A331801 - OEIS

%I #34 Dec 04 2024 14:00:33

%S 8,12,13,16,17,18,20,21,22,24,25,26,27,28,29,30,31,32,33,34,35,36,37,

%T 38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,

%U 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

%N Integers that are sum of two nonsquarefree numbers.

%C Proposition: All integers > 23 are terms of this sequence (see link Prime Curios!).

%C Proof by exhaustion:

%C 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;

%C 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;

%C 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;

%C 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.

%C Conclusion: every integer > 23 is sum of two nonsquarefree numbers (QED).

%H G. L. Honaker, Jr. and Chris K. Caldwell, <a href="https://primes.utm.edu/curios/page.php?curio_id=5071">Prime Curios! 23 (Rupinski)</a>

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

%t max = 85; Union @ Select[Total /@ Tuples[Select[Range[max], !SquareFreeQ[#] &], 2], # <= max &] (* _Amiram Eldar_, Feb 04 2020 *)

%t 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 *)

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

%Y Cf. A005117 (squarefree), A013929 (nonsquarefree), A331802 (complement).

%Y Cf. A000404 (sum of 2 nonzero squares), A018825 (not the sum of 2 nonzero squares).

%Y Cf. A001694 (squareful), A052485 (not squareful), A076871 (sum of 2 squareful), A085253 (not the sum of 2 squareful).

%K nonn

%O 1,1

%A _Bernard Schott_, Jan 26 2020