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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 A332689 A342320 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 October 2 02:58 EDT 2023. Contains 365831 sequences. (Running on oeis4.)