[seqfan] Re: Is This Sequence Ever Negative?

[Richard Mathar]

lq> Date: Sun, 7 Mar 2010 06:46:13 -0800 (PST)
lq> From: Leroy Quet <q1qq2qqq3qqqq at yahoo.com>
lq> To: seqfan at seqfan.eu
lq> Subject: [seqfan]  Is This Sequence Ever Negative?
lq> 
lq> Is sequence A175150 ever negative?
lq> 
lq> In other words, is there an n where
lq> (the number of k <= n where d(k) < d(k-1))
lq> is greater than
lq> (the number of k <= n where d(k) > d(k-1)),
lq> 

This is essentially decomposing A000005, tau(.), in Dyck-Motzkin
terminology into levels L (tau(k)=tau(k+1)), down-steps D (tau(k) > tau(k+1)
and up-steps U (tau(k)< tau(k+1)). If one plots these cumulative counts
as a function of the upper limit n, in the sense of
  n versus #D(k<=n) versus #L(k<=n) versus #U(k<=n),
the number of up-steps is larger than the number of down-steps until roughly 629,
but then many changes of this type of "tau-races" follow. In a table with
(i) n, (ii) number of D for k<=n, (iii) number of L for k<=n, (iv) number of D for k<=n,
(v) difference #U-#D we observe this as sign changes in the last column:

100 40 15 44 4
200 82 29 88 6
300 123 45 131 8
400 167 57 175 8
500 214 63 222 8
600 262 72 265 3
700 307 84 308 1
800 351 95 353 2
900 398 105 396 -2
1000 442 118 439 -3
1100 486 133 480 -6
1200 531 144 524 -7
1300 572 158 569 -3
1400 617 169 613 -4
1500 657 184 658 1
1600 702 195 702 0
1700 746 206 747 1
1800 791 216 792 1
1900 836 228 835 -1
2000 879 240 880 1
2100 926 249 924 -2
2200 972 261 966 -6
2300 1017 271 1011 -6
2400 1055 284 1060 5
2500 1104 293 1102 -2
2600 1146 306 1147 1
2700 1187 322 1190 3
2800 1228 336 1235 7
2900 1277 344 1278 1
3000 1322 356 1321 -1
3100 1368 365 1366 -2
3200 1413 375 1411 -2
3300 1457 385 1457 0
3400 1503 396 1500 -3
3500 1549 405 1545 -4
3600 1595 415 1589 -6
3700 1639 430 1630 -9
3800 1683 440 1676 -7
3900 1727 449 1723 -4
4000 1770 462 1767 -3
4100 1815 471 1813 -2
4200 1861 479 1859 -2
4300 1905 493 1901 -4
4400 1952 499 1948 -4
4500 1994 513 1992 -2
4600 2036 529 2034 -2
4700 2079 542 2078 -1
4800 2124 550 2125 1
4900 2164 563 2172 8
5000 2210 572 2217 7
5100 2253 583 2263 10
5200 2298 588 2313 15
5300 2339 602 2358 19

To be precise, the first transition is monitored from tau(1) to tau(2). Details
would change if one would start from tau(0).

The last column becomes +11 for values near 440, -9 for values near 1130,
+4 near 2000, -8 near 2273, +20 near 5230, -4 near 5880 etc.

On a side note, I prefer to write tau(.), never d(.), for the number of divisors
of an argument. It is a good idea to stick to sigma(.), tau(.),
phi(.), prime(.), Fibonacci(.), rad(.), mu(.), pi(.), binomial(.,.) for the basic
number-theoretical functions. If one needs to use *a* divisor
or *a* prime, one would likely use d, like in sum_{d|n}, or p, q, r etc for
individual primes.

Maple generating the table above (as usual a rich reservoir to discuss my coding styles etc):
lvl := vector(3,[0,0,0]) :
for i from 2 to 10000 do
        if numtheory[tau](i) > numtheory[tau](i-1) then
                lvl[3] := lvl[3]+ 1 ;
        elif numtheory[tau](i) = numtheory[tau](i-1) then
                lvl[2] := lvl[2]+ 1 ;
        else
                lvl[1] := lvl[1]+ 1 ;
        end if ;
        if i mod 100 = 0 then
                printf("%d %d %d %d %d\n",i,lvl[1],lvl[2],lvl[3],lvl[3]-lvl[1]) ;
        end if;
end do: