A066793
The start of a record-breaking run of consecutive integers with an even number of prime factors.
Original entry on oeis.org
1, 9, 14, 33, 54, 140, 213, 1934, 35811, 38405, 200938, 389409, 1792209, 5606457, 8405437, 68780189, 880346227, 85910903650, 87635944146, 284340107954, 2195312273836, 2864568747364, 3155591650029
Offset: 1
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Table[SequencePosition[If[EvenQ[#],1,0]&/@PrimeOmega[Range[40000]],PadRight[{},n,1],1][[All,1]],{n,14}]//Flatten//Union (* The program generates the first 10 terms of the sequence. *) (* Harvey P. Dale, Sep 06 2022 *)
a(12) and a(14) corrected and a(18)-a(20) from
Donovan Johnson, Oct 11 2010
A175202
a(n) is the smallest k such that the n consecutive values L(k), L(k+1), ..., L(k+n-1) = -1, where L(m) is the Liouville function A008836(m).
Original entry on oeis.org
2, 2, 11, 17, 27, 27, 170, 279, 428, 5879, 5879, 13871, 13871, 13871, 41233, 171707, 1004646, 1004646, 1633357, 5460156, 11902755, 21627159, 21627159, 38821328, 41983357, 179376463, 179376463, 179376463, 179376463, 179376463, 179376463, 179376463
Offset: 1
a(1) = 2 and L(2) = -1;
a(2) = 2 and L(2) = L(3)= -1;
a(3) = 11 and L(11) = L(12) = L(13) = -1;
a(4) = 17 and L(17) = L(18) = L(19) = L(20) = -1.
- H. Gupta, On a table of values of L(n), Proceedings of the Indian Academy of Sciences. Section A, 12 (1940), 407-409.
- H. Gupta, A table of values of Liouville's function L(n), Research Bulletin of East Panjab University, No. 3 (Feb. 1950), 45-55.
- Donovan Johnson and Giovanni Resta, Table of n, a(n) for n = 1..43 (terms < 10^13, first 38 terms from Donovan Johnson)
- Peter Borwein, Ron Ferguson, and Michael J. Mossinghoff, Sign changes in sums of the Liouville function. Math. Comp. 77 (2008), 1681-1694.
- R. S. Lehman, On Liouville's function, Math. Comp., 14 (1960), 311-320.
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with(numtheory):for k from 0 to 30 do : indic:=0:for n from 1 to 1000000000 while (indic=0)do :s:=0:for i from 0 to k do :if (-1)^bigomega(n+i)= -1 then s:=s+1: else fi:od:if s=k+1 and indic=0 then print(n):indic:=1:else fi:od:od:
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Table[k=1;While[Sum[LiouvilleLambda[k+i],{i,0,n-1}]!=-n,k++];k,{n,1,30}]
A233445
Start of record runs with lambda(k) = lambda(k+1) = ..., where lambda is Liouville's function A008836.
Original entry on oeis.org
1, 2, 11, 17, 27, 140, 213, 1934, 13871, 38405, 171707, 200938, 389409, 1633357, 5460156, 8405437, 41983357, 68780189, 179376463, 130292951546, 393142151459, 2100234982892, 5942636062287
Offset: 1
Lambda(1) = 1 is the first (record) run, so a(1) = 1.
Lambda(2) = lambda(3) = -1 is the second record run, so a(2) = 2.
Lambda(11) = lambda(12) = lambda(13) = -1 is the third record run, so a(3) = 11.
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sz[n_] := Module[{t = LiouvilleLambda[n], k = n}, While[LiouvilleLambda[k++] == t]; k - n]; r = 0; Reap[For[n = 1, n <= 10^6, n++, t = sz[n]; If[t > r, r = t; Print[t, " ", n]; Sow[n]]]][[2, 1]] (* Jean-François Alcover, Oct 17 2016, adapted from PARI *)
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L(n)=(-1)^bigomega(n);
sz(n)=my(t=L(n),k=n);while(L(k++)==t,);k-n
r=0;for(n=1,1e9,t=sz(n);if(t>r,r=t;print(t" "n)))
Showing 1-3 of 3 results.
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