A077640 Smallest term of a run of at least 7 consecutive integers which are not squarefree.

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

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. A013929, A045882, A049535, A051681, A188347.

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

Adjacent sequences: A077637 A077638 A077639 * A077641 A077642 A077643

Keyword

nonn

Author

Labos Elemer, Nov 14 2002

Status

approved