A006988 -id:A006988 - cp's OEIS frontend

A123233 Difference between the (10^n)-th prime and the Riemann-Gram approximation of the (10^n)-th prime.

1, 0, 5, -4, -39, -24, 1823, -6566, -1844, -34087, 84846, -449836, -1117632, -3465179, -1766196, -11290074, 105510354, -208774399, 704933861

Offset: 0

Cino Hilliard, Oct 06 2006

The algorithm in the PARI script below produces the 10^n-th prime accurate to first n/2 places. Conjecture: The sign of the terms in this sequence changes infinitely often. Based on the small sample presented here, it appears the negative terms occur much more often.

			a(1) = prime(10) - primeGR(10) = 29 - 29 = 0.

Cf. A006988, A121046.

primeGR(n) =
\\ A good approximation for the n-th prime number using
\\ the Gram-Riemann approximation of Pi(x)
{ local(x,px,r1,r2,r,p10,b,e); b=10; p10=log(n)/log(10); if(Rg(b^p10*log(b^(p10+1)))< b^p10,m=p10+1,m=p10); r1 = 0; r2 = 7.18281828; for(x=1,400, r=(r1+r2)/2; px = Rg(b^p10*log(b^(m+r))); if(px <= b^p10,r1=r,r2=r); r=(r1+r2)/2; ); floor(b^p10*log(b^(m+r))+.5); }
Rg(x) =
\\ Gram's Riemann's Approx of Pi(x)
{ local(n=1,L,s=1,r); L=r=log(x); while(s<10^40*r, s=s+r/zeta(n+1)/n; n=n+1; r=r*L/n); (s) }

prime(10^x)-primeRG(10^x), where prime(n) is the n-th prime and primeRG(n)is an approximation of the n-th prime number based on an exponential bisection routine that uses the Riemann-Gram approximation of Pi(x). The flow of the routine is evident in the PARI script below.

a(n) = A006988(n) - A121046(n) for n >= 1. - Amiram Eldar, Jul 04 2024

a(17)-a(18) from Amiram Eldar, Jul 04 2024

A129870 Difference between the (10^n)-th and the (10^n-1)-th prime.

Original entry on oeis.org

6, 18, 12, 6, 20, 6, 2, 4, 12, 12, 20, 12, 22, 26, 30, 6, 72, 152, 72, 24, 30, 96, 124, 50

Offset: 1

Cino Hilliard, Jun 04 2007

It is interesting that the number 2 occurs deep into the sequence indicating a twin prime pair. It is reasonable to ask if this will ever occur again. Similarly, the analogous sequence A074383, "Difference between (1+10^n)-th and (10^n)-th primes" has 2 occurring shallow into the sequence. It is reasonable to ask if the number 2 will ever occur again in that sequence. The link provides an excellent algorithm, primex(n), that I developed to find the n-th prime using Gram's approximation of Riemann's approximation R(x) for Pi(x). Primex(n) will give about n/2 exact digits for prime(n). For A006988 (18), primex(18) is 44211790234127235469.62904554...This is only as good as R (x) but nevertheless is superior to the exact formulas out there from a practical stand point. If we apply the code gpx(n) = for(x=1,n,y=nextprime(primex(10^x))-nextprime (primex(10^x-1));print1(floor(y)",")), we will get the erratic concoction 2,0,8,14,22,28,26,0,72,18,22,0,0,0,0,0,32,0,80,78,60,0 as an analytical counterpart of the sequence given.

			The (10^18)-th prime or A006988(18) = 44211790234832169331.
Using PARI, precprime(A006988(18)-1) = 44211790234832169179.
The difference is a(18) = 152.

C. Hilliard, Nth prime approx [broken link].

Cf. A006988, A074383.

a(n) = A006988(n)-A151799(A006988(n))

a(19) from Max Alekseyev, May 13 2009
a(20) from Max Alekseyev, May 30 2013
a(21),a(22) from Max Alekseyev, Dec 04 2014
a(23)-a(24) from Chai Wah Wu using terms in A006988, Sep 18 2018

A229661 Rounded percentage of primes less than 10^n.

Original entry on oeis.org

0, 40, 25, 17, 12, 10, 8, 7, 6, 5, 5, 4, 4, 3, 3, 3, 3, 3, 2, 2, 2, 2, 2, 2, 2, 2

Offset: 0

Jean-François Alcover, Sep 27 2013

Please refer to the explanations and comments given in A006879 and A006880.

			There are 4 primes less than 10 (i.e., 2, 3, 5, 7), so a(1) = 40.

