

A286288


Least number to start a run of exactly n nondecreasing values of (big) Omega (A001222).


5



46, 5, 43, 1, 2021, 121, 25202, 2521, 162121, 460801, 27268546, 23553169, 244068841, 913535283, 3195380866, 2088087121, 5988790769809, 2601212829601
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OFFSET

1,1


COMMENTS

a(11) > 10^7.  M. F. Hasler, May 16 2017
a(19) > 7.5*10^12.  Giovanni Resta, Nov 12 2019


REFERENCES

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


LINKS

Table of n, a(n) for n=1..18.


EXAMPLE

Omega(1..5) = (0, 1, 1, 2, 1), therefore the first run of 4 numbers with nondecreasing Omega (= A001222) starts at a(4) = 1.
Omega(4..7) = (2, 1, 2, 1), so the first run of 2 numbers with nondecreasing Omega starts at a(2) = 5.
A run of subsequent numbers with nondecreasing Omega is of length 1 if it consists of a single number n with Omega(n1) > Omega(n) > Omega(n+1) (else n belongs to a run of length >= 2). This happens first for a(1) = 46.


MATHEMATICA

Prepend[#, Module[{k = 2}, While[Sign@ Differences@ PrimeOmega[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[PrimeOmega, 10^7], 2, 1]], Sign], w_ /; First@ w < 0] (* Michael De Vlieger, May 19 2017 *)


PROG

(PARI) From M. F. Hasler, May 16 2017: (Start)
alias('A, 'A286288); \\ A bug in PARI 2.9.2 requires the alias() to be issued on a line on itself.
A=vector(19); apply(scan(N, s=1, t=bigomega(s))=for(k=s+1, N, t>(t=bigomega(k))next; ks>#AA[ks]printf(" a(%d)=%d, ", ks, s)A[ks]=s; s=k); done, [1e7]) \\ Then the search may be extended using scan(END, START).  (End)


CROSSREFS

Cf. A001222, A284597 (analog for sigma_0 = A000005), A285893 (analog for sigma = A000203), A286287 (analog for omega = A001221).
See also A286289.
Sequence in context: A036204 A270814 A284597 * A051161 A260512 A033366
Adjacent sequences: A286285 A286286 A286287 * A286289 A286290 A286291


KEYWORD

nonn,more,hard


AUTHOR

N. J. A. Sloane, May 16 2017


EXTENSIONS

Edited by M. F. Hasler, May 16 2017
a(11)a(13) from Jon E. Schoenfield, Jul 16 2017
a(14)a(18) from Giovanni Resta, Nov 12 2019


STATUS

approved



