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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.

%I A253174 #47 Feb 16 2025 08:33:24
%S A253174 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,
%T A253174 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,
%U A253174 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
%N 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.
%C A253174 Let n = p^a*q^b*...r^c be the prime power factorization of b(n). Then lambda(n) is
%C A253174 (-1)^s, where s is the sum of exponents a + b + ... + c.
%C A253174 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.
%C A253174 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.
%H A253174 Peter J. C. Moses, <a href="/A253174/b253174.txt">Table of n, a(n) for n = 1..1000</a>
%H A253174 M. Tanaka, <a href="http://dx.doi.org/10.3836%2Ftjm%2F1270216093">A Numerical Investigation on Cumulative Sum of the Liouville Function</a>, Tokyo J. Math. 3 (1980), 187-189.
%H A253174 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LiouvilleFunction.html">Liouville Function</a>
%F A253174 a(n+1) - a(n) = lambda(A098550(n+1)).
%t A253174 (* 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 *)
%Y A253174 Cf. A008836, A002819, A098550, A189229, A247085.
%K A253174 sign
%O A253174 1,8
%A A253174 _Vladimir Shevelev_, Jan 08 2015