A331801 Integers that are sum of two nonsquarefree numbers.

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

1,1

Comments

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).

Links

Table of n, a(n) for n=1..71.

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

Example

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

Mathematica

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 *)

Prog

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

Crossrefs

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).

Sequence in context: A378689 A335160 A295782 * A143813 A290135 A225533

Adjacent sequences: A331798 A331799 A331800 * A331802 A331803 A331804

Keyword

nonn

Author

Bernard Schott, Jan 26 2020

Status

approved