A073249 -id:A073249 - cp's OEIS frontend

A073247 Squarefree numbers k such that k-1 and k+1 are not squarefree.

17, 19, 26, 51, 53, 55, 89, 91, 97, 127, 149, 151, 161, 163, 170, 197, 199, 233, 235, 241, 249, 251, 269, 271, 293, 295, 305, 307, 337, 339, 341, 349, 362, 377, 379, 413, 415, 449, 451, 485, 487, 489, 491, 521, 523, 530, 551, 557, 559, 577, 579, 593, 595

Offset: 1

Reinhard Zumkeller, Jul 22 2002

nonn

Probably 11*n < a(n) < 12*n for n > 189. - Charles R Greathouse IV, Nov 05 2017

The asymptotic density of this sequence is 1/zeta(2) - 2 * Product_{p prime} (1 - 2/p^2) + Product_{p prime} (1 - 3/p^2) = A059956 - 2*A065474 + A206256 = 0.088145884881346585838... . - Amiram Eldar, Aug 30 2024

Christopher E. Thompson, Table of n, a(n) for n = 1..10000

Cf. A005117, A073249, A073248, A007675.
Cf. A268331, A268332, A268333, A268334 (squarefree numbers isolated by more than 2, 3, etc.).
Cf. A059956, A065474, A206256.

sf:= select(numtheory:-issqrfree,[$1..1000]):
map(t -> `if`(sf[t-1]=sf[t]-1 or sf[t+1]=sf[t]+1,NULL,sf[t]), [$2..nops(sf)-1]); # Robert Israel, Feb 01 2016

Reap[For[n = 0, n <= 1000, n++, If[SquareFreeQ[n] && !SquareFreeQ[n-1] && !SquareFreeQ[n+1], Sow[n]]]][[2, 1]] (* Jean-François Alcover, Feb 26 2019 *)

is(n)=!issquarefree(n-1) && issquarefree(n) && !issquarefree(n+1) \\ Charles R Greathouse IV, Nov 05 2017

list(lim)=my(v=List(),l1,l2); forfactored(k=9,lim\1+1, if(!issquarefree(k) && !issquarefree(l2) && issquarefree(l1), listput(v,l1[1])); l2=l1; l1=k); Vec(v) \\ Charles R Greathouse IV, Nov 27 2024

A073250 Nonprime squarefree numbers n such that n+1 is also squarefree and nonprime, but not n-1 and n+2.

Original entry on oeis.org

14, 21, 38, 57, 65, 69, 77, 105, 110, 114, 118, 122, 129, 133, 145, 154, 158, 165, 177, 182, 194, 205, 209, 221, 230, 237, 246, 258, 273, 290, 298, 309, 318, 326, 329, 334, 345, 354, 357, 365, 370, 381, 385, 390, 398, 402, 406, 410, 417, 426, 429, 434, 437

Offset: 1

Reinhard Zumkeller, Jul 22 2002

nonn

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A000469, A005117, A073248, A073251, A073249.

tQ[n_]:=!PrimeQ[n+1]&&SquareFreeQ[n+1]&&(PrimeQ[n-1]||!SquareFreeQ[n-1])&&(PrimeQ[n+2]||!SquareFreeQ[n+2])
Select[Select[Complement[Range[500],Prime[Range[PrimePi[500]]]],SquareFreeQ],tQ]  (* Harvey P. Dale, Feb 14 2011 *)
SequencePosition[Table[If[SquareFreeQ[n]&&!PrimeQ[n],1,0],{n,500}],{0,1,1,0}][[;;,1]]+1 (* Harvey P. Dale, Feb 27 2023 *)

A073251 Numbers k such that k, k+1 and k+2 are nonprime and squarefree.

Original entry on oeis.org

33, 85, 93, 141, 185, 201, 213, 217, 253, 265, 285, 301, 321, 393, 445, 453, 469, 481, 517, 533, 553, 581, 589, 609, 633, 669, 697, 705, 713, 753, 777, 789, 793, 805, 813, 869, 893, 897, 901, 913, 921, 933, 957, 985, 993, 1001, 1005, 1041, 1045, 1065, 1113

Offset: 1

Reinhard Zumkeller, Jul 22 2002

nonn

k-1 and k+3 are not squarefree. Proof: k is odd, otherwise k or k+2 would be divisible by 4. Thus k+1 is even and not divisible by 4, hence k-1 and k+3 are divisible by 4.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000469, A005117, A007675, A073249, A073250, A121495.

f[upto_]:=Module[{pp=PrimePi[upto],n},lst=Partition[Complement[Range[upto], Prime[Range[pp]]],3,1];Transpose[Select[lst,And@@SquareFreeQ/@#&]][[1]]]; f[1200] (* Harvey P. Dale, Mar 21 2011 *)

isok1(k) = !isprime(k) && issquarefree(k); \\ A000469
isok(k) = isok1(k) && isok1(k+1) && isok1(k+2); \\ Michel Marcus, Mar 25 2021

Edited by Klaus Brockhaus, Aug 07 2006

cp's OEIS Frontend

A073247 Squarefree numbers k such that k-1 and k+1 are not squarefree.

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A073250 Nonprime squarefree numbers n such that n+1 is also squarefree and nonprime, but not n-1 and n+2.

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A073251 Numbers k such that k, k+1 and k+2 are nonprime and squarefree.

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