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A255251 Decimal expansion of d(n) = log(n) + log(log(n)) - prime(n)/n at n = 2688, a (local?) maximum. 0
9, 8, 3, 4, 4, 0, 3, 0, 8, 5, 2, 3, 4, 6, 4, 2, 6, 0, 1, 0, 3, 9, 2, 0, 9, 0, 8, 9, 5, 5, 6, 1, 2, 8, 2, 6, 7, 5, 6, 1, 6, 9, 3, 9, 0, 7, 2, 6, 4, 2, 0, 9, 3, 1, 6, 9, 2, 6, 2, 7, 3, 4, 4, 2, 2, 9, 2, 8, 5, 9, 2, 0, 3, 3, 1, 9, 0, 0, 5, 3, 2, 4 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

Variations on the Prime Number Theorem, as enhanced in recent years (e.g., by Pierre Dusart), have proved: prime(n) > n*(log(n) + log(log(n)) - 1) for n>=2, and prime(n) <  n*(log(n) + log(log(n))) for n>=6.

This implies d(n), defined above, lies between 0 and 1 for n>=6.

d(2688) is, at least, a "local" maximum, checked through n=5,000,000,000,000.

Prime(2688) = 24137.

This prime owes its modest distinction to the fact that the 48 primes before it have average prime gaps of ~8, below the average gap of ~10, which is normal for this region if n*(log(n) + log(log(n))-1) is a guide. It is this cumulative effect that matters. Likewise, "local" minimums of d(n) at much larger n, discussed below, are due to the accumulated deficits from a preceding region of above normal prime gaps.

Checking maximums and minimums on d(n) is simplified greatly by the fact that as n increases both the shorter and longer range rates of variation in d(n) decrease. This is seen below in the summary of means and standard deviations in d(n). These are calculated on all values in a range of size 2*sqrt(n) centered on upper end of each 10^j bracket (e.g., from 90 to 110 for j = 2 below) where the value following "+-" is one standard deviation:

From n=6.....10^2  d(n) increases rapidly from ~.2 to .78 +- 6*10^-2

From n=10^2..10^3  d(n) increases slowly to 0.92 +- 1*10^-2

From n=10^3..10^4  d(n) peaks at d(2688)=0.98344.. ending at 0.955 +- 2*10^-3

From n=10^4..10^5  d(n) is nearly flat at  ~0.9600 +- 5*10^-4

From n=10^5..10^6  d(n) decreases to 0.9555 +- 1*10^-4

From n=10^6..10^7  d(n) increases to 0.95556 +- 2*10^-5

From n=10^7..10^8  d(n) decreases to 0.953401 +- 6*10^-6

From n=10^8..10^9  d(n) decreases to 0.952760 +- 1*10^-6

From n=10^9..10^10 d(n) reaches a "broad minimum" between 2*10^9 and 5*10^9, with a mean in this range of ~0.95265 +- 8*10^-6. This has higher deviations since four different ranges of 1000 were averaged to cover this broad minimum. In this broad range there is a single point "local" minimum of d(n)=0.95258698490936147316314.. at n=2128963733 and p(n)=50229461677. A d(n) value this low has not occurred since n=85050. NOTE: There is one "nearby" value of d(n) almost as low, d(2128964024)=0.952586985014, and two more, slightly higher, at d(4468035499) = 0.952597.., and d(4508851221) = 0.952595..

Then as n -> 10^10, d(n) increases to 0.9526885 +-2*10^-7.

From n=10^10..10^11 d(n) increases to 0.95309061 +- 5*10^-8.

From 10^11 to 10^12 d(n) increases to 0.953735934 +- 6*10-9.

From 10^12 to 5*10^12 d(n) increases to 0.954283767 +- 3*10-9.

The rate of decrease in standard deviations of d(n) appears to be around (1/n)^(2/3) on average over these 12 orders of magnitude. But the rate of decrease also appears to be accelerating. Thus a variable exponent may be needed.  A formula such as (1/n)^(1-1/log(1+log(n))), appears to be an equally good fit, as there is some variation in standard deviations (e.g., by a factor of 2 or 3) depending on where the sample range is selected for a given order of magnitude.

