A077645 Sum of all primes having n decimal digits.

17, 1043, 75067, 5660269, 448660141, 37096005486, 3165774592333, 276006465392920, 24460302301867259, 2196082920489474703, 199246255311162951776, 18234121474806961230363, 1680810854825228712978117, 155890014267359161122671527, 14534809256197269457684141345, 1361418455796443892761407164186

Offset: 1

Author

Labos Elemer, Nov 18 2002

Keywords

• base
• nonn

Comments

Also the sum of the primes between 10^(n-1) and 10^n.

a(12) to a(20) were computed from A046731(12)-A046731(11) to A046731(20)-A046731(19). - Cino Hilliard, May 31 2008

A good estimate for the sum of the primes < k is k^2/(2*log(k)-1). Using this formula, a(20)~(10^20)^2/(2*log(10^20)-1) -(10^19)^2/(2*log(10^19)-1) = 108609290005707493265628731014013409909. The relative error this formula produces for the last 5 terms is a(16): -0.00019454, a(17): -0.00017176, a(18): -0.00015275, a(19): -0.00013674, a(20): -0.00012312. - Cino Hilliard, May 31 2008

Examples

			a(1) = 2 + 3 + 5 + 7 = 17, sum of four 1-digit primes.
		

Links

• Hugo Pfoertner, Table of n, a(n) for n = 1..26 (terms 1..20 from Cino Hilliard).
• Cino Hilliard, Count,Sum primes in a range Win32 Gcc+Gmp.

Crossrefs

Cf. A000040, A000720, A006879, A007504.

Programs

Maple

a:=proc(n) local tot,b,j: tot:=nextprime(10^(n-1)): b:=nextprime(10^(n-1)): for j while nextprime(b) < 10^n do tot:=tot+nextprime(b): b:=nextprime(b) end do:tot end proc: # Emeric Deutsch, Oct 08 2007

Mathematica

Prepend[Table[Apply[Plus,Table[Prime[w],{w,PrimePi[10^(n-1)]+1,PrimePi[10^n]}]],{n,2,7}],17] (* corrected by Ivan N. Ianakiev, Aug 12 2016 *)

Formula

a(n) = Sum_{10^(n-1) <= p <= 10^n, p prime} p = A007504(A000720(10^n)) - A007504(A000720(10^(n-1))).

Extensions

2 more terms from Lior Manor, Sep 11 2007

Corrected and extended by Emeric Deutsch, Oct 08 2007

More terms from Cino Hilliard, May 31 2008