cp's OEIS Frontend

A260981 Primes p that are equal to the sum of the first k primes where p=prime(prime(k)).

%I A260981 #38 Sep 14 2015 03:02:39
%S A260981 5,17,41
%N A260981 Primes p that are equal to the sum of the first k primes where p=prime(prime(k)).
%C A260981 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).
%C A260981 From _Jon E. Schoenfield_, Aug 19 2015: (Start)
%C A260981 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.)
%C A260981 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:
%C A260981      k        S(k)         PP(k)     S(k)/PP(k)
%C A260981   ======  ===========    ========  ==============
%C A260981        1            2           3     0.666666...
%C A260981        2            5  =        5     1
%C A260981        3           10          11     0.909090...
%C A260981        4           17  =       17     1
%C A260981        5           28          31     0.903225...
%C A260981        6           41  =       41     1
%C A260981        7           58          59     0.983050...
%C A260981        8           77          67     1.149253...
%C A260981        9          100          83     1.204819...
%C A260981       10          129         109     1.183486...
%C A260981      ...
%C A260981      100        24133        3911     6.170544...
%C A260981     1000      3682913       80917    45.514700...
%C A260981    10000    496165411     1366661   363.049367...
%C A260981   100000  62260698721    20491057  3038.432752... (End)
%e A260981 k=3: prime(3) = 5 = 2+3 = prime(1) + prime(2).
%e A260981 k=7: prime(7) = 17 = 2+3+5+7 = prime(1) + prime(2) + prime(3) + prime(4).
%e A260981 k=13: prime(13) = 41 = 2+3+5+7+11+13 = prime(1) + prime(2) + prime(3) + prime(4) + prime(5) + prime(6).
%o A260981 (C#) // The code is provided by Ali Adams (www.heliwave.com)
%o A260981 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(); } } }
%Y A260981 Cf. A006450, A007504, A013918.
%K A260981 nonn,fini,bref,less
%O A260981 1,1
%A A260981 _Waleed Mohammed_, Ali Adams, Aug 06 2015