A006988 -id:A006988 - cp's OEIS frontend

A092252 a(n) = prime(prime(10^n)).

3, 109, 3911, 80917, 1366661, 20491057, 285058399, 3767321791, 47991893393, 594421377761, 7201814452873, 85713609222697, 1005238339412819, 11644468481142713, 133472665317708923, 1516047373452105311

Offset: 0

Cino Hilliard, Feb 19 2004

Kim Walisch, Fast C++ prime counting function implementation (primecount).

Cf. A000040, A006450, A006988, A096359.

Table[Prime[Prime[10^n]], {n, 0, 10}] (* Amiram Eldar, Jun 29 2024 *)

a(n) = prime(prime(10^n)); \\ Amiram Eldar, Jun 29 2024

a(n) = A006450(10^n) = A000040(A006988(n)). - Amiram Eldar, Jun 29 2024

a(9)-a(10) from Dennis Kluk, Nov 27 2005

a(11)-a(15) using Kim Walisch's primecount added by Amiram Eldar, Jun 29 2024

A095178 Final digit of the (10^n)-th prime.

Original entry on oeis.org

2, 9, 1, 9, 9, 9, 3, 3, 3, 9, 3, 7, 3, 1, 7, 7, 9, 9, 1, 9, 7, 3, 9, 3, 3

Offset: 0

Cino Hilliard, Jun 21 2004

Cf. A006988, A010879.

a(n)=prime(10^n)%10 \\ Charles R Greathouse IV, Dec 29 2012

a(n) = A010879(A006988(n)).

a(19)-a(24) added from the b-file at A006988 by Amiram Eldar, Mar 24 2021

A099261 Length in bits of (10^n)-th prime number.

Original entry on oeis.org

2, 5, 10, 13, 17, 21, 24, 28, 31, 35, 38, 42, 45, 49, 52, 56, 59, 62, 66, 69, 73, 76, 79, 83, 86, 89, 93, 96, 100, 103, 106, 110, 113, 116, 120, 123, 127, 130, 133, 137, 140, 143, 147, 150, 153, 157, 160, 163, 167, 170, 173, 177, 180, 184, 187, 190, 194, 197, 200, 204

Offset: 0

Rick L. Shepherd, Oct 11 2004

			a(1) = 5 because A006988(1) = prime(10^1) = 29 = 11101 (base 2) has five bits.

Pierre Dusart, Estimates of Some Functions Over Primes Without R.H.

Cf. A006988 ((10^n)-th prime), A006880 (pi(10^n)), A007053 (pi(2^n)), A099260 (decimal digit lengths).

a(n)=if(n<3,return([2,5,10][n+1]));my(l=n*log(10),ll=log(l),x=n*log(10)/log(2),lb=ceil(x+log(l+ll-1+(ll-2.2)/l)/log(2)),ub=ceil(x+log(l+ll-1+(ll-2)/l)/log(2)));if(lb==ub,lb,error("Cannot determine a("n")"))

Extension, program, and reference from Charles R Greathouse IV, Aug 03 2010

A105463 The 10^n-th irregular prime.

Original entry on oeis.org

37, 257, 1669, 22349, 294001, 3553267, 41892163, 481303351

Offset: 0

Robert G. Wilson v, Apr 07 2005

Cf. A006988, A011557, A105468.

ip={ the list of irregular primes to 12 million }; Table[ ip[[10^n]], {n, 0, 5}]

a(n) = A000928(A011557(n)) = A000928(10^n). - Amiram Eldar, Mar 05 2019

Offset changed and a(6)-a(7) added by Amiram Eldar, Mar 05 2019

A105468 Number of irregular primes less than or equal to the 10^n-th prime.

Original entry on oeis.org

0, 30, 392, 3935, 39400, 393737, 3933421, 39339651

Offset: 1

Robert G. Wilson v, Apr 07 2005

Limit_{n->inf.} a(n)/10^n -> 1-e^(-1/2).

Cf. A000928, A006988, A105463.

ip={ the list of irregular primes to 12 million }; Table[ Length[ Select[ip, # <= Prime[10^n] &]], {n, 5}]

Data corrected and a(6)-a(8) added by Amiram Eldar, Mar 05 2019

A119780 a(n) = (100^n)-th prime.

Original entry on oeis.org

2, 541, 104729, 15485863, 2038074743, 252097800623, 29996224275833, 3475385758524527, 394906913903735329, 44211790234832169331, 4892055594575155744537, 536193870744162118627429, 58310039994836584070534263

Offset: 0

Jim Snow (jsnow(AT)mitre.org), Jun 22 2006

nonn

UTM, The Nth Prime Page.

Bisection of A006988.

a(n) = A006988(2*n).

a(7)-a(9) copied from A006988. - Max Alekseyev, May 11 2009

a(10)-a(12) copied from A006988. - Chai Wah Wu, Sep 19 2018

A120843 Initial digit of the (10^n)-th prime.

Original entry on oeis.org

2, 2, 5, 7, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 6, 6, 6, 6, 7, 7, 7, 7, 7, 8, 8, 8, 8, 9, 9, 9, 9, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2

Offset: 0

Cino Hilliard, Aug 18 2006

The algorithm in the PARI program approximates the (10^n)-th prime to an accuracy of roughly n/2 + 1 digits. As a result, we are almost certain to get the initial digit correctly. It remains to prove this. Since the Riemann approximation of pi(x) is used as a boundary in the exponential bisection routine, it would seem that a proof is possible in view of the fact that bisection almost always guarantees convergence. "Almost" is an appropriate term here, as will be demonstrated when we let the initial parameter r2 = 1. For example, we can toggle print(dx) to check the convergence. For primex(1e116), we get 9.999999999999999999999999970 E115.

