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A245071
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a(n) = 12n - prime(n).
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2
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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
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OFFSET
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1,1
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COMMENTS
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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.
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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
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LINKS
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FORMULA
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a(n) = 12*n - prime(n).
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EXAMPLE
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a(3) = 12*3 - prime(3) = 36 - 5 = 31.
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MATHEMATICA
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PROG
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(PARI) vector(133, n, 12*n-prime(n) )
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CROSSREFS
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A000040 (prime(n)), A038605 (Floor(n-th prime/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)).
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KEYWORD
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sign,easy
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AUTHOR
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STATUS
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approved
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