A342814 Numbers k such that k - 1 and floor(k/5) are both prime.

12, 14, 18, 38, 68, 98, 158, 308, 338, 368, 398, 488, 548, 758, 788, 908, 968, 998, 1118, 1568, 1658, 1748, 1868, 1988, 2288, 2438, 2618, 2708, 2858, 2888, 3038, 3068, 3218, 3308, 3458, 3548, 3638, 3698, 3848, 4058

Offset: 1

Author

Claude H. R. Dequatre, Mar 22 2021

Keywords

• nonn
• easy

Comments

Except for a(1) and a(2), all terms == 8 (mod 10).

The first three absolute differences (d) between two consecutive floor(k/5) are respectively equal to 0, 1 and 4 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*A023217(n-2) + 3. Higher terms also coincide with A265767 + 1. - Hugo Pfoertner, Mar 22 2021

Examples

			12 is a term because 12 - 1 = 11 and 11 is prime and 12/5 = 2.4 whose floor value is 2 and 2 is also prime.
97 is not a term because 97 - 1 = 96 and 96 is not prime although floor(97/5) = 19 is prime.
Initial terms, associated primes and d:
          k       k - 1     floor(k/5)     d
a(1)     12        11          2
a(2)     14        13          2           0
a(3)     18        17          3           1
a(4)     38        37          7           4
a(5)     68        67         13           6
a(6)     98        97         19           6
a(7)    158       157         31          12
a(8)    308       307         61          30
a(9)    338       337         67           6
a(10)   368       367         73           6
		

Links

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

Crossrefs

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

Cf. A008864, A023217, A265767.

Programs

Maple

R:= NULL:
p:= 1: count:= 0:
while count < 100 do
  p:= nextprime(p);
  if isprime(floor((p+1)/5)) then
     R:= R,p+1; count:= count+1
  fi
od:
R; # Robert Israel, May 22 2024

Mathematica

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

PARI

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