A003604 -id:A003604

A145532

Number of numbers removed in each step of Eratosthenes's sieve for 5!.

+10
24

59, 19, 7, 4

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

The number of steps in Eratosthenes's sieve for n! is A133228(n).

The number of primes less than 5! is 120 - 59 - 19 - 7 - 4 - 1 = 30 = A003604(5).

LINKS

Table of n, a(n) for n=1..4.

EXAMPLE

a(1)=59 because in first step we removed all numbers divisible by 2 (=60) with the exception of the first one, i.e., 2.

a(2)=19 because the number of numbers divisible by 3 and not divisible by 2 is 20 and we remove all with the exception of the first one, i.e., 3.

CROSSREFS

Cf. A003604, A133228, A145532-A145540.

KEYWORD

fini,nonn,full

AUTHOR

Artur Jasinski, Oct 12 2008

STATUS

approved

A000849

Number of primes <= product of first n primes, A002110(n).

+10
21

0, 1, 3, 10, 46, 343, 3248, 42331, 646029, 12283531, 300369796, 8028643010, 259488750744, 9414916809095, 362597750396740, 15397728527812858, 742238179058722891, 40068968501510691894, 2251262473052300960826, 139566579945945392719413

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

LINKS

David Baugh, Table of n, a(n) for n = 0..19 (terms n = 18..19 found using Kim Walisch's primecount program).

R. Mestrovic, An elementary proof of an estimate for a number of primes less than the product of the first n primes, arXiv:1211.4571 [math.NT], 2012.

C. D. Pruitt, A Theorem & Proof on the Density of Primes Utilizing Primorials

Tomás Oliveira e Silva, Tables of values of pi(x) and of pi2(x)

FORMULA

a(n) = A000720(A002110(n)). - Michel Marcus, Aug 25 2014

MAPLE

seq(numtheory:-pi(mul(ithprime(i), i=1..n)), n=0..10); # Robert Israel, Aug 25 2014

MATHEMATICA

a=1; Table[a=a*Prime[n]; PrimePi[a], {n, 12}]

Join[{0}, PrimePi/@FoldList[Times, Prime[Range[12]]]] (* Harvey P. Dale, Jan 28 2019 *)

PROG

(PARI) t=1; forprime(p=2, 66, print1(primepi(t), ", "); t*=p); \\ Joerg Arndt, Aug 25 2014

(Sage) [prime_pi(sloane.A002110(n)) for n in range (14)] # Giuseppe Coppoletta, Mar 02 2015

CROSSREFS

Cf. A000720, A002110, A003604.

KEYWORD

nonn,hard

AUTHOR

James D. Ausfahl, gandalf(AT)hrn.office.ssi.net

EXTENSIONS

More terms from David W. Wilson

a(10)-a(13) from Paul Zimmermann

a(14)-a(15) from Donovan Johnson, Mar 01 2010

a(16)-a(17) from Henri Lifchitz, Aug 25 2014

a(18)-a(19) from David Baugh, Sep 29 2020

STATUS

approved

A133228

Number of primes < square root of n!.

+10
10

0, 0, 1, 2, 4, 9, 19, 46, 110, 291, 822, 2455, 7740, 25635, 88849, 320749, 1202674, 4670156, 18741145, 77553119, 330321299, 1445829174, 6493985903, 29891948760, 140843699641, 678576973614, 3339785593878

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,4

LINKS

Table of n, a(n) for n=1..27.

MATHEMATICA

Table[PrimePi[Sqrt[n! ]], {n, 27}] (* Robert G. Wilson v *)

PROG

(PARI) a(n)=primepi(sqrtint(n!)) \\ Charles R Greathouse IV, Sep 15 2015

CROSSREFS

Cf. A003604.

KEYWORD

nonn

AUTHOR

Giovanni Teofilatto, Oct 13 2007

EXTENSIONS

More terms from Stefan Steinerberger and Robert G. Wilson v, Oct 14 2007

STATUS

approved

A145537

a(n) is the number of numbers removed in each step of Eratosthenes's sieve for 10!.

+10
8

1814399, 604799, 241919, 138239, 75402, 58003, 40941, 34478, 26982, 20473, 18496, 15008, 13184, 12266, 10957, 9492, 8342, 7920, 7057, 6538, 6248, 5667, 5317, 4874, 4414, 4181, 4057, 3866, 3752, 3582, 3166, 3054, 2911, 2856, 2675, 2640, 2544, 2455, 2399

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

Number of steps in Eratosthenes's sieve for n! is A133228(n).

Number of primes less than 10! is 10! - (sum all numbers in this sequence) - 1 = A003604(10).

LINKS

Nathaniel Johnston, Table of n, a(n) for n = 1..291 (full sequence)

MAPLE

A145537:=Array([seq(0, j=1..291)]): lim:=10!: p:=Array([seq(ithprime(j), j=1..291)]): for n from 4 to lim do if(isprime(n))then n:=n+1: fi: for k from 1 to 291 do if(n mod p[k] = 0)then A145537[k]:=A145537[k]+1: break: fi: od: od: seq(A145537[j], j=1..291); # 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 = 10; kk = PrimePi[Sqrt[nn! ]]; t3 = f3[nn!, kk] (* Bob Hanlon (hanlonr(AT)cox.net) *)

CROSSREFS

Cf. A003604, A133228, A145532-A145540.