			The (10^3)-th prime is 7919, so a(3)=7.

Robert G. Wilson v, Table of n, a(n) for n = 0..1000

Cf. A000030, A006988, A077648.

f[n_] := RealDigits[ n (Log[n] + Log[Log[n]] - 1 + (Log[Log[n]] - 2)/Log[n] - (Log[Log[n]]^2 - 6 Log[Log[n]] + 11)/(2 Log[n]^2)), 10, 10][[1, 1]]; f[1] = f[10] = 2; f[100] = 5; Array[ f[10^#] &, 105, 0] (* Robert G. Wilson v, Jan 15 2017 *)

g(n) = print1(2", "); for(x=1, n, y=Vec(Str(primex(10^x))); print1(y[1]", "))
primex(n) = /* Efficient Algorithm to accurately approximate the n-th prime */ { local(x, px, r1, r2, r, p10, b, e); b=10; /*Select base*/ p10=log(n)/log(10); /*p10=pow of 10 n is to adjust in b^p10*/ if(Rg(b^p10*log(b^(p10+1)))< b^p10, m=p10+1, m=p10); r1 = 0; r2 = 2.718281828; /*Real kicker. if 1, it fails at 1e117*/ for(x=1, 360, r=(r1+r2)/2; px = Rg(b^p10*log(b^(m+r))); if(px <= b^p10, r1=r, r2=r); r=(r1+r2)/2.; ); (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<10^100*r, s=s+r/zeta(n+1)/n; n=n+1; r=r*L/n); (s) }

a(n) = most significant digit of A006988(n). - Robert G. Wilson v, Jan 17 2017

a(n) = A000030(A006988(n)). - Michel Marcus, Jan 18 2017

A215609 Smallest prime p congruent to 1 modulo prime(10^n).

Original entry on oeis.org

3, 59, 9739, 63353, 209459, 15596509, 154858631, 1794246731, 4076149487, 45603526979, 11092303227413, 93864728285579, 1319833868136653, 11656098322067917, 27803086068196217, 1781976386163140977, 32382366940106296979, 100447117955224696057, 707388643757314709297

Offset: 0

Zak Seidov, Aug 17 2012

nonn

			a(0) = 3 because p = prime(1) = 2, 3 = 1+p.
a(1) = 59 because p = prime(10) = 29, 59 = 1+2*p.
a(2) = 9739 because p = prime(10) = 29, 59 = 1+18*p.

Amiram Eldar, Table of n, a(n) for n = 0..24

Cf. A035095, A006988.

a[n_]:=(If[n<2,3,p=Prime[n];r=2p+1;While[!PrimeQ[r],r=r+2p];r]);Table[a[10^n],{n,0,12}]

a(n) = A035095(10^n).

a(13)-a(18) from Amiram Eldar, Apr 30 2024

A110900 Sum of the lesser of twin primes <= prime(10^n).

Original entry on oeis.org

65, 4803, 584914, 59273184, 6098138012, 616905504491, 62144914085485, 6267868143394323, 630501829011138256, 63335169132014778363

Offset: 1

Cino Hilliard, Sep 20 2005

prime(n) = n-th prime number.

After the 3rd term, the next term is roughly 100 times the previous term.

			3, 5, 11, 17, and 29 are lesser members of twin primes <= prime(10^1) = 29. These add up to 65, the first term of this sequence.

Cf. A001359, A006988, A007508.

lista(pmax) = {my(s = 0, pow = 10, prev = 2, k = 1); forprime(p = 3, pmax, k++; if(p == prev + 2, s += prev); if(k > pow, print1(s, ", "); pow *= 10); prev = p);} \\ Amiram Eldar, Jun 30 2024

a(9)-a(10) from Amiram Eldar, Jun 30 2024

A117324 Prime(10^n) modulo semiprime(10^n).

Original entry on oeis.org

2, 3, 227, 729, 22965, 380555, 156346, 10920166, 202913258, 2973399074, 39284376410, 489544827463, 5874954672992

Offset: 0

Jonathan Vos Post, Mar 08 2006

			prime(10^0) modulo semiprime(10^0) = 2 mod 4 = 2.
prime(10^1) modulo semiprime(10^1) = 29 mod 26 = 3.
prime(10^2) modulo semiprime(10^2) = 541 mod 314 = 227.

a(n) = A000040(10^n) modulo A001358(10^n). a(n) = A117322(10^n). a(n) = A006988(n) modulo A114125(n).

a(12) from Zak Seidov

cp's OEIS Frontend

A092252 a(n) = prime(prime(10^n)).

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A095178 Final digit of the (10^n)-th prime.

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A099261 Length in bits of (10^n)-th prime number.

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A105463 The 10^n-th irregular prime.

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A105468 Number of irregular primes less than or equal to the 10^n-th prime.

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A119780 a(n) = (100^n)-th prime.

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A120843 Initial digit of the (10^n)-th prime.

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A215609 Smallest prime p congruent to 1 modulo prime(10^n).

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