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A304111 Partial sums of f(n) = (-1)^(1-A304109(n)). 4
0, 1, 2, 3, 2, 1, 2, 3, 2, 3, 2, 3, 2, 3, 4, 3, 2, 1, 2, 3, 2, 1, 2, 3, 2, 3, 4, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 4, 3, 2, 3, 2, 3, 2, 1, 2, 3, 2, 3, 4, 3, 2, 3, 2, 3, 2, 1, 2, 3, 2, 3, 4, 3, 2, 1, 2, 3, 2, 1, 2, 3, 2, 3, 4, 3, 2, 3, 2, 3, 2, 1, 2, 3, 2, 1, 2, 3, 2, 3, 2, 3, 2, 3, 4, 3, 2, 3, 4, 3, 2, 3, 2, 3, 2, 1, 2, 1, 0, 1, 2, 3, 2, 3, 2, 3, 2, 3, 4, 3, 2 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Start from the initial value a(0) = 0, after which, for n > 0, each successive term a(n) is either one more or one less than the previous term a(n-1), depending on whether the binary expansion of n encodes a squarefree (0,1)-polynomial when the factorization is done in polynomial ring GF(2)[X]], or whether it encodes a polynomial where at least one of its irreducible divisors occurs more than once.

The first negative term occurs as a(153) = -1. See also comments at A304010.

LINKS

Antti Karttunen, Table of n, a(n) for n = 0..65537

Index entries for sequences related to polynomials in ring GF(2)[X]

FORMULA

a(0) = 0, and for n > 0, a(n) = a(n-1) + (-1)^(1-A304109(n)).

For n >= 1, a(n) = (2*A304110(n)) - n.

PROG

(PARI)

up_to = 128;

A304109(n) = { my(fm=factor(Pol(binary(n))*Mod(1, 2))); for(k=1, #fm~, if(fm[k, 2] > 1, return(0))); (1); };

prepare_v304110(up_to) = { my(v=vector(up_to), c=0); for(n=1, up_to, c += A304109(n); v[n] = c); (v); };

v304110 = prepare_v304110(up_to);

A304110(n) = v304110[n];

A304111(n) = ((2*A304110(n)) - n);

\\ Or just as:

c=0; for(n=0, up_to, if(n>0, c+=((-1)^(1-A304109(n)))); print1(c, ", "));

CROSSREFS

Cf. A304107, A304108, A304109, A304110.

Sequence in context: A084199 A277745 A320858 * A030314 A269624 A098236

Adjacent sequences:  A304108 A304109 A304110 * A304112 A304113 A304114

KEYWORD

sign

AUTHOR

Antti Karttunen, May 13 2018

STATUS

approved

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Last modified October 26 06:08 EDT 2021. Contains 348257 sequences. (Running on oeis4.)