cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Previous Showing 11-17 of 17 results.

A145586 a(n) = number of numbers removed in each step of Eratosthenes's sieve for 2^8.

Original entry on oeis.org

127, 42, 16, 8, 5, 3
Offset: 1

Views

Author

Artur Jasinski with assistance from Bob Hanlon (hanlonr(AT)cox.net), Oct 14 2008

Keywords

Comments

Number of steps in Eratosthenes's sieve for 2^n is A060967(n).
Number of primes less than 2^8 is equal to 2^8 - (sum all of numbers in this sequence) - 1 = A007053(8).

Crossrefs

Cf. A006880, A122121, A145532-A145540, A145583-A145592.

Programs

  • Mathematica
    f3[k_Integer?Positive, i_Integer?Positive] := Module[{f, m, r, p}, p = Transpose[{r = Range[2, i], Prime[r]}];f[x_] := Catch[Fold[If[Mod[x, #2[[2]]] == 0, Throw[m[ #2[[1]]] = m[ #2[[1]]] + 1], #1] &, If[Mod[x, 2] == 0, Throw[m[1] = m[1] + 1]], p]]; Table[m[n] = -1, {n, i}]; f /@ Range[k]; Table[m[n], {n, i}]];nn = 8; kk = PrimePi[Sqrt[2^nn]]; t3 = f3[2^nn, kk] (* Bob Hanlon (hanlonr(AT)cox.net), Oct 14 2008 *)

A145587 a(n) = number of numbers removed in each step of Eratosthenes's sieve for 2^9.

Original entry on oeis.org

255, 84, 33, 19, 10, 7, 4, 2
Offset: 1

Views

Author

Artur Jasinski with assistance from Bob Hanlon (hanlonr(AT)cox.net), Oct 14 2008

Keywords

Comments

Number of steps in Eratosthenes's sieve for 2^n is A060967(n).
Number of primes less than 2^9 is equal to 2^9 - (sum all of numbers in this sequence) - 1 = A007053(9).

Crossrefs

Cf. A006880, A122121, A145532-A145540, A145583-A145592.

Programs

  • Mathematica
    f3[k_Integer?Positive, i_Integer?Positive] := Module[{f, m, r, p}, p = Transpose[{r = Range[2, i], Prime[r]}];f[x_] := Catch[Fold[If[Mod[x, #2[[2]]] == 0, Throw[m[ #2[[1]]] = m[ #2[[1]]] + 1], #1] &, If[Mod[x, 2] == 0, Throw[m[1] = m[1] + 1]], p]]; Table[m[n] = -1, {n, i}]; f /@ Range[k]; Table[m[n], {n, i}]];nn = 9; kk = PrimePi[Sqrt[2^nn]]; t3 = f3[2^nn, kk] (* Bob Hanlon (hanlonr(AT)cox.net), Oct 14 2008 *)

A145588 a(n) = number of numbers removed in each step of Eratosthenes's sieve for 2^10.

Original entry on oeis.org

511, 170, 67, 38, 20, 16, 11, 9, 6, 2, 1
Offset: 1

Views

Author

Artur Jasinski with assistance from Bob Hanlon (hanlonr(AT)cox.net), Oct 14 2008

Keywords

Comments

Number of steps in Eratosthenes's sieve for 2^n is A060967(n).
Number of primes less than 2^10 is equal to 2^10 - (sum all of numbers in this sequence) - 1 = A007053(10).

Crossrefs

Cf. A006880, A122121, A145532-A145540, A145583-A145592.

Programs

  • Mathematica
    f3[k_Integer?Positive, i_Integer?Positive] := Module[{f, m, r, p}, p = Transpose[{r = Range[2, i], Prime[r]}];f[x_] := Catch[Fold[If[Mod[x, #2[[2]]] == 0, Throw[m[ #2[[1]]] = m[ #2[[1]]] + 1], #1] &, If[Mod[x, 2] == 0, Throw[m[1] = m[1] + 1]], p]]; Table[m[n] = -1, {n, i}]; f /@ Range[k]; Table[m[n], {n, i}]];nn = 10; kk = PrimePi[Sqrt[2^nn]]; t3 = f3[2^nn, kk] (* Bob Hanlon (hanlonr(AT)cox.net), Oct 14 2008 *)

A145589 a(n) = number of numbers removed in each step of Eratosthenes's sieve for 2^11.

Original entry on oeis.org

1023, 340, 136, 77, 41, 32, 24, 21, 16, 10, 8, 5, 3, 2
Offset: 1

Views

Author

Artur Jasinski with assistance from Bob Hanlon (hanlonr(AT)cox.net), Oct 14 2008

Keywords

Comments

Number of steps in Eratosthenes's sieve for 2^n is A060967(n).
Number of primes less than 2^11 is equal to 2^11 - (sum all of numbers in this sequence) - 1 = A007053(11).

Crossrefs

Cf. A006880, A122121, A145532-A145540, A145583-A145592.

Programs

  • Mathematica
    f3[k_Integer?Positive, i_Integer?Positive] := Module[{f, m, r, p}, p = Transpose[{r = Range[2, i], Prime[r]}];f[x_] := Catch[Fold[If[Mod[x, #2[[2]]] == 0, Throw[m[ #2[[1]]] = m[ #2[[1]]] + 1], #1] &, If[Mod[x, 2] == 0, Throw[m[1] = m[1] + 1]], p]]; Table[m[n] = -1, {n, i}]; f /@ Range[k]; Table[m[n], {n, i}]];nn = 11; kk = PrimePi[Sqrt[2^nn]]; t3 = f3[2^nn, kk] (* Bob Hanlon (hanlonr(AT)cox.net), Oct 14 2008 *)

A145590 a(n)=number of numbers removed in each step of Eratosthenes's sieve for 2^12.

