A162870 -id:A162870 - cp's OEIS frontend

A086708 Primes p such that p-1 and p+1 are both divisible by cubes (other than 1).

271, 487, 593, 751, 809, 919, 1249, 1567, 1783, 1889, 1999, 2647, 2663, 2753, 2969, 3079, 3511, 3617, 3727, 3833, 3943, 4049, 4159, 4481, 4591, 4751, 4801, 5023, 6857, 6967, 7937, 8263, 8369, 9127, 9343, 10289, 10313, 10529, 10639, 11071, 11177

Offset: 1

Jason Earls and Amarnath Murthy, Jul 28 2003

nonn

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

Cf. A162870 (subsequence).

isA086708 := proc(n)
    if isprime(n) then
        isA046099(n-1) and isA046099(n+1) ;
    else
        false;
    end if;
end proc:
n := 1:
for c from 1 to 50000 do
    if isA086708(c) then
        printf("%d %d\n",n,c) ;
        n := n+1 ;
    end if;
end do: # R. J. Mathar, Dec 08 2015
Res:= NULL: count:= 0:
p:= 1:
while count < 100 do
  p:= nextprime(p);
  if max(seq(t[2],t=ifactors(p-1)[2]))>=3 and max(seq(t[2],t=ifactors(p+1)[2]))>=3 then
    count:= count+1; Res:= Res, p;
  fi
od:
Res; # Robert Israel, Jul 11 2018

f[n_]:=Max[Last/@FactorInteger[n]]; lst={};Do[p=Prime[n];If[f[p-1]>=3&&f[p+1]>=3,AppendTo[lst,p]],{n,6!}];lst (* Vladimir Joseph Stephan Orlovsky, Oct 03 2009 *)
dbcQ[p_]:=AnyTrue[Surd[#,3]&/@Rest[Divisors[p-1]],IntegerQ]&&AnyTrue[Surd[#,3]&/@Rest[ Divisors[ p+1]],IntegerQ]; Select[ Prime[Range[1500]],dbcQ] (* Harvey P. Dale, Sep 21 2024 *)

\\ Input no. of iterations n, power p and number to subtract and add k.
powerfreep4(n,p,k) = { c=0; pc=0; forprime(x=2,n, pc++; if(!ispowerfree(x-k,p) && !ispowerfree(x+k,p), c++; print1(x","); ) ); print(); print(c","pc","c/pc+.0) }
ispowerfree(m,p1) = { flag=1; y=component(factor(m),2); for(i=1,length(y), if(y[i] >= p1,flag=0;break); ); return(flag) } \\ Cino Hilliard, Dec 08 2003

{p in A000040: p+1 in A046099 and p-1 in A046099}. - R. J. Mathar, Dec 08 2015

A089199 INTERSECT A089200. - R. J. Mathar, Dec 08 2015

Definition clarified by Harvey P. Dale, Sep 21 2024

A162872 Primes p such that p-1 and p+1 each contain at least one squared prime in their prime factorization.

Original entry on oeis.org

19, 149, 197, 199, 293, 307, 349, 491, 523, 557, 577, 739, 773, 883, 1013, 1051, 1061, 1151, 1171, 1277, 1451, 1493, 1531, 1549, 1601, 1637, 1667, 1693, 1709, 1733, 1747, 1861, 1949, 2069, 2141, 2179, 2251, 2351, 2357, 2467, 2549, 2683, 2789, 2843, 2851

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jul 15 2009

nonn

The selection criterion is that p-1 and p+1 are in the subsequence 4=2^2, 9=3^2, 12=2^2*3, 18=2*3^2, ... of nonsquarefree numbers (A013929) that actually display at least one square in their standard prime factorization.

So at least one of the e_i in p-1=product p_i^e_i, and at least one of the e_j in p+1=product p_j^e_j must equal 2. This is more stringent than being nonsquarefree, and the sequence becomes a subsequence of A075432.

			19 is in the sequence because 19 - 1 = 2*3^2 contains 3^2 and because 19 + 1 = 2^2*5 contains 2^2 in the factorization.

