A045882
Smallest term of first run of (at least) n consecutive integers which are not squarefree.
Original entry on oeis.org
4, 8, 48, 242, 844, 22020, 217070, 1092747, 8870024, 221167422, 221167422, 47255689915, 82462576220, 1043460553364, 79180770078548, 3215226335143218, 23742453640900972, 125781000834058568
Offset: 1
a(3) = 48 as 48, 49 and 50 are divisible by squares.
n=5 -> {844=2^2*211; 845=5*13^2; 846=2*3^2*47; 847=7*11^2; 848=2^4*53}.
- J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 242, p. 67, Ellipses, Paris 2008.
-
cnt = 0; k = 0; Table[While[cnt < n, k++; If[! SquareFreeQ[k], cnt++, cnt = 0]]; k - n + 1, {n, 7}]
-
a(n)=my(s);for(k=1,9^99,if(issquarefree(k),s=0,if(s++==n,return(k-n+1)))) \\ Charles R Greathouse IV, May 29 2013
a(12)-a(15) from Louis Marmet (louis(AT)marmet.org) and David Bernier (ezcos(AT)yahoo.com), Nov 15 1999
a(16) was obtained as a result of a team effort by Z. McGregor-Dorsey et al. [Louis Marmet (louis(AT)marmet.org), Jul 27 2000]
a(17) was obtained as a result of a team effort by E. Wong et al. [Louis Marmet (louis(AT)marmet.org), Jul 13 2001]
a(18) was obtained as a result of a team effort by L. Marmet et al.
A049535
Starts of runs of exactly 6 consecutive nonsquarefree numbers.
Original entry on oeis.org
22020, 24647, 30923, 47672, 55447, 57120, 73447, 74848, 96675, 105772, 121667, 121847, 152339, 171348, 179972, 182347, 185247, 190447, 200848, 204323, 215303, 229172, 233223, 234375, 240424, 268223, 274547, 310120, 327424, 338920
Offset: 1
Squares dividing the numbers in the starting at 22020 are 4, 361, 121, 9, 4, 25, respectively.
The smallest members of such strings of length k are listed in
A045882.
Cf.
A045882 (min terms),
A068781 (2-chains),
A070258 (3-chains),
A070284 (4-chains),
A078144 (5-chains),
A049535 (6-chains),
A077640 (7-chains),
A077647 (8-chains),
A078143 (9-chains),
A268313 (10-chains),
A268314 (11-chains).
-
Res:= NULL:
st:= 0:
for n from 1 to 500000 do
if numtheory:-issqrfree(n) then
if st = 6 then Res:= Res, n-6 fi;
st:= 0;
else
st:= st+1;
fi
od:
Res; # Robert Israel, Feb 08 2017
-
Select[Range[400000], !SquareFreeQ[#] && !SquareFreeQ[#+1] && !SquareFreeQ[#+2] && !SquareFreeQ[#+3] && !SquareFreeQ[#+4] && !SquareFreeQ[#+5] && SquareFreeQ[#+6]&] (* Vladimir Joseph Stephan Orlovsky, Mar 30 2011 *)
Flatten[Position[Partition[SquareFreeQ/@Range[60000],6,1],?(Union[#] == {False}&),{1},Heads->False]] (* _Harvey P. Dale, May 24 2014 *)
A077647
Smallest term of a run of at least 8 consecutive integers which are not squarefree.
Original entry on oeis.org
1092747, 7216618, 8870024, 8870025, 14379271, 22635347, 24816974, 25047846, 33678771, 33908368, 33908369, 34394371, 34682346, 37923938, 49250144, 49250145, 53379270, 69147868, 69147869, 70918820, 70918821, 71927247, 72913022, 83605071, 85972019, 90571646
Offset: 1
n=8870024: squares dividing n+j (j=0...8) i.e. 9 consecutive integers are as follows {4,25,121,841,4,49,961,9,16}
Cf.
A045882 (first k-chain),
A068781 (2-chains),
A070258 (3-chains),
A070284 (4-chains),
A078144 (5-chains),
A049535 (6-chains),
A077640 (7-chains),
A077647 (8-chains),
A078143 (9-chains),
A268313 (10-chains),
A268314 (11-chains).
