A268332 -id:A268332 - 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

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

A268331 Squarefree numbers differing by more than 2 from any other squarefree number.

Original entry on oeis.org

26, 170, 362, 530, 638, 727, 874, 926, 962, 1027, 1126, 1423, 1574, 1774, 1814, 1826, 1861, 2059, 2402, 2526, 2674, 2726, 2782, 2874, 3178, 3482, 3574, 3719, 3774, 3970, 4166, 4474, 4490, 4526, 4654, 5042, 5045, 5374, 5426, 5914, 5930, 6026, 6173, 6254, 6274, 6326, 6418, 6626, 6649, 6726, 7138, 7174

Offset: 1

Christopher E. Thompson, Feb 01 2016

nonn

			26 is a term because 26 is squarefree but 24,25,27,28 are not.

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

Cf. A073247, A268332, A268333, A268334.

SequencePosition[Table[If[SquareFreeQ[n],1,0],{n,7200}],{0,0,1,0,0}][[All,1]]+2 (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jun 19 2018 *)

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.

A369238 Tetraprime numbers differing by more than 3 from any other squarefree number.

Original entry on oeis.org

72474, 106674, 193026, 237522, 261478, 308649, 342066, 370785, 391674, 491322, 604878, 865974, 885477, 931022, 938598, 1005630, 1070727, 1152822, 1186926, 1206822, 1289978, 1306878, 1363326, 1371774, 1392726, 1412918, 1455249, 1528111, 1634227, 1654678, 1688478

Offset: 1

Massimo Kofler, Jan 19 2024

nonn

Tetraprimes are the product of four distinct prime numbers (cf. A046386).

			72474 = 2 * 3 * 47 * 257 is a tetraprime; 72471 = 3 * 7^2 * 17 * 29, 72472 = 2^3 * 9059, 72473 = 23^2 * 137, 72475 = 5^2 * 13 * 223, 72476 = 2^2 * 18119, 72477 = 3^2 * 8053 are all nonsquarefree numbers, so 72474 is a term.

Robert Israel, Table of n, a(n) for n = 1..5415

Cf. A046386, A013929. Subsequence of A268332.

N:= 3*10^6: # for terms <= N
P:= select(isprime,[2,seq(i,i=3 .. N/30,2)]): nP:= nops(P):
filter:= proc(x) not ormap(numtheory:-issqrfree, [x-3,x-2,x-1,x+1,x+2,x+3]) end proc:
R:= NULL:
for i1 from 1 to nP do
  r1:= P[i1];
  for i2 from 1 to i1-1 do
    r2:= r1 * P[i2]; if r2 > N/6 then break fi;
    for i3 from 1 to i2-1 do
      r3:= r2 * P[i3]; if r3 > N/2 then break fi;
      for i4 from 1 to i3-1 do
        r:= r3 * P[i4];
        if r > N then break fi;
        if filter(r) then R:= R,r; fi
od od od od:
sort([R]); # Robert Israel, Jan 19 2025

f[n_] := Module[{e = FactorInteger[n][[;; , 2]], p}, p = Times @@ e; If[p > 1, 0, If[e == {1, 1, 1, 1}, 1, -1]]]; SequencePosition[Array[f, 2*10^6], {0, 0, 0, 1, 0, 0, 0}][[;; , 1]] + 3 (* Amiram Eldar, Jan 19 2024 *)

A369521 Sphenic numbers differing by more than 3 from any other squarefree number.

Original entry on oeis.org

2526, 44405, 209674, 220209, 234622, 328877, 375823, 409737, 428947, 473673, 540026, 569427, 611174, 736077, 748673, 758423, 781747, 800022, 863722, 889251, 914878, 927622, 973927, 982398, 988478, 994061, 1003474, 1021602, 1072477, 1088877, 1150077, 1157822, 1158451, 1211822

Offset: 1

Massimo Kofler, Jan 25 2024

nonn

Sphenic numbers are the product of three distinct primes (cf. A007304).

			2526 = 2 * 3 * 421 is a sphenic number; 2523 = 3 * 29^2, 2524 = 2^2 * 631, 2525 = 5^2 * 101, 2527 = 7 * 19^2, 2528 = 2^5 * 79, 2529 = 3^2 * 281 are all nonsquarefree numbers, so 2526 is a term.

Robert Israel, Table of n, a(n) for n = 1..3273 (all terms < 10^8)

Cf. A007304, A013929. Subsequence of A268332.

N:= 2*10^6: # to get all terms <= N
P:= select(isprime, [2,seq(i,i=3..N/6,2)]):
nP:= nops(P): R:= NULL:
for i from 1 do
 p:= P[i]; if p^3 >= N then break fi;
 for j from i+1 do
   q:= P[j]: if p*q^2 >= N then break fi;
   for k from j+1 to nP do
     x:= p*q*P[k];
     if x > N then break fi;
     if not ormap(numtheory:-issqrfree, [x-3,x-2,x-1,x+1,x+2,x+3]) then R:= R,x fi
od od od:
sort([R]); # Robert Israel, Feb 25 2024

f[n_] := Module[{e = FactorInteger[n][[;; , 2]], p}, p = Times @@ e; If[p > 1, 0, If[e == {1, 1, 1}, 1, -1]]]; SequencePosition[Array[f, 2*10^6], {0, 0, 0, 1, 0, 0, 0}][[;; , 1]] + 3 (* Amiram Eldar, Jan 25 2024 *)

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

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

Views

Author

Keywords

Links

Crossrefs

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

MATLAB

Mathematica

Extensions

A268331 Squarefree numbers differing by more than 2 from any other squarefree number.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Links

Crossrefs

A369238 Tetraprime numbers differing by more than 3 from any other squarefree number.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

A369521 Sphenic numbers differing by more than 3 from any other squarefree number.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica