A360659 - OEIS

%I #72 Jun 17 2024 15:31:33

%S 0,1,0,-1,0,-1,0,-1,-2,-1,0,-1,-2,-3,-4,-3,-2,-3,-4,-5,-4,-5,-4,-5,-6,

%T -5,-6,-7,-8,-9,-8,-9,-8,-7,-8,-7,-6,-7,-8,-7,-8,-9,-10,-11,-12,-13,

%U -14,-15,-14,-13,-12,-13,-14,-15,-14,-13,-14,-15,-16,-17,-18,-19

%N a(n) is the minimum sum of a completely multiplicative sign sequence of length n.

%C A completely multiplicative sign sequence is a sequence S of numbers -1 and +1 such that S(a*b) = S(a)*S(b).

%C Directly from the definition, the first term of every multiplicative sign sequence is +1.

%C The number of completely multiplicative sign sequences of length n is 2^A000720(n), since every multiplicative sign sequence S is uniquely determined by the values of S on prime indexes.

%C Liouville's function A002819(n) gives the sum of the multiplicative sign sequence of length n in which every prime is assigned the value of -1. This is, therefore, an upper bound on a(n). The first difference occurs at n = 14.

%C Conjecture: For every n >= 22 there exists completely multiplicative sign sequence of length n with the minimal sum, which have +1's on initial primes and -1's on the remaining primes, i.e., its subsequence on primes is ++...+--...- for some number of +1's.

%C First differences are either 1 or -1. This is because optimum sequences for n cannot have a sum less than one less than those for n-1 and similarly optimum sequences for n cannot have a sum greater than one more than those for n-1. A zero difference is not possible because terms alternate between even and odd. - _Andrew Howroyd_, Feb 16 2023

%C Liouville's function (A002819) is unbounded below, hence this sequence is also unbounded below and every negative number occurs in the sequence. - _Rémy Sigrist_, Feb 17 2023

%C From _David A. Corneth_, May 28 2024: (Start)

%C One might ease the search by setting values with prime indices in (n/2, n] to -1 in the multiplicative function.

%C Furthermore one can use the squarefree part of n to evaluate the value of composites in the sequence. For example the value of index 60 has the same value as the one at index 15. (End)

%H David A. Corneth, <a href="/A360659/b360659.txt">Table of n, a(n) for n = 0..181</a> (terms to n = 99 from Bartlomiej Pawlik)

%F a(n) <= A002819(n).

%F Conjecture: a(n) < A002819(n) for n > 20.

%F |a(n) - a(n-1)| = 1 for n > 0. - _Andrew Howroyd_, Feb 16 2023

%F From _David A. Corneth_, May 28 2024: (Start)

%F a(k^2) = a(k^2-1) + 1 for k >= 1.

%F a(p) = a(p-1) - 1 for prime p. (End)

%e There are eight completely multiplicative sign sequences of length 5: +--+-, +--++, +-++-, +-+++, ++-+-, ++-++, ++++- and +++++. The sums of those sequences are, respectively, -1, 1, 1, 3, 1, 3, 3 and 5. Hence, the minimum sum is equal to -1 and so a(5) = -1.

%o (PARI)

%o F(n, b)={vector(n, k, my(f=factor(k)); prod(i=1, #f~, if(bittest(b, primepi(f[i, 1])-1), 1, -1)^f[i, 2]))}

%o a(n)={my(m=oo); for(b=0, 2^primepi(n)-1, m=min(m, vecsum(F(n,b)))); m} \\ _Andrew Howroyd_, Feb 16 2023

%o (Python)

%o from itertools import product

%o from sympy import primerange, primepi, factorint

%o def A360659(n):

%o     a = dict(zip(primerange(n+1),range(c:=primepi(n))))

%o     return (min(sum(sum(e for p,e in factorint(m).items() if b[a[p]])&1^1 for m in range(1,n+1)) for b in product((0,1),repeat=c))<<1)-n # _Chai Wah Wu_, May 31 2024

%Y Cf. A000720, A002819, A007913, A008836.

%K sign

%O 0,9

%A _Bartlomiej Pawlik_, Feb 15 2023