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