A162874 -id:A162874 - cp's OEIS frontend

A162875 Twin primes p and r such that p - 1, p + 1 and r + 1 are cubefree.

3, 5, 11, 13, 59, 61, 179, 181, 227, 229, 347, 349, 419, 421, 659, 661, 827, 829, 1019, 1021, 1091, 1093, 1427, 1429, 1451, 1453, 1667, 1669, 1787, 1789, 1931, 1933, 2027, 2029, 2339, 2341, 3299, 3301, 3371, 3373, 3467, 3469, 3539, 3541, 3851, 3853, 4019

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jul 15 2009

nonn

Apart from the first two terms, a(2n+1) = 11 mod 24 and a(2n) = 13 (mod 24). - Charles R Greathouse IV, Oct 12 2009

			179 and 181 are in the sequence because they are twin primes and 178 = 2*89, 180 = 2^2*3^2*5, 182 = 2*7*13 have no factors that are cubes.

Amiram Eldar, Table of n, a(n) for n = 1..5000

Cf. A089189, A089194, A162870, A162872, A162873, A162874.

f[n_]:=Module[{a=m=0},Do[If[FactorInteger[n][[m,2]]>2,a=1],{m,Length[FactorInteger[n]]}];a]; lst={};Do[p=Prime[n];r=p+2;If[PrimeQ[r],If[f[p-1]==0&&f[p+1]==0&&f[r+1]==0,AppendTo[lst,p];AppendTo[lst,r]]],{n,2*6!}];lst

A162989 Lesser of twin primes p such that none of p-1, p+1 and p+3 are cubefree.

Original entry on oeis.org

69497, 416501, 474497, 632501, 960497, 1068497, 1226501, 1402871, 1464101, 1635497, 1716497, 1919429, 1986497, 2114249, 2144501, 2283497, 2645189, 3120497, 3174497, 3232751, 3305501, 3332501, 3525497, 3637169, 3998537

Offset: 1

Zak Seidov, Jul 19 2009

nonn

			p=69497 and p+2=69499 are twin primes, also:
p-1=69496=2^3*7*17*73
p+1=69498=2*3^5*11*13
p+3=69500=2^2*5^3*139.

Robert Israel, Table of n, a(n) for n = 1..1000

Cf. A046099.

See A162874 for another version.

cf:= proc(n) local F;
    F:= ifactors(n)[2];
    max(map(t->t[2],F))>=3
end proc:
select(t -> isprime(t) and isprime(t+2) and cf(t-1) and cf(t+1) and cf(t+3), [seq(i,i=5..10^7,6)]); # Robert Israel, Nov 24 2020

f[m_]:=Max[Last/@FactorInteger[m]]>=3;
S={};Do[If[PrimeQ[p=6x-1]&&PrimeQ[p+2]&&
f[p-1]==f[p+1]==f[p+3]==True,AppendTo[S,p]],{x,1,10^6}];S

Definition clarified by Robert Israel, Nov 24 2020

A162876 Twin prime pairs p, p+2 such that p-1 and p+3 are both squarefree.

Original entry on oeis.org

3, 5, 11, 13, 59, 61, 71, 73, 107, 109, 179, 181, 191, 193, 227, 229, 311, 313, 419, 421, 431, 433, 599, 601, 659, 661, 827, 829, 1019, 1021, 1031, 1033, 1091, 1093, 1319, 1321, 1427, 1429, 1487, 1489, 1607, 1609, 1619, 1621, 1787, 1789, 1871, 1873, 1931

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jul 15 2009

nonn

By definition, the lower member, here at the odd-indexed positions, is in A089188.

p+1 must be divisible by 4. - Robert Israel, Jul 24 2015

			(179,181) are in the sequence because 179-1=2*89 is squarefree and 181+1=2*7*13 is also squarefree.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A089189, A089194, A162870, A162872, A162873, A162874, A162875

f:= p -> if isprime(p) and isprime(p+2) and numtheory:-issqrfree(p-1) and numtheory:-issqrfree(p+3) then (p,p+2) else NULL fi:
map(f, [4*k-1 $ k=1..1000]); # Robert Israel, Jul 24 2015

f[n_]:=Module[{a=m=0},Do[If[FactorInteger[n][[m,2]]>1,a=1],{m,Length[FactorInteger[n]]}]; a]; lst={};Do[p=Prime[n];r=p+2;If[PrimeQ[r],If[f[p-1]==0&&f[r+1]==0, AppendTo[lst,p];AppendTo[lst,r]]],{n,7!}];lst

{(p,p+2) : p in A001359, and p-1 in A005117, and p+3 in A005117}.

Definition rephrased by R. J. Mathar, Jul 27 2009

cp's OEIS Frontend

A162875 Twin primes p and r such that p - 1, p + 1 and r + 1 are cubefree.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A162989 Lesser of twin primes p such that none of p-1, p+1 and p+3 are cubefree.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Extensions

A162876 Twin prime pairs p, p+2 such that p-1 and p+3 are both squarefree.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions