A253174 For a permutation of the positive integers B={b(k)}, denote L_B(n) = Sum_{k=1..n} lambda(b(k)), where lambda(n) is Liouville's function (A008836). In this sequence, a(n) = L_B(n) in case B = A098550.

1, 0, -1, 0, 1, 0, 1, 2, 1, 2, 3, 2, 3, 4, 3, 4, 5, 4, 3, 4, 5, 4, 3, 4, 5, 4, 3, 4, 3, 2, 1, 2, 3, 4, 5, 6, 7, 8, 7, 8, 9, 8, 7, 6, 7, 6, 7, 8, 9, 8, 7, 6, 7, 6, 5, 6, 7, 8, 9, 10, 9, 8, 9, 10, 11, 10, 9, 10, 11, 12, 13, 14, 13, 14, 13, 12, 13, 12, 11, 10, 11, 10, 9, 8, 9, 10, 9, 8, 9, 10, 11, 12, 11, 10, 9, 10, 11, 12, 13, 12

Offset

1,8

Comments

Let n = p^a*q^b*...r^c be the prime power factorization of b(n). Then lambda(n) is

(-1)^s, where s is the sum of exponents a + b + ... + c.

In case B=A000027 (the natural numbers), G. Polya (1919) conjectured that L_B(n)<=0, for n>=2. But this was disproved in 1958 by B. Haselgrove, and in 1980 M. Tanaka found the smallest counterexample, 906150257.

However, for this sequence we conjecture that a(n)>=0 for all n other than 3. A reason for our conjecture is the later appearance of primes in A098550 than in A000027. By our conjecture, among the first N terms of A098550, the terms with odd s are never in the majority, if N is other than 3. Peter J. C. Moses verified the conjecture up to 2.5*10^5 and, moreover, in this range a(n)>0 for n>6.

Links

Peter J. C. Moses, Table of n, a(n) for n = 1..1000

M. Tanaka, A Numerical Investigation on Cumulative Sum of the Liouville Function, Tokyo J. Math. 3 (1980), 187-189.

Eric Weisstein's World of Mathematics, Liouville Function

Formula

a(n+1) - a(n) = lambda(A098550(n+1)).

Mathematica

(* b = A098550 *) b[n_ /; n <= 3] := n; b[n_] := b[n] = For[bb = Table[b[j], {j, 1, n-1}]; k=4, True, k++, If[FreeQ[bb, k] && !CoprimeQ[k, b[n-2]] && CoprimeQ[k, b[n-1]], Return[k]]]; a[1]=1; a[n_] := a[n] = a[n-1] + LiouvilleLambda[b[n]]; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Jan 13 2015 *)

Crossrefs

Cf. A008836, A002819, A098550, A189229, A247085.

Sequence in context: A220098 A246017 A116939 * A008611 A025798 A161064

Adjacent sequences: A253171 A253172 A253173 * A253175 A253176 A253177

Keyword

sign

Author

Vladimir Shevelev, Jan 08 2015

Status

approved