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A295243
Sums of two numbers that are both consecutive and squarefree.
1
3, 5, 11, 13, 21, 27, 29, 43, 45, 59, 61, 67, 69, 75, 77, 83, 85, 93, 115, 117, 123, 131, 133, 139, 141, 147, 155, 157, 165, 171, 173, 187, 189, 203, 205, 211, 213, 219, 221, 227, 229, 237, 245, 259, 261, 267, 275, 277, 283, 285, 291, 309, 315, 317, 331
OFFSET
1,1
COMMENTS
From Robert Israel, Nov 20 2017: (Start)
All terms == 3 or 5 (mod 8).
Odd numbers k such that (k-1)/2 and (k+1)/2 are both squarefree.
Intersection of 2*A005117+1 and 2*A005117-1. (End)
The asymptotic density of this sequence is (1/2) * Product_{p prime} (1 - 2/p^2) = A065474 / 2 = 0.161317049469... . - Amiram Eldar, Feb 26 2024
LINKS
FORMULA
a(n) = 2*A007674(n) + 1. - Amiram Eldar, Feb 26 2024
EXAMPLE
a(3) = 11; 11 = 5 + 6, with 5, 6 consecutive and squarefree.
MAPLE
with(numtheory): a:=n->`if`(mobius(n)^2 = 1 and mobius(n+1)^2=1, 2*n+1, NULL): seq(a(n), n=1..300);
MATHEMATICA
Total /@ Select[Partition[Range@ 166, 2, 1], AllTrue[#, SquareFreeQ] &] (* Michael De Vlieger, Nov 18 2017 *)
PROG
(PARI) lista(nn) = for (n=0, nn, if (issquarefree(n) && issquarefree(n+1), print1(2*n+1, ", ")); ); \\ Michel Marcus, Nov 19 2017
CROSSREFS
KEYWORD
nonn
AUTHOR
Wesley Ivan Hurt, Nov 18 2017
STATUS
approved