cp's OEIS Frontend

A245071 a(n) = 12n - prime(n).

%I A245071 #46 Nov 10 2024 02:22:12
%S A245071 10,21,31,41,49,59,67,77,85,91,101,107,115,125,133,139,145,155,161,
%T A245071 169,179,185,193,199,203,211,221,229,239,247,245,253,259,269,271,281,
%U A245071 287,293,301,307,313,323,325,335,343,353,353,353,361,371,379,385,395,397,403,409,415,425
%N A245071 a(n) = 12n - prime(n).
%C A245071 Prime(n) > n for n > 0. Let prime(n) = k*n with k as an even integer constant, for example, k = 12; then a(n) = k*n - prime(n) is a sequence of odd integers that are positive as long as k*n > prime(n). This is the case up to a(40072) = 11. If k*n < prime(n) then a(n) < 0, a(40073) = -5 up to a(40083) = -5. From a(40084) = 5 up to a(40121) = 5, a(n) > 0 again, but a(n) < 0 for n >= 40122. For k = 12 the table shows this result compared with floor(prime(n)/n) and (prime(n) mod n) <= (prime(n+1) mod (n+1)) for n >= 1. Observations:
%C A245071 (1) If k > floor(prime(n)/n) then a(n) is positive.
%C A245071 (2) If k <= floor(prime(n)/n) and (prime(n) mod n) < (prime(n+1) mod (n+1)) and n > 1 then a(n) is negative.
%C A245071 (3) If k <= floor(prime(n)/n) and (prime(n) mod n) > (prime(n+1) mod (n+1)) then a(n) is positive.
%C A245071 .
%C A245071 n     prime(n) floor(prime(n)/n) (prime(n) mod n)  a(n)
%C A245071 40072 480853        12                 5            11
%C A245071 40073 480881        12                23            -5
%C A245071 40083 481001        11             40079            -5
%C A245071 40084 481003        11             40074             5
%C A245071 40121 481447        12                 5             5
%C A245071 40122 481469        12                13            -5
%H A245071 Freimut Marschner, <a href="/A245071/b245071.txt">Table of n, a(n) for n = 1..100000</a>
%F A245071 a(n) = 12*n - prime(n).
%e A245071 a(3) = 12*3 - prime(3) = 36 - 5 = 31.
%t A245071 Table[12n - Prime[n], {n, 60}] (* _Alonso del Arte_, Jul 27 2014 *)
%o A245071 (PARI) vector(133, n, 12*n-prime(n) )
%Y A245071 A000040 (prime(n)), A038605 (floor(prime(n)/n)), A004648 (prime(n) mod n), A038606 (Least k such that k-th prime > n * k),  A038607 (the smallest prime number k such that k > n*pi(k)),  A102281 (the largest number m such that m = pi(n*m)).
%K A245071 sign,easy
%O A245071 1,1
%A A245071 _Freimut Marschner_, Jul 21 2014