Cf. A000720, A006879, A006880, A007053, A040014, A006988.

a[n_] := PrimePi[10^n]/10^(n-2) // Round;
(* or *) a[n_] := A006880[[n+1]]/10^(n-2) // Round; Table[Print["10^", n, " ", a[n], "%"]; a[n], {n, 0, 25}] (* Jean-François Alcover, Sep 27 2013 *)

a(n) = pi(10^n)/10^(n-2) rounded.

A235642 a(n) = prime(10^n) mod 10^n.

Original entry on oeis.org

0, 9, 41, 919, 4729, 99709, 485863, 9424673, 38074743, 801763489, 2097800623, 60727302517, 996224275833, 3780508946331, 75385758524527, 124508045065437, 4906913903735329, 85296581467695669, 211790234832169331, 5675465116607065549

Offset: 0

Zak Seidov, Apr 20 2014

nonn

Last (n+1) digits of A006988(n).

Cf. A004648, A006988.

a(n) = prime (10^n) % 10^n; \\ Michel Marcus, Apr 21 2014

a(n) = A004648(10^n) = A006988(n) mod 10^n.

A297424 a(n) = tau((10^n)-th prime) where tau(n)=A000594(n) is Ramanujan's tau function.

Original entry on oeis.org

-24, 128406630, -1695266465052058, -4467161474022023509680, 2643202687128887204371152330, 5631587063815097155948902224731910, -6022712388820421179671063354886818730888, 3519878895631571325515625172934456394359869922, 452493700789420934344344052367209646628330983653992

Offset: 0

Seiichi Manyama, Dec 30 2017

sign

Olivier Rozier, Ramanujan's tau function

Cf. A000594, A006988, A076847, A297127.

PARI
```
{a(n) = ramanujantau(prime(10^n))}
```

a(n) = A076847(10^n) = A000594(A006988(n)).

A301599 Numbers k at which the ratio r(k) = (k-th prime) / (average of first k primes) reaches a record high.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 9, 10, 12, 17, 25, 31, 35, 48

Offset: 1

Jon E. Schoenfield, Mar 24 2018

Equivalently, define the function f(k) = k*prime(k)/Sum_{j=1..k} prime(j); sequence lists numbers k such that f(k) > f(m) for all m < k.
a(14)=48 is the final term. Beyond k=48, r(k) decreases fairly smoothly (although nonmonotonically); see the Example section.
For m = 4..18, the first k > 48 at which r(k) < 2 - 1/m is 50, 53, 61, 775, 2678, 8973, 23483, 63535, 159863, 431988, 1091840, 2753459, 7186422, 18479367, 47260890, respectively. Does lim_{k->inf} r(k) equal 2? - Jon E. Schoenfield, Mar 27 2018

			The table below shows k, prime(k), the sum and average of the first k primes, and r(k), for each number k in the sequence, and also for k = 100, 1000, ..., 10^7.
.
   n|   a(n)=k  prime(k)             sum         avg    r(k)
  --+--------------------------------------------------------
   1|        1         2               2        2.000 1.00000
   2|        2         3               5        2.500 1.20000
   3|        3         5              10        3.333 1.50000
   4|        4         7              17        4.250 1.64706
   5|        5        11              28        5.600 1.96429
   6|        7        17              58        8.286 2.05172
   7|        9        23             100       11.111 2.07000
   8|       10        29             129       12.900 2.24806
   9|       12        37             197       16.417 2.25381
  10|       17        59             440       25.882 2.27955
  11|       25        97            1060       42.400 2.28774
  12|       31       127            1720       55.484 2.28895
  13|       35       149            2276       65.029 2.29130
  14|       48       223            4661       97.104 2.29650
           100       541           24133      241.330 2.24174
          1000      7919         3682913     3682.913 2.15020
         10000    104729       496165411    49616.541 2.11077
        100000   1299709     62260698721   622606.987 2.08753
       1000000  15485863   7472966967499  7472966.967 2.07225
      10000000 179424673 870530414842019 87053041.484 2.06110

Cf. A000040 (primes), A007504 (sum of first n primes), A006988 ((10^n)-th prime), A099824 (sum of first 10^n primes).

cp's OEIS Frontend

A123233 Difference between the (10^n)-th prime and the Riemann-Gram approximation of the (10^n)-th prime.

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A129870 Difference between the (10^n)-th and the (10^n-1)-th prime.

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A229661 Rounded percentage of primes less than 10^n.

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A235642 a(n) = prime(10^n) mod 10^n.

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A297424 a(n) = tau((10^n)-th prime) where tau(n)=A000594(n) is Ramanujan's tau function.

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A301599 Numbers k at which the ratio r(k) = (k-th prime) / (average of first k primes) reaches a record high.

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