cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Previous Showing 11-17 of 17 results.

A139564 Number of primes < 10^(2n) - sum of primes < 10^n.

Original entry on oeis.org

0, 8, 169, 2371, 25059, 655974, 57509995, 1616756446, 28550646649, 442195486325, -2830020810398, 208945571225276, 11214798494956489, 307494657722438109, 8565236469871178577
Offset: 0

Views

Author

Cino Hilliard, Jun 11 2008

Keywords

Comments

Pi(x) is the prime counting function or the number of primes <= x. SumP(x) = sum of primes <= x.
For 2n = 20, we get a negative value -2830020810398. This seems strange to me.
I emailed Marc Deleglise on this. The links shows the Pi(x) and SumP(x) values.

Links

Crossrefs

Cf. A000720, A006880, A046731.

Formula

a(n) = A006880(2*n) - A046731(n). - Amiram Eldar, Jun 30 2024

Extensions

a(12)-a(14) from Amiram Eldar, Jun 30 2024

A145065 a(n) = sum of primes <= n*10^9.

Original entry on oeis.org

24739512092254535, 95673602693282040, 211169315361937456, 370412807102643725, 572840944428163514, 818062002011808625, 1105720106635140803, 1435543734762318281, 1807310307011831228, 2220822432581729238, 2675837181777714162, 3172247016440147644
Offset: 1

Views

Author

Jacques Tramu, Feb 20 2009

Keywords

Examples

			a(1) = 2 + 3 + 5 + ... + 999999937 = 24739512092254535.

Links

Crossrefs

Cf. A046731, A178221, A179649, A179660.

Programs

  • Mathematica
    k = 1; p = 2; s = 0; lst = {}; While[k < 10, While[p < 10^9*k, s = s + p; p = NextPrime@p]; k++; AppendTo[lst, s]; Print[{k - 1, s}]] (* Robert G. Wilson v, Jul 23 2010 *)
  • PARI
    A145065(n)=my(s=0);forprime(p=2,n*1e9,s+=p);s

Formula

a(n) = A179660(10*n). - R. J. Mathar, Aug 05 2010

Extensions

Extended by Charles R Greathouse IV, Jul 24 2009
Edited by N. J. A. Sloane, Jul 27 2010

A179660 Sum of the primes < n*10^8.

Original entry on oeis.org

279209790387276, 1075207199997334, 2367646772295462, 4146437503168147, 6404774487532576, 9137303389808024, 12340337443955708, 16010026481858292, 20143329357815162, 24739512092254535, 29795030058292249
Offset: 1

Views

Author

Robert G. Wilson v, Jul 23 2010

Keywords

Links

Crossrefs

Cf. A046731, A178221, A179649, A145065.

Programs

  • Mathematica
    k = 1; p = 2; s = 0; lst = {}; While[k < 12, While[p < 10^8*k, s = s + p; p = NextPrime@p]; k++; AppendTo[lst, s]; Print[{k - 1, s}]]

A077645 Sum of all primes having n decimal digits.

Original entry on oeis.org

17, 1043, 75067, 5660269, 448660141, 37096005486, 3165774592333, 276006465392920, 24460302301867259, 2196082920489474703, 199246255311162951776, 18234121474806961230363, 1680810854825228712978117, 155890014267359161122671527, 14534809256197269457684141345, 1361418455796443892761407164186
Offset: 1

Views

Author

Labos Elemer, Nov 18 2002

Keywords

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

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

A139043 Sum of the composite numbers <= 10^n.

Original entry on oeis.org

37, 3989, 424372, 44268603, 4545653462, 462450097976, 46796680005643, 4720790259612723, 475260488407745464, 47779177572418270761, 4798532922306255318985, 481564411447949294088622, 48300753556627220581110505, 4842410739289313059458438978, 485307601483092493601774297633
Offset: 1

Views

Author

Cino Hilliard, Jun 01 2008

Keywords

Comments

Conjecture: 10^n(10^n+1)/2 - 1 -(10^n)^2/(2*log(10^n)-1) -> a(n) as n -> infinity. Here (10^n)^2/(2*log(10^n)-1) is also conjectured to -> sum of primes < 10^n and is a very good approximation for the sum of primes < 10^n. We know that k^2/(2log(k)-1) diverges as k -> infinity. So if we can prove this limit approaches the sum of the primes <= k, then this implies the sum of primes is infinite and therefore the number of primes is infinite.

