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A225854 Frequency of prime numbers between consecutive partial sums of primes.

%I A225854 #18 May 04 2017 00:12:40
%S A225854 1,2,1,3,2,4,3,5,4,6,6,8,6,9,6,9,10,10,8,12,12,11,12,12,15,14,14,14,
%T A225854 14,17,17,16,17,19,19,22,16,24,21,20,20,20,28,22,26,21,24,28,23,31,23,
%U A225854 30,28,28,32,28,31,30,27,36,29,32,31,39,33,38,36,36,37
%N A225854 Frequency of prime numbers between consecutive partial sums of primes.
%C A225854 Gives the numbers of primes between adjacent numbers in the sequence A014284, that is, primes in the half-open interval [A014284(k), A014284(k+1)).
%C A225854 The plot of this sequence follows a linear curve.
%H A225854 Giovanni Resta, <a href="/A225854/b225854.txt">Table of n, a(n) for n = 1..10000</a>
%e A225854 List the numbers with an increment of 1 beginning at n=1, and stop when the number of numbers reaches a prime, in this case the list would be {1,2} since its size is 2. Find the number of primes in that interval and add it to the sequence. In this case, there is 1 prime in the list. Continue counting from the last number in the previous list and apply the same rules, the next list will be {3,4,5} of size 3 and contains 2 prime numbers. The list after that will be {6,7,8,9,10} of size 5 and contains 1 prime number.
%t A225854 numberOfLines = 100; (*How many elements desired in the sequence*) a = {0}; distribution = {}; last = 0; For[j = 1, j <= numberOfLines, j++, frequency = 0; b = {}; For[i = 1, i <= Prime[j], i++, b = Append[b, last + i]; If[PrimeQ[b[[i]]], frequency += 1];];last += Prime[j]; distribution = Append[distribution, frequency];]; Print["Distribution = ", distribution]; ListPlot[distribution]; (*original program*)
%t A225854 seq[n_] := Block[{a=0, b=2, p=2, v}, Table[v = PrimePi@b-PrimePi@a; p = NextPrime@p; a = b; b += p; v, {n}]]; seq[100] (* faster version, _Giovanni Resta_, May 18 2013 *)
%Y A225854 Cf. A014284.
%K A225854 nonn
%O A225854 1,2
%A A225854 _Victor Phan_, May 17 2013