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A321211 Let S be the sequence of integer sets defined by these rules: S(1) = {1}, and for any n > 1, S(n) = {n} U S(pi(n)) U S(n - pi(n)) (where X U Y denotes the union of the sets X and Y and pi is the prime counting function); a(n) = the number of elements of S(n). 1
1, 2, 3, 3, 4, 4, 5, 4, 6, 6, 6, 7, 7, 7, 8, 7, 8, 9, 9, 9, 9, 8, 10, 9, 10, 11, 11, 11, 11, 12, 12, 12, 11, 12, 11, 12, 13, 13, 13, 14, 14, 14, 13, 14, 14, 14, 15, 14, 14, 12, 14, 15, 14, 15, 16, 17, 17, 16, 16, 16, 16, 17, 16, 16, 17, 17, 16, 16, 15, 17, 19 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

The prime counting function corresponds to A000720.

This sequence has similarities with A294991; a(n) gives approximately the number of intermediate terms to consider in order to compute A316434(n) using the formula of its definition.

LINKS

Rémy Sigrist, Table of n, a(n) for n = 1..10000

Rémy Sigrist, Illustration of a(42)

Rémy Sigrist, Density plot of the first 100000000 terms

Rémy Sigrist, C++ program for A321211

EXAMPLE

The first terms, alongside pi(n) and S(n), are:

  n   a(n)  pi(n)  S(n)

  --  ----  -----  ----------------------

   1     1      0  {1}

   2     2      1  {1, 2}

   3     3      2  {1, 2, 3}

   4     3      2  {1, 2, 4}

   5     4      3  {1, 2, 3, 5}

   6     4      3  {1, 2, 3, 6}

   7     5      4  {1, 2, 3, 4, 7}

   8     4      4  {1, 2, 4, 8}

   9     6      4  {1, 2, 3, 4, 5, 9}

  10     6      4  {1, 2, 3, 4, 6, 10}

  11     6      5  {1, 2, 3, 5, 6, 11}

  12     7      5  {1, 2, 3, 4, 5, 7, 12}

PROG

(C++) See Links section.

(PARI) a(n) = my (v=Set([-1, -n]), i=1); while (v[i]!=-1, my (pi=primepi(-v[i])); v=setunion(v, Set([v[i]+pi, -pi])); i++); #v

CROSSREFS

Cf. A000720, A294991, A316434.

Sequence in context: A294991 A300118 A256544 * A336515 A130500 A072073

Adjacent sequences:  A321208 A321209 A321210 * A321212 A321213 A321214

KEYWORD

nonn

AUTHOR

Altug Alkan and Rémy Sigrist, Oct 31 2018

STATUS

approved

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Last modified June 15 11:10 EDT 2021. Contains 345048 sequences. (Running on oeis4.)