A066794
Start of a record-breaking run of consecutive integers with an odd number of prime factors.
Original entry on oeis.org
2, 11, 17, 27, 170, 279, 428, 5879, 13871, 41233, 171707, 1004646, 1633357, 5460156, 11902755, 21627159, 38821328, 41983357, 179376463, 130292951546, 393142151459, 2100234982892, 5942636062287
Offset: 1
A175201
a(n) is the smallest k such that the n consecutive values lambda(k), lambda(k+1), ..., lambda(k+n-1) = 1, where lambda(m) is the Liouville function A008836(m).
Original entry on oeis.org
1, 9, 14, 33, 54, 140, 140, 213, 213, 1934, 1934, 1934, 35811, 38405, 38405, 200938, 200938, 389409, 1792209, 5606457, 8405437, 8405437, 8405437, 8405437, 68780189, 68780189, 68780189, 68780189, 880346227, 880346227, 880346227, 880346227, 880346227
Offset: 1
a(1) = 1 and L(1) = 1;
a(2) = 9 and L(9) = L(10)= 1;
a(3) = 14 and L(14) = L(15) = L(16) = 1;
a(4) = 33 and L(33) = L(34) = L(35) = L(36) = 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..44 (terms < 10^13, first 37 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}]
With[{c=LiouvilleLambda[Range[841*10^4]]},Table[SequencePosition[c,PadRight[ {},n,1],1][[All,1]],{n,24}]//Flatten] (* The program generates the first 24 terms of the sequence. *) (* Harvey P. Dale, Jul 27 2022 *)
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)))
A066963
Start of a record-breaking run of consecutive composite integers with an odd number of prime factors.
Original entry on oeis.org
8, 27, 170, 242, 2522, 5882, 18238, 48513, 61532, 506517, 752714, 1213848, 3098613, 5481504, 78214964, 103886546, 118689518, 1608906624, 3939877246, 7964728742, 166384601987, 250060982098
Offset: 1
a(3)=170 because this is the start of third record breaking run of consecutive composite integers (170,171,172) with an odd number of prime factors (170=2*5*17, 171=3*3*19, 172=2*2*43).
Showing 1-4 of 4 results.
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