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

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, 14, 17, 17, 16, 17, 19, 19, 22, 16, 24, 21, 20, 20, 20, 28, 22, 26, 21, 24, 28, 23, 31, 23, 30, 28, 28, 32, 28, 31, 30, 27, 36, 29, 32, 31, 39, 33, 38, 36, 36, 37

Offset

1,2

Comments

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)).

The plot of this sequence follows a linear curve.

Links

Giovanni Resta, Table of n, a(n) for n = 1..10000

Example

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.

Mathematica

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*)
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 *)

Crossrefs

Cf. A014284.

Sequence in context: A241825 A029162 A325132 * A371973 A005044 A266755

Adjacent sequences: A225851 A225852 A225853 * A225855 A225856 A225857

Keyword

nonn

Author

Victor Phan, May 17 2013

Status

approved