Original entry on oeis.org

2047, 682, 272, 155, 83, 65, 46, 40, 32, 25, 22, 18, 13, 11, 9, 6, 3, 2
Offset: 1

Views

Author

Artur Jasinski with assistance from Bob Hanlon (hanlonr(AT)cox.net), Oct 14 2008

Keywords

Comments

Number of steps in Eratosthenes's sieve for 2^n is A060967(n).
Number of primes less than 2^12 is equal to 2^12 - (sum all of numbers in this sequence) - 1 = A007053(12).

Crossrefs

Cf. A006880, A122121, A145532-A145540, A145583-A145592.

Programs

  • Mathematica
    f3[k_Integer?Positive, i_Integer?Positive] := Module[{f, m, r, p}, p = Transpose[{r = Range[2, i], Prime[r]}];f[x_] := Catch[Fold[If[Mod[x, #2[[2]]] == 0, Throw[m[ #2[[1]]] = m[ #2[[1]]] + 1], #1] &, If[Mod[x, 2] == 0, Throw[m[1] = m[1] + 1]], p]]; Table[m[n] = -1, {n, i}]; f /@ Range[k]; Table[m[n], {n, i}]];nn = 12; kk = PrimePi[Sqrt[2^nn]]; t3 = f3[2^nn, kk] (* Bob Hanlon (hanlonr(AT)cox.net) *)

A145591 a(n)=number of numbers removed in each step of Eratosthenes's sieve for 2^13.

Original entry on oeis.org

4095, 1364, 545, 311, 170, 130, 91, 77, 63, 51, 46, 36, 34, 29, 26, 21, 17, 15, 12, 11, 9, 6, 3, 1
Offset: 1

Views

Author

Artur Jasinski with assistance from Bob Hanlon (hanlonr(AT)cox.net), Oct 14 2008

Keywords

Comments

Number of steps in Eratosthenes's sieve for 2^n is A060967(n).
Number of primes less than 2^13 is equal to 2^13 - (sum all of numbers in this sequence) - 1 = A007053(13).

Crossrefs

Cf. A006880, A122121, A145532-A145540, A145583-A145592.

Programs

  • Mathematica
    f3[k_Integer?Positive, i_Integer?Positive] := Module[{f, m, r, p}, p = Transpose[{r = Range[2, i], Prime[r]}];f[x_] := Catch[Fold[If[Mod[x, #2[[2]]] == 0, Throw[m[ #2[[1]]] = m[ #2[[1]]] + 1], #1] &, If[Mod[x, 2] == 0, Throw[m[1] = m[1] + 1]], p]]; Table[m[n] = -1, {n, i}]; f /@ Range[k]; Table[m[n], {n, i}]];nn = 13; kk = PrimePi[Sqrt[2^nn]]; t3 = f3[2^nn, kk] (* Bob Hanlon (hanlonr(AT)cox.net) *)

A247073 Triangle read by rows: T(n,k) is the number of k-th prime powers up to 2^n, for k = 1 to n.

Original entry on oeis.org

1, 2, 1, 4, 1, 1, 6, 2, 1, 1, 11, 3, 2, 1, 1, 18, 4, 2, 1, 1, 1, 31, 5, 3, 2, 1, 1, 1, 54, 6, 3, 2, 2, 1, 1, 1, 97, 8, 4, 2, 2, 1, 1, 1, 1, 172, 11, 4, 3, 2, 2, 1, 1, 1, 1, 309, 14, 5, 3, 2, 2, 1, 1, 1, 1, 1, 564, 18, 6, 4, 3, 2, 2, 1, 1, 1, 1, 1, 1028, 24, 8, 4, 3, 2, 2, 2, 1, 1, 1, 1, 1
Offset: 1

Views

Author

Michel Marcus, Nov 18 2014

Keywords

Examples

			Up to 16, there are 6 primes (2, 3, 5, 7, 11, 13), 2 squared primes (4,9), 1 cube (8), and 1 fourth power (16), so 4th row is 6, 2, 1, 1.
Triangle starts:
1;
2, 1;
4, 1, 1;
6, 2, 1, 1;
11, 3, 2, 1, 1;
18, 4, 2, 1, 1, 1;
...

Links

Crossrefs

Cf. A000961 (prime powers), A007053 (first column), A060967 (second column).
Cf. A025474.

Programs

  • Haskell
    import Data.List (sort, groupBy); import Data.Function (on)
a247073 n k = a247073_tabl !! (n-1) !! (k-1)
a247073_tabl = map a247073_row [1..]
a247073_row n = map length $ groupBy ((==) `on` fst) $ sort $
   takeWhile ((<= 2^n). snd) $ tail $ zip a025474_list a000961_list
-- Reinhard Zumkeller, Nov 18 2014
  • PARI
    tabl(nn) = {for (n=1, nn, v = vector(2^n, i, i); vr = vector(n); for (k=1, #v, if (pp = isprimepower(v[k]), vr[pp] ++);); for (k=1, n, print1(vr[k], ", ");); print(););}
Previous Showing 11-17 of 17 results.

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