A294280 - OEIS

A294280

a(n) = least positive k such that omega(n+k) > max(omega(n), omega(k)), where omega(m) = A001221(m), the number of distinct primes dividing m.

1, 4, 3, 2, 1, 24, 3, 2, 1, 20, 1, 18, 1, 16, 15, 2, 1, 12, 1, 10, 9, 8, 1, 6, 1, 4, 1, 2, 1, 180, 2, 1, 9, 8, 7, 6, 1, 4, 3, 2, 1, 168, 1, 16, 15, 14, 1, 12, 1, 10, 9, 8, 1, 6, 5, 4, 3, 2, 1, 150, 1, 4, 3, 1, 1, 144, 1, 2, 1, 140, 1, 6, 1, 4, 3, 2, 1, 132, 1

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

For any n > 0, a(n) <= n * (A053669(n) - 1).

Apparently, a(n) = n * (A053669(n) - 1) iff n belongs to A077011.

a(n) = 1 iff omega(n) < omega(n+1).

a(p) = 1 for any prime power p not in A006549.

The scatterplot of the sequence shows segments of slope -1, corresponding to frequent values of n+a(n); these segments correspond to the strands in the plot of the ordinal transform of n+a(n) (see plots in Links section).

LINKS

Table of n, a(n) for n=1..79.

Rémy Sigrist, Logarithmic scatterplot of the first 10000 terms

Rémy Sigrist, Pin plot of the first 10000 terms

Rémy Sigrist, Ordinal transform of the first 10000 terms of n+a(n)

EXAMPLE

For n=2:

- omega(2+1) = 1 = omega(2),

- omega(2+2) = 1 = omega(2),

- omega(2+3) = 1 = omega(2),

- omega(2+4) = 2 > max(omega(2), omega(4)) = 1,

- hence, a(2) = 4.