-
s8[x_] := Apply[Plus, Table[Abs[MoebiusMu[x+j]], {j, 0, 7}]]; Do[If[s8[n] == 0, Print[n]], {n, 10^8}]
Flatten[Position[Partition[SquareFreeQ/@Range[91000000],8,1],_?(Union[#]=={False}&),{1},Heads->False]]
-
for(n=1,10^8,forstep(k=7,0,-1,issquarefree(n+k)&&(n+=k)&&next(2));print1(n",")) \\ M. F. Hasler, Feb 03 2016
A078143
Smallest term of a run of at least 9 consecutive integers which are not squarefree.
Original entry on oeis.org
8870024, 33908368, 49250144, 69147868, 70918820, 111500620, 112931372, 164786748, 167854344, 200997948, 203356712, 207543320, 211014920, 216785256, 221167422, 221167423, 221167424, 236645624, 240574368, 262315467, 262315468
Offset: 1
-
s9[x_] := Apply[Plus, Table[Abs[MoebiusMu[x+j]], {j, 0, 8}]]; Do[If[Equal[s9[n], 0], Print[n]], {n, 8000000, 1000000000}]
-
is(n)=for(i=n,n+8, if(!issquarefree(i), return(0))); 1 \\ Charles R Greathouse IV, Nov 05 2017
A268313
First term of a run of at least 10 consecutive integers which are not squarefree.
Original entry on oeis.org
221167422, 221167423, 262315467, 363504972, 463239475, 1202800371, 1407472722, 1407472723, 1557947844, 1609077723, 1911823144, 2217728772, 2695179044, 2737800168, 2847305571, 3639720042, 3639720043, 3672883247, 3865964268, 3865964269, 3982659575, 4246929267, 4818537743, 4982931368
Offset: 1
-
s10[x_] := Apply[Plus, Table[Abs[MoebiusMu[x+j]], {j, 0, 9}]]; Do[If[Equal[s10[n], 0], Print[n]], {n, 10^8, 10^10}]
A268314
First term of a run of at least 11 consecutive integers which are not squarefree.
Original entry on oeis.org
221167422, 1407472722, 3639720042, 3865964268, 4982931368, 5005996146, 7108776620, 8044261244, 10249558974, 12766690268, 13585489166, 19792784322, 26995377572, 30410811296, 30477326444, 32070270968, 34317891368, 39956560824, 40841363528, 42216508746, 43133805944, 46295514872, 47255689915
Offset: 1
-
s11[x_] := Apply[Plus, Table[Abs[MoebiusMu[x+j]], {j, 0, 10}]]; Do[If[Equal[s11[n], 0], Print[n]], {n, 10^8, 10^13}]
A188347
Central element of a series of 7 successive nonsquarefree numbers.
Original entry on oeis.org
217073, 671349, 826827, 1092750, 1092751, 1427373, 2097051, 2779373, 3112822, 3306447, 3597726, 3994823, 4063777, 4442877, 4630547, 4842477, 5436378, 5479622, 5610647, 5634125, 6315022, 6474223, 6626322, 6677867, 7128474, 7216621, 7216622, 7295451, 7507926
Offset: 1
217073=17*113^2, 217073-1=2^4*13567, 217073-2=3^2*89*271, 217073-3=2*5*7^2*443, 217073+1=2*3*11^2*13*23, 217073+2=5^2*19*457, 217073+3=2^2*54269.
A268270
Least prime that is at distance > n from the nearest squarefree number.
Original entry on oeis.org
2, 17, 727, 47527, 29002021, 494501773, 44633461423, 7489886099881
Offset: 0
a(0)=2 is the least prime and it is at distance 1 from the nearest squarefree numbers (1 and/or 3).
a(1)=17 is the least prime that has no squarefree neighbor: both 16 and 18 are divisible by a square; the nearest squarefree numbers, 15 and 19, are both at distance 2.
a(2)=727 is the least prime p such that p-2 and p+1 are (two consecutive terms) in A068781, namely A068781(75..76).
a(3)=47527 is the least prime p such that p-3 and p+1 are (two consecutive terms) in A070258, namely A070258(878..879).
a(4)=29002021 is the least prime p such that p-4 and p+1 are (two consecutive terms) in A070284.
a(5)=494501773 is the least prime p such that p-5 and p+1 are (two consecutive terms) in A078144.
Similarly, for n = 6, 7, 8 and 9, a(n) is the least prime p such that p-n and p+1 are (two consecutive terms) in A049535, A077640, A077647 and A078143, respectively.
-
a(n)=forprime(p=n,,for(s=1,n,(issquarefree(p-s)||issquarefree(p+s)) && next(2)); return(p))
Showing 1-8 of 8 results.
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