A386623.

filter:= proc(n) local t;
   issqr(n + add(t[1]^3*t[2],t=ifactors(n)[2]))
end proc:
select(filter, [$1..10^6]);

lim=184000;f[{p_,e_}]:=e*p^3;a224787[k_]:=If[k==1,0,Total[f/@FactorInteger[k]]];q[k_]:=IntegerQ[Sqrt[k+a224787[k]]];Select[Range[lim],q[#]&] (* James C. McMahon, Jul 30 2025 *)

cp's OEIS Frontend

A386304 Numbers k such that k - A067666(k) is a square.

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