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

1,1

Comments

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

Links

Table of n, a(n) for n=1..26.

Example

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

Mathematica

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 *)

Prog

(Python)
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

Crossrefs

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

Sequence in context: A190379 A205758 A205588 * A162989 A052195 A089218

Adjacent sequences: A162871 A162872 A162873 * A162875 A162876 A162877

Keyword

nonn

Author

Vladimir Joseph Stephan Orlovsky, Jul 15 2009

Extensions

Terms corrected by Zak Seidov, Jul 19 2009

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

Status

approved