A162873 -id:A162873 - cp's OEIS frontend

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

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

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

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

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

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