A089194 -id:A089194 - cp's OEIS frontend

A089189 Primes p such that p-1 is cubefree.

2, 3, 5, 7, 11, 13, 19, 23, 29, 31, 37, 43, 47, 53, 59, 61, 67, 71, 79, 83, 101, 103, 107, 127, 131, 139, 149, 151, 157, 167, 173, 179, 181, 191, 197, 199, 211, 223, 227, 229, 239, 263, 269, 277, 283, 293, 307, 311, 317, 331, 347, 349, 359, 367, 373, 383

Offset: 1

Cino Hilliard, Dec 08 2003 and Reinhard Zumkeller, Aug 11 2004

The ratio of the count of primes p <= n such that p-1 is cubefree to the count of primes <= n converges to 0.69.. . This implies that roughly 70% of the primes less one are cubefree. This compares to about 0.37 of the primes less one are squarefree.

More accurately, the density of this sequence within the primes is Product_{p prime} (1-1/(p^2*(p-1))) = 0.697501... (A065414) (Mirsky, 1949). - Amiram Eldar, Feb 16 2021

			43 is included because 43-1 = 2*3*7.
41 is omitted because 41-1 = 2^3*5.
97 is omitted because 96 = 2^5*3 since higher powers are also tested for exclusion.

Reinhard Zumkeller and Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Leon Mirsky, The number of representations of an integer as the sum of a prime and a k-free integer, American Mathematial Monthly 56:1 (1949), pp. 17-19.

Cf. A004709, A039787, A065414, A097380, A089194 (subsequence).

a097375 n = a097375_list !! (n-1)
a097375_list = filter ((== 1) . a212793 . (subtract 1)) a000040_list
-- Reinhard Zumkeller, May 27 2012

filter:= p -> isprime(p) and max(seq(t[2],t=ifactors(p-1)[2]))<=2:
select(filter, [2,seq(2*i+1,i=1..1000)]); # Robert Israel, Sep 11 2014

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]==0,AppendTo[lst,p]],{n,6!}];lst (* Vladimir Joseph Stephan Orlovsky, Jul 15 2009 *)
Select[Prime[Range[100]],Max[Transpose[FactorInteger[#-1]][[2]]]<3&] (* Harvey P. Dale, Feb 05 2012 *)

lista(nn) = forprime(p=2, nn, f = factor(p-1)[,2]; if ((#f == 0) || vecmax(f) < 3, print1(p, ", "));) \\ Michel Marcus, Sep 11 2014

A212793(a(n) - 1) = 1. - Reinhard Zumkeller, May 27 2012

Corrected and extended by Harvey P. Dale, Feb 05 2012

A162870 Primes p such that p-1 and p+1 each contain at least one cubed prime in their prime factorization.

Original entry on oeis.org

919, 1999, 2647, 2663, 2969, 3511, 3833, 3943, 4751, 6857, 9127, 10313, 11287, 11719, 12041, 12583, 13033, 13337, 13879, 14249, 14633, 15497, 15607, 16903, 18089, 18199, 18251, 18521, 19751, 20249, 20359, 20681, 21751, 21977, 22409

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jul 15 2009

nonn

The selection criterion is that p-1 and p+1 are in the subsequence 8=2^3, 24=2^3*3, 27=3^3, 40=2^3*5, 54=2*3^3,... of cubeful numbers (A046099) which actually display at least one cube in their standard prime factorization (A176297).

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 3. This is more restrictive than being cubeful, so the sequence becomes a subsequence of A086708.

			271 is not in the sequence although 271 - 1 = 2*3^3*5 contains a third cube in the prime factorization, because 271 + 1 = 2^4*17 does not.
919 is in the sequence because 919 - 1 = 2*3^3*17 contains a third cube in the prime factorization and so does 919 + 1 = 2^3*5*23.

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

Cf. A089189, A089194.

isA162870 := proc(n)
    if isprime(n) then
        isA176297(n-1) and isA176297(n+1) ;
    else
        false;
    end if;
end proc:
for n from 1 to 40000 do
    if isA162870(n) then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, Dec 08 2015
N:= 10^6: # to get all terms < N, where N is even
V:= Vector(N/2):
for i from 1 do
  p:= ithprime(i);
  if p^3 > N+1 then break fi;
  if p = 2 then inds:= 4*[seq(i,i=1..floor(N/8),2)]
  else inds:= p^3*select(t -> t mod p <> 0, [$1..floor(N/2/p^3)])
  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]]==3,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

Role of cubefree 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

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A089189 Primes p such that p-1 is cubefree.

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A162870 Primes p such that p-1 and p+1 each contain at least one cubed prime in their prime factorization.

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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.

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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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