

A334194


a(n) = n  d1*d2, where d1, d2 are the distances from n to the previous and the next prime number respectively.


0



0, 0, 1, 3, 1, 5, 1, 5, 5, 7, 3, 11, 5, 11, 11, 13, 9, 17, 11, 17, 17, 19, 1, 19, 17, 17, 19, 23, 17, 29, 19, 27, 25, 25, 27, 31, 13, 35, 35, 37, 33, 41, 35, 41, 41, 43, 23, 43, 41, 41, 43, 47, 17, 49, 47, 47, 49, 53, 47, 59, 49, 57, 55
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OFFSET

1,4


COMMENTS

Except of a(1), a(2), a(7) and a(23) that are negative or zero, it seems that the density of the prime numbers is such that; n is greater than the product d1 * d2 for each n.
So I conjecture that n > d1 * d2 for every N  {1, 2, 7, 23}. If it is true, means that there is at least 1 prime number in space [n  sqrt(n), n + sqrt(n)].
I also conjecture that lim_{n > oo} (n  d1*d2)/n = a(n)/n = 1.


LINKS

Table of n, a(n) for n=1..63.


EXAMPLE

For n = 9, a(9) = 9  (9  7) * (11  9) = 9  2 * 2 = 5, because the previous prime is 7 and the next prime is 11.
For n = 23, the previous prime is 19 so d1 = 23  19 = 4. The next prime is 29 so d2 = 29  23 = 6. The product d1 * d2 = 4 * 6 = 24 (maybe the last time that the product d1 * d2 > n). So a(23) = 23  24 = 1.


PROG

(PARI) for(n = 1, 100, d1 = n  precprime(n  1); d2 = nextprime(n + 1)  n; print1(n  d1*d2", "))


CROSSREFS

Cf. A000040, A151799, A151800.
Sequence in context: A114567 A001051 A214737 * A046730 A002972 A324896
Adjacent sequences: A334191 A334192 A334193 * A334195 A334196 A334197


KEYWORD

sign


AUTHOR

Dimitris Valianatos, Apr 18 2020


STATUS

approved



