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.

Showing 1-4 of 4 results.

A145538 Number of numbers removed in each step of Eratosthenes's sieve for 10^5.

Original entry on oeis.org

49999, 16666, 6666, 3808, 2077, 1597, 1127, 949, 741, 555, 499, 405, 358, 335, 305, 274, 248, 242, 219, 203, 199, 184, 175, 165, 148, 141, 137, 131, 128, 124, 108, 104, 97, 95, 87, 86, 79, 75, 70, 67, 62, 60, 57, 54, 52, 50, 45, 39, 37, 35, 32, 29, 28, 25, 23, 20
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 10^n is A122121(n).
Number of primes less than 10^5 equals 10^5 - A065894(5) (sum of all numbers in this sequence) - 1 = A006880(5).
a(n) is the number of composite numbers m <= 10^5 whose least prime factor (A020639(m)) is prime(n).

Links

Crossrefs

Cf. A006880, A122121, A145532-A145540.
Cf. A065894, A020639.

Programs

  • Maple
    A145538:=Array([seq(0,j=1..65)]): lim:=10^5: p:=Array([seq(ithprime(j),j=1..65)]): for n from 4 to lim do if(isprime(n))then n:=n+1: fi: for k from 1 to 65 do if(n mod p[k] = 0)then A145538[k]:=A145538[k]+1: break: fi: od: od: seq(A145538[j],j=1..65); # Nathaniel Johnston, Jun 23 2011
  • 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 = 5; kk = PrimePi[Sqrt[10^nn]]; t3 = f3[10^nn, kk] (* Bob Hanlon (hanlonr(AT)cox.net) *)

Extensions

Edited by Rick L. Shepherd, Mar 02 2013

A145539 Number of numbers removed in each step of Eratosthenes's sieve for 10^6.

Original entry on oeis.org

499999, 166666, 66666, 38094, 20778, 15983, 11283, 9502, 7434, 5646, 5098, 4136, 3617, 3356, 2982, 2575, 2261, 2143, 1910, 1775, 1700, 1553, 1460, 1354, 1244, 1195, 1171, 1130, 1109, 1074, 964, 937, 898, 886, 832, 820, 794, 763, 745, 719, 697, 689, 654
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 10^n is A122121(n).
Number of primes less than 10^6 equals 10^6 - A065894(6) (sum of all numbers in this sequence) - 1 = A006880(6).
a(n) is the number of composite numbers m <= 10^6 whose least prime factor (A020639(m)) is prime(n). - Rick L. Shepherd, Mar 02 2013

Links

Crossrefs

Cf. A006880, A122121, A145532-A145540.
Cf. A065894, A020639.

Programs

  • Maple
    A145539:=Array([seq(0,j=1..168)]): lim:=10^6: p:=Array([seq(ithprime(j),j=1..168)]): for n from 4 to lim do if(isprime(n))then n:=n+1: fi: for k from 1 to 168 do if(n mod p[k] = 0)then A145539[k]:=A145539[k]+1: break: fi: od: od: seq(A145539[j],j=1..168); # Nathaniel Johnston, Jun 23 2011
  • 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 = 6; kk = PrimePi[Sqrt[10^nn]]; t3 = f3[10^nn, kk] (* Bob Hanlon (hanlonr(AT)cox.net) *)

A092853 Number of composites > mean composite (=A092802(n)) below 10^n.

Original entry on oeis.org

2, 37, 418, 4398, 45288, 461339, 4671939, 47150884, 474823446, 4774453663, 47957215384, 481331669604, 4828116680970, 48407394207052, 485163305702187
Offset: 1

Views

Author

Enoch Haga, Mar 07 2004

Keywords

Examples

			At 10^1 there are 4 composites: 4+6+8+9=27. The rounded mean is 27\4=7. there are 2 composites over 7: 8 and 9, so a(1)=2.

Crossrefs

Cf. A092802, A092853, A092854.

Formula

a(n) = A092871(n) - A092852(n) = A065894(n) - 1 - A092852(n). - Max Alekseyev, Aug 15 2013

Extensions

Terms a(9)-a(15) from Max Alekseyev, Aug 15 2013

A092871 Number of composites < 10^n.

Original entry on oeis.org

0, 4, 73, 830, 8769, 90406, 921500, 9335419, 94238543, 949152464, 9544947487, 95881945185, 962392087980, 9653934463159, 96795058249196, 970155429577329, 9720761658966073, 97376442842345765, 975260045712259138, 9765942332723655391, 97779180397439081158
Offset: 0

Views

Author

Enoch Haga, Mar 08 2004

Keywords

Comments

The number 1 is omitted from the count as it is neither prime nor composite

Examples

			10^3 = 1000. 1000-2 = 998. a(3) = 830 because the 830 composites+168 primes must total 998.

Links

Crossrefs

Cf. A065894, A006880, A092801, A092802.

Programs

  • Mathematica
    Table[10^i-PrimePi[10^i]-2,{i,14}] (* Harvey P. Dale, Oct 01 2011 *) (* Mathematica's implementation of PrimePi does not work for 10^15 or above *)

Formula

For n>0, a(n) = A065894(n) - 1 = 10^n - 2 - A006880(n). - Max Alekseyev, Aug 15 2013

Extensions

Edited by Max Alekseyev, Aug 15 2013
Showing 1-4 of 4 results.

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