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

%I A342814 #21 May 22 2024 11:43:20
%S A342814 12,14,18,38,68,98,158,308,338,368,398,488,548,758,788,908,968,998,
%T A342814 1118,1568,1658,1748,1868,1988,2288,2438,2618,2708,2858,2888,3038,
%U A342814 3068,3218,3308,3458,3548,3638,3698,3848,4058
%N A342814 Numbers k such that k - 1 and floor(k/5) are both prime.
%C A342814 Except for a(1) and a(2), all terms == 8 (mod 10).
%C A342814 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 A342814 Subsequence of A008864, by definition. - _Michel Marcus_, Mar 22 2021
%C A342814 For n >= 3, a(n) = 5*A023217(n-2) + 3. Higher terms also coincide with A265767 + 1. - _Hugo Pfoertner_, Mar 22 2021
%H A342814 Robert Israel, <a href="/A342814/b342814.txt">Table of n, a(n) for n = 1..10000</a>
%e A342814 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 A342814 97 is not a term because 97 - 1 = 96 and 96 is not prime although floor(97/5) = 19 is prime.
%e A342814 Initial terms, associated primes and d:
%e A342814           k       k - 1     floor(k/5)     d
%e A342814 a(1)     12        11          2
%e A342814 a(2)     14        13          2           0
%e A342814 a(3)     18        17          3           1
%e A342814 a(4)     38        37          7           4
%e A342814 a(5)     68        67         13           6
%e A342814 a(6)     98        97         19           6
%e A342814 a(7)    158       157         31          12
%e A342814 a(8)    308       307         61          30
%e A342814 a(9)    338       337         67           6
%e A342814 a(10)   368       367         73           6
%p A342814 R:= NULL:
%p A342814 p:= 1: count:= 0:
%p A342814 while count < 100 do
%p A342814   p:= nextprime(p);
%p A342814   if isprime(floor((p+1)/5)) then
%p A342814      R:= R,p+1; count:= count+1
%p A342814   fi
%p A342814 od:
%p A342814 R; # _Robert Israel_, May 22 2024
%t A342814 Select[Range[2,5000,2],And@@PrimeQ[{#-1,Floor[#/5]}]&] (* _Giorgos Kalogeropoulos_, Apr 01 2021 *)
%o A342814 (PARI) for(k = 1,10000,if(isprime(k - 1) && isprime(k\5),print1(k", ")))
%Y A342814 Cf. A000040, A007530, A007811, A014561, A259645.
%Y A342814 Cf. A008864, A023217, A265767.
%K A342814 nonn,easy
%O A342814 1,1
%A A342814 _Claude H. R. Dequatre_, Mar 22 2021