A073250 -id:A073250 - cp's OEIS frontend

A073248 Squarefree numbers k such that k+1 is also squarefree, but k-1 and k+2 are not.

10, 46, 61, 73, 82, 118, 122, 133, 145, 154, 173, 190, 205, 226, 246, 262, 273, 277, 290, 298, 313, 326, 334, 370, 373, 385, 406, 421, 426, 442, 457, 473, 478, 493, 505, 514, 526, 537, 565, 573, 586, 601, 606, 622, 626, 658, 673, 694, 709, 730, 733, 745

Offset: 1

Reinhard Zumkeller, Jul 22 2002

nonn

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A005117, A073250, A073247, A007675.

state:= [true,true,true,true]:
R:= NULL: count:= 0:
for n from 1 while count < 100 do
  state:= [state[2],state[3],state[4],numtheory:-issqrfree(n)];
  if state = [false,true,true,false] then
     R:= R, n-2; count:= count+1
  fi
od:
R; # Robert Israel, Mar 02 2022

Transpose[SequencePosition[Table[If[SquareFreeQ[n],1,0],{n,800}],{0,1,1,0}]][[1]]+1 (* The program uses the SequencePosition function from Mathematica version 10 *) (* Harvey P. Dale, Mar 09 2016 *)

A073249 Nonprime squarefree numbers n such that both n-1 and n+1 are not squarefree and not prime.

Original entry on oeis.org

26, 51, 55, 91, 161, 170, 235, 249, 295, 305, 339, 341, 362, 377, 413, 415, 451, 485, 489, 530, 551, 559, 579, 595, 629, 638, 649, 651, 665, 667, 685, 687, 703, 721, 723, 737, 749, 849, 851, 874, 917, 926, 949, 951, 955, 962, 989, 1015, 1027, 1057, 1059

Offset: 1

Reinhard Zumkeller, Jul 22 2002

nonn

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

Cf. A000469, A005117, A073247, A073250, A073251.

npsQ[n_]:=SquareFreeQ[n]&&NoneTrue[n+{1,0,-1},PrimeQ]&&NoneTrue[n+{1,-1},SquareFreeQ]; Select[Range[2000],npsQ] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jan 20 2019 *)
Mean/@SequencePosition[Table[Which[CompositeQ[n]&&SquareFreeQ[n],1,!SquareFreeQ[ n] && CompositeQ[ n],-1,True,0],{n,1100}],{-1,1,-1}] (* Harvey P. Dale, Jun 17 2022 *)

Corrected by Harvey P. Dale, Jan 20 2019

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

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A073248 Squarefree numbers k such that k+1 is also squarefree, but k-1 and k+2 are not.

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A073249 Nonprime squarefree numbers n such that both n-1 and n+1 are not squarefree and not prime.

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Mathematica

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

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