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A225854 Frequency of prime numbers between consecutive partial sums of primes. 1
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 (list; graph; refs; listen; history; text; internal format)
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 * A005044 A266755 A029142

Adjacent sequences:  A225851 A225852 A225853 * A225855 A225856 A225857

KEYWORD

nonn

AUTHOR

Victor Phan, May 17 2013

STATUS

approved

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Last modified July 23 14:44 EDT 2021. Contains 346259 sequences. (Running on oeis4.)