A268331 -id:A268331 - 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

A268332 Squarefree numbers differing by more than 3 from any other squarefree number.

Original entry on oeis.org

2526, 44405, 47527, 47973, 55779, 72474, 101037, 106327, 106674, 109023, 110107, 133577, 153173, 165574, 183553, 186247, 193026, 196747, 208847, 209674, 212127, 218527, 220209, 234622, 237522, 245149, 261478, 266853, 269953, 308649, 328877, 334522, 342066, 364151, 370785, 375823

Offset: 1

Christopher E. Thompson, Feb 01 2016

nonn

Christopher E. Thompson, Table of n, a(n) for n = 1..1000

Cf. A005117, A073247, A268331, A268333, A268334.

SF:= select(numtheory:-issqrfree, [$1..10^6]):
SF[select(i -> SF[i]-SF[i-1]>=4  and SF[i+1]-SF[i]>=4, [$2..nops(SF)-1])]; # Robert Israel, Feb 02 2016

A268334 Squarefree numbers differing by more than 5 from any other squarefree number.

Original entry on oeis.org

8061827, 60529549, 82490423, 213819827, 245990821, 360350923, 364661627, 465966527, 494501773, 531794155, 651012673, 661327077, 665519027, 693770521, 765921647, 800254429, 826112857, 836818573, 885970253, 914588627, 930996623, 936321349, 958710973, 992890427, 1069677149

Offset: 1

Christopher E. Thompson, Feb 01 2016

nonn

Christopher E. Thompson, Table of n, a(n) for n = 1..100

Cf. A073247, A268331, A268332, A268333.

A268330 Least squarefree number differing by more than n from any other squarefree number.

Original entry on oeis.org

1, 17, 26, 2526, 5876126, 8061827, 8996188226, 2074150570370

Offset: 0

Christopher E. Thompson, Feb 01 2016

1.8*10^12 < a(7) <= 10735237201449 - Robert Israel, Mar 18 2016

a(8) > 5*10^12. - Giovanni Resta, Apr 11 2016

			a(2) = 26 because 26 is squarefree but 24,25,27,28 are not.

Cf. A073247, A268331, A268332, A268333, A268334.

B = 10^8; % blocks of size B
nB = 1000; % nB blocks
A = [1];
P = primes(floor(sqrt(nB*B)));
mmax = 1;
i0 = 1;
for k = 0:nB-1  % search squarefrees from i0+1 to i0 + B
  V = true(1, B);
  for i = 1:numel(P)
    p = P(i);
    V([(p^2 - mod(i0,p^2)):p^2:B]) = false;
  end
  SF = find(V) + i0;
  DSF = SF(2:end) - SF(1:end-1);
  i0 = SF(end-2);
  M = min(DSF(1:end-1), DSF(2:end));
  newmax = max(mmax,max(M));
  for i = mmax+1:newmax
    A(i) = SF(1 + find(M>=i, 1, 'first'));
  end
  mmax = newmax;
end
for i=1:mmax
  fprintf('%d ',A(i));
end
fprintf('\n');  % Robert Israel, Mar 16 2016

(* implementation assumes a(n) is increasing *)
nsfRun[n_]:=Module[{i=n}, While[!SquareFreeQ[i], i++]; i-n]
a268330[{low_, high_}, width_]:=Module[{k=width, i, next, r, s, list={}}, For[i=low, i<=high, i+=next, r=nsfRun[i]; If[r0 (* Hartmut F. W. Hoft, Mar 15 2016 *)
a268330[{0,10000000},1] (* computes a(1)...a(5) *)

a(6) from Hartmut F. W. Hoft, Mar 15 2016

a(7) from Giovanni Resta, Apr 11 2016

A268333 Squarefree numbers differing by more than 4 from any other squarefree number.

Original entry on oeis.org

5876126, 8061827, 19375679, 27926071, 29002021, 29850943, 39224453, 39728861, 54427974, 56389147, 60529549, 63520174, 67806346, 71987374, 75239979, 82490423, 87011827, 91332377, 97447622, 99368949, 100842249, 107447838, 110196277, 118389143, 134817726, 149840974, 159140777, 161651279

Offset: 1

Christopher E. Thompson, Feb 01 2016

nonn

Christopher E. Thompson, Table of n, a(n) for n = 1..200

Cf. A073247, A268331, A268332, A268334.

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

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A268332 Squarefree numbers differing by more than 3 from any other squarefree number.

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A268334 Squarefree numbers differing by more than 5 from any other squarefree number.

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A268330 Least squarefree number differing by more than n from any other squarefree number.

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A268333 Squarefree numbers differing by more than 4 from any other squarefree number.

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