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

Offset: 1

Author

Rémy Sigrist, Oct 26 2017

Keywords

• nonn

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).

Examples

			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.
		

Links

• 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)

Crossrefs

Cf. A001221, A006549, A053669, A077011, A294277.

Programs

PARI

a(n) = my (on=omega(n)); for (k=1, oo, if (omega(n+k) > max(on, omega(k)), return (k)))