A294280 - OEIS

%I #9 Oct 28 2017 09:26:18

%S 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,

%T 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,

%U 1,4,3,1,1,144,1,2,1,140,1,6,1,4,3,2,1,132,1

%N 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.

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

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

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

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

%C 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).

%H Rémy Sigrist, <a href="/A294280/a294280.png">Logarithmic scatterplot of the first 10000 terms</a>

%H Rémy Sigrist, <a href="/A294280/a294280_1.png">Pin plot of the first 10000 terms</a>

%H Rémy Sigrist, <a href="/A294280/a294280_2.png">Ordinal transform of the first 10000 terms of n+a(n)</a>

%e For n=2:

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

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

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

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

%e - hence, a(2) = 4.

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

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

%K nonn

%O 1,2

%A _Rémy Sigrist_, Oct 26 2017