A342814 - OEIS

A342814

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

%I #21 May 22 2024 11:43:20

%S 12,14,18,38,68,98,158,308,338,368,398,488,548,758,788,908,968,998,

%T 1118,1568,1658,1748,1868,1988,2288,2438,2618,2708,2858,2888,3038,

%U 3068,3218,3308,3458,3548,3638,3698,3848,4058

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

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

%C 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.

%C Subsequence of A008864, by definition. - _Michel Marcus_, Mar 22 2021

%C For n >= 3, a(n) = 5*A023217(n-2) + 3. Higher terms also coincide with A265767 + 1. - _Hugo Pfoertner_, Mar 22 2021

%H Robert Israel, <a href="/A342814/b342814.txt">Table of n, a(n) for n = 1..10000</a>

%e 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.

%e 97 is not a term because 97 - 1 = 96 and 96 is not prime although floor(97/5) = 19 is prime.

%e Initial terms, associated primes and d:

%e k k - 1 floor(k/5) d

%e a(1) 12 11 2

%e a(2) 14 13 2 0

%e a(3) 18 17 3 1

%e a(4) 38 37 7 4

%e a(5) 68 67 13 6

%e a(6) 98 97 19 6

%e a(7) 158 157 31 12

%e a(8) 308 307 61 30

%e a(9) 338 337 67 6

%e a(10) 368 367 73 6

%p R:= NULL:

%p p:= 1: count:= 0:

%p while count < 100 do

%p p:= nextprime(p);

%p if isprime(floor((p+1)/5)) then

%p R:= R,p+1; count:= count+1

%p fi

%p od:

%p R; # _Robert Israel_, May 22 2024

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

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

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

%Y Cf. A008864, A023217, A265767.

%K nonn,easy

%O 1,1

%A _Claude H. R. Dequatre_, Mar 22 2021