A286287 Least number to start a run of exactly n nondecreasing values of little omega (A001221).

106, 11, 13, 7, 512, 1, 1941, 141, 6847, 211, 195031, 82321, 808083, 534077, 3355906, 526093, 526889774, 127890361, 22529949392, 118968284927, 164159173895, 244022049199, 3022058317713, 585927201061

Offset: 1

Author

N. J. A. Sloane, May 16 2017

Keywords

• nonn
• more

Comments

a(17) > 10^7.

a(18) = 127890361; a(n) > 4*10^8 for n=17 and for n >= 19. - Jon E. Schoenfield, Jul 16 2017

a(n) > 6*10^12 for n >= 25. - Giovanni Resta, Jul 16 2019

Examples

			We have omega(10) = 2, omega(11) = 1, omega(12) = 2, omega(13) = 1. Therefore 11 starts a run of exactly 2 consecutive integers (11, 12) which have nondecreasing (here: strictly increasing) values of omega.
The 6 numbers from 1 through 6 yield values (0, 1, ..., 1, 2) for omega, therefore a(6) = 1. The 4 numbers from 7 through 10 yield values (1, 1, 1, 2) for omega, therefore a(4) = 7.
A run of length 1 is a single number n such that omega(n-1) > omega(n) > omega(n+1). (If we had "<=" in one of the cases, it would be part of a run of at least 2 numbers with nondecreasing omega.) This first happens for a(1) = 106.
		

References

• M. F. Hasler, Posting to Sequence Fans Mailing List, May 06 2017

Crossrefs

Cf. A001221, A284597 (analog for tau = A000005), A285893 (analog for sigma = A000203).

See also A286288, A286289.

Programs

Mathematica

Prepend[#, Module[{k = 2}, While[Sign@ Differences@ PrimeNu[k + {-1, 0, 1}] != {-1, -1}, k++]; k]] &@ Function[s, Function[r, If[Length@ # > 0, #[[1, 1]], -1] &@ Select[s, Length@ # == r &]] /@ Range@ Max@ Map[Length, s]]@ DeleteCases[SplitBy[MapIndexed[Function[k, (2 Boole[#1 <= #2] - 1) k & @@ #1]@ First@ #2 &, Partition[Array[PrimeNu, 10^7], 2, 1]], Sign], w_ /; First@ w < 0] (* Michael De Vlieger, May 19 2017 *)

PARI

alias("A","A286287"); A=vector(19); apply(scan(N, s=1, t=omega(s))=for(k=s+1, N, t>(t=omega(k))||next; k-s>#A||A[k-s]||printf(" a(%d)=%d, ", k-s, s)||A[k-s]=s; s=k); done, [4e6]) \\ Then the search may be extended using scan(END,START). - M. F. Hasler, May 16 2017

Extensions

Edited by M. F. Hasler, May 16 2017

a(17)-a(24) from Giovanni Resta, Jul 16 2019