A331802 -id:A331802 - cp's OEIS frontend

A378896 Numbers k such that k - p^2 is squarefree for every prime p < sqrt(k).

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 14, 15, 19, 23, 26, 30, 35, 38, 39, 42, 46, 47, 51, 55, 62, 66, 71, 78, 82, 83, 86, 87, 91, 95, 110, 111, 114, 118, 119, 122, 127, 131, 138, 143, 155, 158, 163, 167, 182, 183, 186, 190, 191, 195, 203, 206, 210, 215, 222, 226, 227, 230, 231, 235, 239, 255, 258, 262

Offset: 1

Robert Israel, Dec 14 2024

nonn

Numbers k such that there is no solution to k = p^2 + m * q^2 with p and q prime and m > 0.

Numbers k such that A379018(k) = -1.

			a(10) = 11 is a term because both 11 - 2^2 = 7 and 11 - 3^2 = 2 are squarefree, while 11 - 5^2 < 0.

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

Cf. A005117, A379018, A331802.

filter:= proc(n) local p;
  p:= 2;
  while p^2 <= n do
    if not numtheory:-issqrfree(n-p^2) then return false fi;
    p:= nextprime(p);
  od;
  true
end proc:
select(filter, [$1..300]);

sfpQ[n_]:=With[{prs=Select[Prime[Range[PrimePi[Sqrt[n]]]],#Harvey P. Dale, Mar 24 2025 *)

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

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A378896 Numbers k such that k - p^2 is squarefree for every prime p < sqrt(k).

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