A316434 -id:A316434 - cp's OEIS frontend

A317442 a(1) = a(2) = 1; for n >= 3, a(n) = a(t(n)) + a(n-t(n)) where t = A000195.

1, 1, 2, 3, 4, 5, 6, 6, 7, 7, 8, 8, 9, 9, 10, 10, 11, 11, 12, 12, 13, 14, 14, 15, 16, 16, 17, 18, 18, 19, 20, 20, 21, 22, 22, 23, 24, 24, 25, 26, 26, 27, 28, 28, 29, 30, 30, 31, 32, 32, 33, 34, 34, 35, 36, 37, 37, 38, 39, 40, 40, 41, 42, 43, 43, 44, 45, 46, 46, 47, 48, 49, 49, 50, 51, 52

Offset: 1

Altug Alkan, Jul 28 2018

nonn

This sequence hits every positive integer.

Altug Alkan, A log plot of a(n)/n

Cf. A000195, A316434.

Nest[Function[{a, n}, Append[a, a[[Floor@ Log@ n]] + a[[n - Floor@ Log@ n]] ] ] @@ {#, Length@ # + 1} &, {1, 1}, 74] (* Michael De Vlieger, Jul 30 2018 *)

q=vector(100); for(n=1, 2, q[n]=1); for(n=3, #q, q[n]=q[floor(log(n))] + q[n-floor(log(n))]); q

a(n+1) - a(n) = 0 or 1 for all n >= 1.

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).

Original entry on oeis.org

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

Offset: 1

Altug Alkan and Rémy Sigrist, Oct 31 2018

nonn

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.

			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}

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

Cf. A000720, A294991, A316434.

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

A316942 a(n) = n - a(pi(n)) - a(n-pi(n)) with a(1) = a(2) = 1, where pi = A000720.

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 4, 4, 4, 4, 4, 5, 5, 6, 7, 8, 9, 10, 11, 11, 12, 12, 13, 13, 13, 13, 13, 13, 14, 15, 16, 16, 17, 17, 18, 19, 19, 20, 21, 22, 23, 23, 23, 23, 23, 24, 24, 24, 25, 25, 25, 26, 26, 26, 26, 26, 26, 27, 27, 28, 28, 29, 30, 30, 31, 32, 32, 32, 33, 34, 35, 35, 35, 36, 37

Offset: 1

Altug Alkan, Jul 17 2018

nonn

This sequence hits every positive integer.

Cf. A000720, A062298, A316434.

Nest[Append[#2, #1 - #2[[PrimePi[#1] ]] - #2[[#1 - PrimePi[#1] ]] ] & @@ {Length@ # + 1, #} &, {1, 1}, 73] (* Michael De Vlieger, Jul 20 2018 *)

q=vector(75); for(n=1, 2, q[n] = 1); for(n=3, #q, q[n] = n - q[primepi(n)] - q[n-primepi(n)]); q

a(n) = n - a(A000720(n)) - a(A062298(n)) with a(1) = a(2) = 1.
a(n+1) - a(n) = 0 or 1 for all n >= 1.
Conjecture : lim_{n->infinity} a(n)/n = 1/2.

A336095 a(n) = a(f(n)) + a(n-f(n)) with a(1) = a(2) = 1 (f = A000006).

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 4, 4, 5, 6, 7, 8, 8, 8, 9, 9, 10, 11, 11, 12, 12, 12, 13, 14, 14, 15, 16, 17, 17, 18, 19, 19, 19, 20, 21, 22, 22, 23, 24, 24, 25, 25, 26, 27, 27, 27, 27, 28, 29, 30, 31, 31, 32, 33, 33, 33, 34, 34, 35, 36, 36, 37, 37, 37, 38, 39, 40, 41, 42, 42, 43, 44, 44, 44, 44

Offset: 1

Altug Alkan, Jul 08 2020

nonn

If Legendre's conjecture is true, then this sequence hits every positive integer.

Does the lim_{n->infinity} a(n)/n exist? If it exists, what is its value?

Wikipedia, Legendre's conjecture

Cf. A000006, A316434.

f[n_] := IntegerPart[Sqrt[Prime[n]]]; a[1] = a[2] = 1; a[n_] := a[n] = a[(f1 = f[n])] + a[n - f1]; Array[a, 100] (* Amiram Eldar, Jul 08 2020 *)

q=vector(10^2); for(n=1, 2, q[n] = 1); for(n=3, #q, q[n] = q[sqrtint(prime(n))] + q[n- sqrtint(prime(n))]); q

cp's OEIS Frontend

A317442 a(1) = a(2) = 1; for n >= 3, a(n) = a(t(n)) + a(n-t(n)) where t = A000195.

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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).

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A316942 a(n) = n - a(pi(n)) - a(n-pi(n)) with a(1) = a(2) = 1, where pi = A000720.

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A336095 a(n) = a(f(n)) + a(n-f(n)) with a(1) = a(2) = 1 (f = A000006).

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