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A331801
Integers that are sum of two nonsquarefree numbers.
1
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
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 *)
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: A173461 A335160 A295782 * A143813 A290135 A225533
KEYWORD
nonn
AUTHOR
Bernard Schott, Jan 26 2020
STATUS
approved