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A077640 Smallest term of a run of at least 7 consecutive integers which are not squarefree. 9
217070, 671346, 826824, 1092747, 1092748, 1427370, 2097048, 2779370, 3112819, 3306444, 3597723, 3994820, 4063774, 4442874, 4630544, 4842474, 5436375, 5479619, 5610644, 5634122, 6315019, 6474220, 6626319, 6677864, 7128471 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..488 from Robert Israel)
FORMULA
A077640 = { A078144[k] | A078144[k+2] = A078144[k]+2 } = { A070284[k] | A070284[k+3] = A070284[k]+3 } etc. Note that A049535 is defined differently. - M. F. Hasler, Feb 01 2016
a(n) = A188347(n) - 3. - Amiram Eldar, Feb 09 2021
EXAMPLE
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}.
MATHEMATICA
s7[x_] := Apply[Plus, Table[Abs[MoebiusMu[x+j]], {j, 0, 6}]] If[Equal[s, 0], Print[n]], {n, 217000, 100000000}]
Flatten[Position[Partition[SquareFreeQ/@Range[7000000], 7, 1], _?(Union[#] == {False}&), {1}, Heads->False]] (* Harvey P. Dale, May 24 2014 *)
SequencePosition[Table[If[SquareFreeQ[n], 0, 1], {n, 72*10^5}], {1, 1, 1, 1, 1, 1, 1}][[All, 1]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Aug 15 2017 *)
PROG
(MATLAB)
N = 10^7; % to get all terms <= N-6
T = zeros(1, N);
for m = 2:floor(sqrt(N))
T([m^2 : m^2 : N]) = 1;
end
S = T(1:N-6).*T(2:N-5).*T(3:N-4).*T(4:N-3).*T(5:N-2).*T(6:N-1).*T(7:N);
find(S) % Robert Israel, Feb 03 2016
(PARI) {my(N=10^6, M=0, t, m2); for(m=2, sqrtint(N), t=1; m2=m^2; M=bitor(sum(i=1, N\m^2, t<<=m2), M)); for(i=1, 6, M=bitand(M, M>>1)); for(i=0, N, M||break; print1(i+=t=valuation(M, 2), ", "); M>>=t+1)} \\ Works but is much slower than the following (16s for 10^6 vs. 3s for 10^7). Should scale better (~sqrt(n) vs linear) but doesn't because of inefficient implementation of binary operations (copies & re-allocation of very large bitmaps): increasing N from 10^5 to 10^6 multiplies CPU time by a factor of 100!
(PARI) for(n=1, 10^7, forstep(k=6, 0, -1, issquarefree(n+k)&&(n+=k)&&next(2)); print1(n", ")) \\ M. F. Hasler, Feb 03 2016
CROSSREFS
Cf. 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).
Sequence in context: A351662 A251804 A184668 * A188347 A078520 A019291
KEYWORD
nonn
AUTHOR
Labos Elemer, Nov 14 2002
STATUS
approved

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Last modified March 3 14:16 EST 2024. Contains 370512 sequences. (Running on oeis4.)