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A162874
Twin primes p and r (p < r) such that p-1, p+1 and r+1 are not cubefree.
3
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
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
KEYWORD
nonn
AUTHOR
EXTENSIONS
Terms corrected by Zak Seidov, Jul 19 2009
Edited by N. J. A. Sloane, Aug 12 2009
STATUS
approved