

A303545


For any n > 0 and prime number p, let d_p(n) be the distance from n to the nearest psmooth number; a(n) = Sum_{i prime} d_i(n).


5



0, 0, 1, 0, 2, 2, 3, 0, 1, 3, 6, 4, 7, 5, 2, 0, 6, 2, 9, 6, 9, 11, 14, 8, 8, 10, 5, 6, 12, 4, 10, 0, 4, 9, 5, 4, 15, 16, 13, 12, 24, 18, 28, 18, 16, 22, 28, 16, 17, 16, 20, 20, 25, 10, 12, 12, 17, 22, 24, 8, 21, 13, 3, 0, 5, 8, 26, 18, 16, 10, 25, 8, 28, 21
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OFFSET

1,5


COMMENTS

For any n > 0 and prime number p >= A006530(n), d_p(n) = 0; hence the series in the name contains only finitely many nonzero terms and is well defined.
See also A303548 for a similar sequence.


LINKS



FORMULA

a(n) = 0 iff n is a power of 2.
a(2 * n) <= 2 * a(n).


EXAMPLE

For n = 42:
 d_2(42) = 42  32 = 10,
 d_3(42) = 42  36 = 42  48 = 6,
 d_5(42) = 42  40 = 2,
 d_p(42) = 0 for any prime number p >= 7,
 hence a(42) = 10 + 6 + 2 = 18.


PROG

(PARI) gpf(n) = if (n==1, 1, my (f=factor(n)); f[#f~, 1])
a(n) = my (v=0, pi=primepi(gpf(n))); for (d=0, oo, my (o=min(primepi(gpf(nd)), primepi(gpf(n+d)))); if (o<pi, v += d*(pio); pi=o); if (o<=1, return (v)))


CROSSREFS



KEYWORD



AUTHOR



STATUS

approved



