cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

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

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Author

G. L. Honaker, Jr., Jan 18 2002

Keywords

Comments

a(22) > 10^12. [From Donovan Johnson, Oct 11 2010]
a(24) > 10^13. - Giovanni Resta, Aug 01 2013

Crossrefs

Cf. A066793.

Extensions

Except for first 4 or 5 terms, computed by Shyam Sunder Gupta, Jan 26 2002
Corrected by Don Reble, Nov 20 2006
a(20)-a(21) from Donovan Johnson, Oct 11 2010
a(22)-a(23) from Giovanni Resta, Aug 01 2013

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

Views

Author

Michel Lagneau, Mar 04 2010

Keywords

Comments

Short history of conjecture L(n) <= 0 for all n >= 2 by Deborah Tepper Haimo, where L(n) is the summatory Liouville function A002819(n). George Polya conjectured 1919 that L(n) <= 0 for all n >= 2. The conjecture was generally deemed true for nearly 40 years, until 1958, when C. B. Haselgrove proved that L(n) > 0 for infinitely many n. In 1962, R. S. Lehman found that L(906180359) = 1 and in 1980, M. Tanaka discovered that the smallest counterexample of the Polya conjecture occurs when n = 906150257.

Examples

			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.
		

References

  • 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.

Crossrefs

Programs

  • Maple
    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:
  • Mathematica
    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 *)

Formula

lambda(n) = (-1)^omega(n) where omega(n) is the number of prime factors of n with multiplicity.

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

Views

Author

Keywords

Examples

			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.
		

Crossrefs

Programs

  • Mathematica
    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 *)
  • PARI
    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

Views

Author

Shyam Sunder Gupta, Jan 27 2002

Keywords

Comments

a(23) > 10^12. [Donovan Johnson, Oct 11 2010]

Examples

			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).
		

Crossrefs

Extensions

a(18)-a(21) from Donovan Johnson, Sep 09 2008
a(22) from Donovan Johnson, Oct 11 2010
Showing 1-4 of 4 results.