A342809 Numbers k such that k-1 and round(k/5) are both prime.

8, 12, 14, 24, 54, 84, 114, 234, 264, 294, 354, 444, 504, 564, 654, 684, 744, 864, 954, 984, 1164, 1194, 1284, 1554, 1584, 1734, 1914, 2004, 2154, 2214, 2244, 2334, 2394, 2544, 2844, 2964, 3084, 3204, 3414, 3594

Offset: 1

Author

Claude H. R. Dequatre, Mar 22 2021

Keywords

• nonn
• easy

Comments

Except for a(1) and a(2), all terms == 4 (mod 10).

The first three absolute differences (d) between two consecutive rounded (k/5) are respectively equal to 0, 1 and 2 and all the others to 6 or a multiple of 6.

Subsequence of A008864, by definition. - Michel Marcus, Mar 22 2021

For n >= 3, a(n) = 5*A158318(n-2) - 1. - Hugo Pfoertner, Mar 22 2021

Examples

			8 is a term because 8 - 1 = 7 and 7 is prime and 8/5 = 1.6 which when rounded gives 2 and 2 is also prime.
235 is not a term because 235 - 1 = 234 and 234 is not a prime although 235/5 = 47 is prime.
Initial terms, associated primes and d:
         k     k - 1   round(k/5)    d
a(1)     8       7         2
a(2)    12      11         2         0
a(3)    14      13         3         1
a(4)    24      23         5         2
a(5)    54      53        11         6
a(6)    84      83        17         6
a(7)   114     113        23         6
a(8)   234     233        47        24
a(9)   264     263        53         6
a(10)  294     293        59         6
		

Crossrefs

Cf. A000040, A007530, A007811, A014561, A259645.

Cf. A008864, A158318.

Programs

Mathematica

Select[Range[2,5000,2],And@@PrimeQ[{#-1,Round[#/5]}]&] (* Giorgos Kalogeropoulos, Apr 01 2021 *)

PARI

for(k = 1,10000,if(isprime(k - 1) && isprime(k\/5),print1(k", ")))

Python

from sympy import isprime
A342809_list = [k for k in range(1,10**5) if isprime(k-1) and isprime(k//5+int(k % 5 > 2))] # Chai Wah Wu, Apr 08 2021