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

Author

Vladimir Joseph Stephan Orlovsky, Jul 15 2009

Keywords

• nonn

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

Examples

			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
		

Crossrefs

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

Programs

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 *)

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

Extensions

Terms corrected by Zak Seidov, Jul 19 2009

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