Examples

			The  sum of the composite numbers <= 10^1 is 4 + 6 + 8 + 9 + 10 = 37, the first entry in the sequence.

Links

Crossrefs

Cf. A046731.

Formula

a(n) = 10^n(10^n+1)/2 - 1 - A046731(n). Note: The b-file from Marc Deleglise was used for A046731(16) to A046731(20).

Extensions

a(13)-a(15) from Amiram Eldar, Jun 30 2024

A282246 Primes p such that the sum of all primes <= p has no prime divisor > p.

Original entry on oeis.org

2, 5, 11, 19, 23, 31, 41, 47, 59, 71, 83, 97, 101, 103, 109, 113, 127, 137, 157, 163, 167, 173, 179, 191, 197, 223, 227, 229, 233, 239, 241, 263, 269, 271, 317, 337, 349, 353, 367, 389, 401, 409, 433, 439, 449, 457, 461, 463, 467, 491, 521, 563, 571, 607, 613, 617, 631, 641, 653, 661, 701, 709, 719, 739, 757, 797
Offset: 1

Views

Author

Emmanuel Vantieghem, Feb 09 2017

Keywords

Comments

Number of terms < 10^k: 2, 12, 79, 523, 4124, 32678, 267850, etc. Compare these to A006880. - Robert G. Wilson v, Feb 09 2017
Primes p such that A006530(A007504(i)) <= p, where i is the index of p in A000040. - Felix Fröhlich, Feb 12 2017

Examples

			5 is in the sequence for the sum of all primes <= 5 is 10, and 10 has no prime divisor > 5.
17 is not in the sequence for the corresponding sum is 58 which has a prime divisor > 17.

Links

Crossrefs

Cf. A007504, A013916, A013917, A013918, A046731.

Programs

  • Mathematica
    p = s = 2; lst = {}; While[p < 1000, If[ FactorInteger[s][[-1, 1]] <= p, AppendTo[lst, p]]; p = NextPrime@ p; s = s + p]; lst (* Robert G. Wilson v, Feb 09 2017 *)
  • PARI
    isok(n) = isprime(n) && (vecmax(factor(sum(k=1, primepi(n), prime(k)))[,1]) <= n); \\ Michel Marcus, Feb 12 2017

A129697 Sum of isolated primes < 10^n.

Original entry on oeis.org

2, 577, 51896, 4009989, 345281974, 30126035380, 2657646397769, 238004015750349, 21505022353019864, 1960179022139638131, 180020101551309284879, 16639947666244921992434, 1546703841471331792075283, 144470014127904647017055666
Offset: 1

Views

Author

Cino Hilliard, Jun 08 2007

Keywords

Comments

Isolated primes are primes that are not twin prime members.

Examples

			The sum of the isolated primes < 100 = 2+23+37+47+53+67+79+83+89+97 = 577, the second term of this sequence.

Links

  • Cino Hilliard, Sum Isolated Primes. [broken link]
  • Cino Hilliard, Gcc code. [broken link]
  • Cino Hilliard, Sum of Isolated primes, message 38 in seqfun Yahoo group, providing code for gcc (needs formatting to become compilable), Jun 5, 2007. [Cached copy]

Crossrefs

Cf. A007510, A046731, A118552, A129542.

Programs

  • PARI
    isoprimes(n) = { local(j,s,x); for(j=1,n, s=0; forprime(x=2,10^j, if(!isprime(x-2)&&!isprime(x+2),s+=x) ); print1(s", ") ) }

Formula

a(n) = A046731(n) - A118552(n) + 5.

Extensions

a(13)-a(14) from Amiram Eldar, Jul 02 2024
Previous Showing 11-17 of 17 results.

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