R. J. Mathar and Robert Israel, Table of n, a(n) for n = 1..10000 (n = 1..536 from R. J. Mathar)

Cf. A038109, A162870.

isA162872 := proc(n)
    if isprime(n) then
        isA038109(n-1) and isA038109(n+1) ;
    else
        false;
    end if;
end proc:
n := 1:
for c from 1 to 50000 do
    if isA162872(c) then
        printf("%d %d\n",n,c) ;
        n := n+1 ;
end if; # R. J. Mathar, Dec 08 2015
N:= 10^5: # to get all terms < N, where N is even
V:= Vector(N/2):
for i from 1 do
  p:= ithprime(i);
  if p^2 > N+1 then break fi;
  if p = 2 then inds:= 2*[seq(i, i=1..floor(N/4), 2)]
  else inds:= p^2*select(t -> t mod p <> 0, [$1..floor(N/2/p^2)])
  fi;
  V[inds]:= 1;
od:
select(t -> V[(t-1)/2] = 1 and V[(t+1)/2] = 1 and isprime(t), [seq(t, t=3..N, 2)]); # Robert Israel, Dec 08 2015

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];If[f[p-1]==1&&f[p+1]==1,AppendTo[lst,p]], {n,7!}];lst
ospQ[n_]:=AnyTrue[FactorInteger[n+1][[;;,2]],#==2&]&&AnyTrue[FactorInteger[n-1][[;;,2]],#==2&]; Select[Prime[Range[500]],ospQ] (* Harvey P. Dale, May 11 2025 *)

{p in A000040: p+1 in A038109 and p-1 in A038109}. - R. J. Mathar, Dec 08 2015

Role of squarefree numbers clarified by R. J. Mathar, Jul 31 2007

A162873 List of pairs (p,r) of twin primes p and r=p+2 such that p-1, p+1 and r+1 are all divisible by squares > 1.

Original entry on oeis.org

17, 19, 149, 151, 197, 199, 269, 271, 521, 523, 809, 811, 881, 883, 1049, 1051, 1061, 1063, 1277, 1279, 1949, 1951, 1997, 1999, 2141, 2143, 2549, 2551, 2789, 2791, 2969, 2971, 3257, 3259, 3329, 3331, 3581, 3583, 3821, 3823, 4049, 4051, 4157, 4159, 4229, 4231

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jul 15 2009

			(197,199) is a pair because all of 196, 198, and 200 are divisible by squares.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A089189, A089194, A162870, A162872

dsQ[n_] := Length[Select[FactorInteger[n][[All, 2]], # > 1 &]] > 0; Select[ Partition[ Prime[ Range[ 250]],2,1],#[[2]]-#[[1]]==2&&AllTrue[ #[[1]]+ {-1,1,3},dsQ]&]//Flatten (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Nov 28 2018 *)

Corrected and previous (incorrect) Mathematica program replaced by Harvey P. Dale, Nov 28 2018

A162874 Twin primes p and r (p < r) such that p-1, p+1 and r+1 are not cubefree.

Original entry on oeis.org

69497, 69499, 416501, 416503, 474497, 474499, 632501, 632503, 960497, 960499, 1068497, 1068499, 1226501, 1226503, 1402871, 1402873, 1464101, 1464103, 1635497, 1635499, 1716497, 1716499, 1919429, 1919431, 1986497, 1986499

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jul 15 2009

nonn

A variant of A162989, which is the main entry. - N. J. A. Sloane, Aug 12 2009

Note that p+1 = r-1. Thus, the sequence describes twin primes whose immediate neighbors are not cubefree. - Tanya Khovanova, Aug 22 2021

			69497 and 69499 twin primes. Moreover, 69496 is divisible by 2^3, 69498 is divisible by 3^3, and 69500 is divisible by 5^3. Thus, 69497 and 69499 are in the sequence. - _Tanya Khovanova_, Aug 22 2021

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

s=Select[Prime@Range[200000],PrimeQ[#+2]&&Min[Max[Last/@FactorInteger[#]]&/@{#-1,#+1,#+3}]>2&];Sort@Join[s,s+2] (* Giorgos Kalogeropoulos, Aug 22 2021 *)

from sympy import nextprime, factorint
def cubefree(n): return max(e for e in factorint(n).values()) <= 2
def auptop(limit):
    alst, p, r = [], 3, 5
    while p < limit:
        if r - p == 2 and not any(cubefree(i) for i in [p-1, p+1, r+1]):
            alst.extend([p, r])
        p, r = r, nextprime(p)
    return alst
print(auptop(2*10**6)) # Michael S. Branicky, Aug 22 2021

Terms corrected by Zak Seidov, Jul 19 2009

Edited by N. J. A. Sloane, Aug 12 2009

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

Original entry on oeis.org

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

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

A086708 Primes p such that p-1 and p+1 are both divisible by cubes (other than 1).

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A162872 Primes p such that p-1 and p+1 each contain at least one squared prime in their prime factorization.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A162873 List of pairs (p,r) of twin primes p and r=p+2 such that p-1, p+1 and r+1 are all divisible by squares > 1.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

A162874 Twin primes p and r (p < r) such that p-1, p+1 and r+1 are not cubefree.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Python

Extensions

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

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