Also note that standard deviations are less when the sample range is reduced, because mid-range to long range "drifts" drive standard deviation much more than point-to-point fluctuations. This is obvious on plots of d(n).

The impact of record large prime gaps at particular n values (as listed below), on the variations in d(n) were also examined.  A comparison of d(n)-d(n+1) with the standard deviation of d(n), calculated on a range of +/-sqrt(n), centered around those same n values, shows the following:

At the smaller n values of 99, 154, 189, 217, 1183, 1831, 2225, 3385, 14357, 30802, 31545, 40933, 103520, 104071, 149689, 325852, 1094422, 1319945, 2850174, where p(n+1) - p(n) is a record breaking gap, the single point variation of d(n)-d(n+1) is usually 2 to 3 times greater than one standard deviation at the lower values here or about equal to one standard deviation (+-25%) at highest and some of the mid-sized n values here.

At larger n values for record breaking prime gaps, such as 6957876, 981765347, 1094330259, 11992433551, 50070452577, 94906079600, 473258870471, d(n)-d(n+1) falls from ~1/2 to ~1/10 of one standard deviation, and will continue to fall because the size of record breaking prime gaps grows increasingly smaller relative to their corresponding primes, and n values.

Thus even record breaking prime gaps become minor "noise" compared to the variations in d(n) due to longer run fluctuations in average prime gaps, even as those long range (i.e., +-sqrt(n)) fluctuations are also decreasing rapidly.

Conjectures:

1.)  d(n) values for n > 2688 are bounded from above by d(2688) = 0.983440..

2.)  d(n) values for n > 2128963733 are bounded from below by d(2128963733) = 0.952586984909361..

3.)  There are many such d(n) values, at specific n, that more tightly bound d(n) from above and below, forming a "funnel" on the values of d(n) as n -> infinity.

These bounds are found by applying the rule that such bounds must alternate between being upper vs. lower bounds:

n =  2688       d(n) = .983440...  upper bound  (this entry)

n =  3428       d(n) = .914144...  lower bound

n =  6046       d(n) = .978971...  upper bound

n =  6934       d(n) = .930067...  lower bound

n = 30490       d(n) = .971848...  upper bound

n = 38685       d(n) = .947442...  lower bound

n = 50461       d(n) = .967684...  upper bound

n = 85021       d(n) = .951368...  lower bound

n = 92607       d(n) = .966237...  upper bound

n = 2128963733  d(n) = .952586...  lower bound

These values would seem arbitrary except that each is the minimum (maximum) for d(n) at all n values following the previous d(n) maximum (minimum), where the first minimum, at d(2) = -1.17, is omitted from the list above.

No subsequent d(n) values, checked for n up 5*10^12, violate any of these bounds.

The next upper bound (or maximum) has not been found yet.

Averaging these upper and lower bounds, or the last two, may indicate the "center of the funnel", without implying convergence, since d(n) will always fluctuate. The average of them all is 0.956379. The average of last two bounds is 0.959412.

LINKS

Table of n, a(n) for n=0..81.

P. Dusart, The kth prime is greater than k(ln k + ln ln k-1) for k>=2, Mathematics of Computation 68 (1999), pp. 411-415.

EXAMPLE

0.98344030852346426010...

MATHEMATICA

RealDigits[N[Log[2688] + Log[Log[2688]] - Prime[2688]/2688, 100]]

CROSSREFS

Sequence in context: A139342 A210649 A144666 * A224236 A118427 A199170

Adjacent sequences:  A255248 A255249 A255250 * A255252 A255253 A255254

KEYWORD

nonn,cons

AUTHOR

Richard R. Forberg, Feb 18 2015

STATUS

approved

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Last modified August 23 22:20 EDT 2019. Contains 326254 sequences. (Running on oeis4.)