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A378458 Squarefree numbers k such that k + 1 is squarefree but k + 2 is not.

%I A378458 #10 Dec 06 2024 11:29:04
%S A378458 2,6,10,14,22,30,34,38,42,46,58,61,66,70,73,78,82,86,94,102,106,110,
%T A378458 114,118,122,130,133,138,142,145,154,158,166,173,178,182,186,190,194,
%U A378458 202,205,210,214,218,222,226,230,238,246,254,258,262,266,273,277,282
%N A378458 Squarefree numbers k such that k + 1 is squarefree but k + 2 is not.
%C A378458 These are the positions of 2 in A378369 (difference between n and the next nonsquarefree number).
%C A378458 The asymptotic density of this sequence is Product_{p prime} (1 - 2/p^2) - Product_{p prime} (1 - 3/p^2) = A065474 - A206256 = 0.19714711803343537224... . - _Amiram Eldar_, Dec 03 2024
%H A378458 Amiram Eldar, <a href="/A378458/b378458.txt">Table of n, a(n) for n = 1..10000</a>
%t A378458 Select[Range[100],NestWhile[#+1&,#,SquareFreeQ[#]&]==#+2&]
%o A378458 (PARI) list(lim) = my(q1 = 1, q2 = 1, q3); for(k = 3, lim, q3 = issquarefree(k); if(q1 && q2 &&!q3, print1(k-2, ", ")); q1 = q2; q2 = q3); \\ _Amiram Eldar_, Dec 03 2024
%Y A378458 Complement of A007675 within A007674.
%Y A378458 The version for prime power instead of nonsquarefree is a subset of A006549.
%Y A378458 Another variation is A073247.
%Y A378458 The version for nonprime instead of squarefree is A179384.
%Y A378458 Positions of 0 in A378369 are A013929.
%Y A378458 Positions of 1 in A378369 are A373415.
%Y A378458 Positions of 2 in A378369 are A378458 (this).
%Y A378458 Positions of 3 in A378369 are A007675.
%Y A378458 A000961 lists the powers of primes, differences A057820.
%Y A378458 A120327 gives the least nonsquarefree number >= n.
%Y A378458 A378373 counts composite numbers between nonsquarefree numbers.
%Y A378458 Cf. A065474, A067535, A081221, A206256, A236575, A377046, A378033, A378039, A378086.
%K A378458 nonn
%O A378458 1,1
%A A378458 _Gus Wiseman_, Dec 02 2024