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

10, 21, 31, 41, 49, 59, 67, 77, 85, 91, 101, 107, 115, 125, 133, 139, 145, 155, 161, 169, 179, 185, 193, 199, 203, 211, 221, 229, 239, 247, 245, 253, 259, 269, 271, 281, 287, 293, 301, 307, 313, 323, 325, 335, 343, 353, 353, 353, 361, 371, 379, 385, 395, 397, 403, 409, 415, 425

Offset: 1

Author

Freimut Marschner, Jul 21 2014

Keywords

• sign
• easy

Comments

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:

(1) If k > floor(prime(n)/n) then a(n) is positive.

(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.

(3) If k <= floor(prime(n)/n) and (prime(n) mod n) > (prime(n+1) mod (n+1)) then a(n) is positive.

.

n prime(n) floor(prime(n)/n) (prime(n) mod n) a(n)

40072 480853 12 5 11

40073 480881 12 23 -5

40083 481001 11 40079 -5

40084 481003 11 40074 5

40121 481447 12 5 5

40122 481469 12 13 -5

Examples

			a(3) = 12*3 - prime(3) = 36 - 5 = 31.
		

Links

• Freimut Marschner, Table of n, a(n) for n = 1..100000

Crossrefs

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)).

Programs

Mathematica

Table[12n - Prime[n], {n, 60}] (* Alonso del Arte, Jul 27 2014 *)

PARI

vector(133, n, 12*n-prime(n) )

Formula

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