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

5, 17, 41

Offset: 1

Author

Waleed Mohammed, Ali Adams, Aug 06 2015

Keywords

• nonn
• fini
• bref
• less

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)

Examples

			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).
		

Crossrefs

Cf. A006450, A007504, A013918.