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A260981 Primes p that are equal to the sum of the first k primes where p=prime(prime(k)). 0
5, 17, 41 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Terms listed are the only three primes p found to satisfy the condition that p = prime(m) = Sum_{i=1..k} prime(i) where m=prime(k).

From Jon E. Schoenfield, Aug 19 2015: (Start)

Let S(k) be the sum of the first k primes, and let PP(k) = prime(prime(k)); then the terms of the sequence are the values of prime(prime(k)) at those values of k at which S(k) = PP(k). (This occurs at k = 2, 4 and 6.)

Given the behavior of the ratio S(k)/PP(k) over the range of values of k shown in the table below, it seems very unlikely that this ratio will return to 1 for any k beyond the values that have been tested, and thus very likely that a(3) = 41 = PP(6) is the final term in the sequence:

     k        S(k)         PP(k)     S(k)/PP(k)

  ======  ===========    ========  ==============

       1            2           3     0.666666...

       2            5  =        5     1

       3           10          11     0.909090...

       4           17  =       17     1

       5           28          31     0.903225...

       6           41  =       41     1

       7           58          59     0.983050...

       8           77          67     1.149253...

       9          100          83     1.204819...

      10          129         109     1.183486...

     ...

     100        24133        3911     6.170544...

    1000      3682913       80917    45.514700...

   10000    496165411     1366661   363.049367...

  100000  62260698721    20491057  3038.432752... (End)

LINKS

Table of n, a(n) for n=1..3.

EXAMPLE

k=3: prime(3) = 5 = 2+3 = prime(1) + prime(2).

k=7: prime(7) = 17 = 2+3+5+7 = prime(1) + prime(2) + prime(3) + prime(4).

k=13: prime(13) = 41 = 2+3+5+7+11+13 = prime(1) + prime(2) + prime(3) + prime(4) + prime(5) + prime(6).

PROG

(C#) // The code is provided by Ali Adams (www.heliwave.com)

using System; using System.Collections.Generic; using System.Text; namespace PrimeSum { class Program { static void Main(string[] args) { Console.WriteLine("Prime\tP\tSum"); // 17 = P7 = Sum[2..7] for (int i = 0; i < 1000000; i++) { // prime = 17 long prime = Numbers.Primes[i]; // i = 6 // order = 7 int order = i + 1; if (Numbers.IsPrime(order)) { int index = Numbers.IndexOfPrime(order); StringBuilder str = new StringBuilder(); long sum = 0L; for (int j = 0; j < index; j++) { long p = Numbers.Primes[j]; sum += p; str.Append(p + "+"); } str.Remove(str.Length - 1, 1); if (sum == prime) { Console.WriteLine(prime + "\t" + order + "\t" + str.ToString()); } } } Console.WriteLine("Press any key to exit ..."); Console.ReadKey(); } } }

CROSSREFS

Cf. A006450, A007504, A013918.

Sequence in context: A147035 A146134 A011931 * A078866 A144620 A217622

Adjacent sequences:  A260978 A260979 A260980 * A260982 A260983 A260984

KEYWORD

nonn,fini,bref,less

AUTHOR

Waleed Mohammed, Ali Adams, Aug 06 2015

STATUS

approved

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Last modified November 15 03:35 EST 2018. Contains 317224 sequences. (Running on oeis4.)