A270434 - OEIS

A270434

a(n) = A270432(n) - A270433(n).

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

The first negative term occurs at a(223) = -1.

After a(2457) = -1 the sequence dips next time to the negative side at n=218351.

No other negative terms after a(2346395) = -1 in range 1 .. 2^25.

In range 1..(2^25) the maximum value is a(23963418) = 8326 and there are 1252224 negative terms in that range (less than 4%).

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..35000

FORMULA

a(n) = A270432(n) - A270433(n).

MATHEMATICA

nn = 200; f[n_] := (Times @@ Power[If[# == 1, 1, NextPrime@ #] & /@ First@ #, Last@ #] + 1)/2 &@ Transpose@ FactorInteger@ n; g[n_] := Times @@ Power[If[# == 1, 1, NextPrime[#, -1]] & /@ First@ #, Last@ #] &@ Transpose@ FactorInteger[2 n - 1]; s = Select[Range@ nn, Xor[EvenQ@ f@ #, OddQ@ g@ #] &]; t = Select[Range@ nn, Xor[EvenQ@ f@ #, EvenQ@ g@ #] &]; Table[Count[s, k_ /; k <= n] - Count[t, k_ /; k <= n], {n, nn/2}] (* Michael De Vlieger, Mar 17 2016 *)

PROG

(PARI)

default(primelimit, 2^30);

A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From Michel Marcus

A048673(n) = (A003961(n)+1)/2;

A064989(n) = {my(f); f = factor(n); if((n>1 && f[1, 1]==2), f[1, 2] = 0); for (i=1, #f~, f[i, 1] = precprime(f[i, 1]-1)); factorback(f)};

A064216(n) = A064989((2*n)-1);

t=0; for(n=1, 2^25, if(!((A048673(n)+A064216(n))%2), t++, t--); write("b270434.txt", n, " ", t));

(Scheme) (define (A270434 n) (- (A270432 n) (A270433 n)))

CROSSREFS

Cf. A270430, A270431, A270432, A270433.

Cf. A270435 (positions of zeros).