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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 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.)