KEYWORD

fini,nonn

AUTHOR

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

STATUS

approved

A061232

Number of primes p with n! < p <= (n+1)!.

+10
5

0, 1, 2, 6, 21, 98, 547, 3556, 26738, 227720, 2170267, 22877331, 264314464, 3320870054, 45076422125, 657316885209, 10247614197601, 170081414212020, 2994059471570761, 55718507205774017, 1092932100469356250, 22536709415953547880, 487361620197926253365

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

COMMENTS

First differences of A003604. - Artur Jasinski, Dec 13 2007

LINKS

Table of n, a(n) for n=0..22.

Andrew R. Booker, The Nth Prime Page

FORMULA

I conjecture that for n>2 we have n + 1/2 <= a(n)/a(n-1) <= n + 2/3. If this conjecture is true we have floor(a(n+1)/a(n)) = n. - Mohammed Bouayoun (mohammed.bouayoun(AT)sanef.com), Apr 03 2006

EXAMPLE

a(3) = 6 as there are 6 primes between 3! = 6 and 4! = 24: 7,11,13,17,19,23; a(4) = 21 as there are 21 primes between 24 and 120.

MATHEMATICA

Table[PrimePi[(n + 1)! ] - PrimePi[n! ], {n, 0, 15}]

CROSSREFS

Cf. A003604.

KEYWORD

nonn,hard

AUTHOR

Amarnath Murthy, Apr 23 2001

EXTENSIONS

Extended from a(6) on by Patrick De Geest, May 29 2001, using A. Booker's 'Nth Prime Page'

a(15) from Robert G. Wilson v, Jan 29 2003

Edited by N. J. A. Sloane, May 15 2008 at the suggestion of R. J. Mathar

a(17)-a(18) from Donovan Johnson, Oct 30 2012

a(19)-a(22) from A003604(n+1) - A003604(n) by Jinyuan Wang, Mar 11 2020

STATUS

approved

A145533

a(n) is the number of numbers removed in each step of Eratosthenes's sieve for 6!.

+10
5

359, 119, 47, 26, 14, 11, 7, 5, 3

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

Number of steps in Eratosthenes's sieve for n! is A133228(n).

Number of primes less than 6! is 720 - 359 - 119 - 47 - 26 - 14 - 11 - 7 - 5 - 3 - 1 = 128 = A003604(6).

LINKS

Table of n, a(n) for n=1..9.

EXAMPLE

a(1)=359 because in the first step we remove all numbers divisible by 2 (= 360) with the exception of the first one, i.e., 2.

a(2)=119 because the number of numbers divisible by 3 and not divisible by 2 is 120 and we remove all such numbers with the exception of the first one, 3.

MAPLE

A145533 := {$(1..6!)}: for n from 1 do p:=ithprime(n): r:=0: lim:=6!/p: for k from 2 to lim do if(member(k*p, A145533))then r:=r+1: fi: A145533 := A145533 minus {k*p}: od: printf("%d, ", r): if(r=0)then break: fi: od: # Nathaniel Johnston, Jun 23 2011

MATHEMATICA

{m1, m2, m3, m4, m5, m6, m7, m8, m9} = {-1, -1, -1, -1, -1, -1, -1, -1, -1};

Do[If[Mod[n, 2] == 0, m1 = m1 + 1,

If[Mod[n, 3] == 0, m2 = m2 + 1,

If[Mod[n, 5] == 0, m3 = m3 + 1,

If[Mod[n, 7] == 0, m4 = m4 + 1,

If[Mod[n, 11] == 0, m5 = m5 + 1,

If[Mod[n, 13] == 0, m6 = m6 + 1,

If[Mod[n, 17] == 0, m7 = m7 + 1,

If[Mod[n, 19] == 0, m8 = m8 + 1,

If[Mod[n, 23] == 0, m9 = m9 + 1]]]]]]]]], {n, 1, 6!}];

Print[{m1, m2, m3, m4, m5, m6, m7, m8, m9}] (* Artur Jasinski *)

CROSSREFS

Cf. A003604, A133228, A145532-A145540.

KEYWORD

fini,full,nonn

AUTHOR

Artur Jasinski, Oct 12 2008

STATUS

approved

A145534

a(n) is the number of numbers removed in each step of Eratosthenes's sieve for 7!.

+10
5

2519, 839, 335, 191, 104, 79, 57, 49, 39, 31, 27, 21, 18, 17, 14, 9, 7, 5, 3

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

Number of steps in Eratosthenes's sieve for n! is A133228(n).

Number of primes less than 7! is 7! - (sum all numbers in this sequence) - 1 = A003604(7).

LINKS

Table of n, a(n) for n=1..19.

MAPLE

A145534 := {$(1..7!)}: for n from 1 do p:=ithprime(n): r:=0: lim:=7!/p: for k from 2 to lim do if(member(k*p, A145534))then r:=r+1: fi: A145534 := A145534 minus {k*p}: od: printf("%d, ", r): if(r=0)then break: fi: od: # 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 = 7; kk = PrimePi[Sqrt[nn! ]]; t3 = f3[nn!, kk] (* Bob Hanlon (hanlonr(AT)cox.net) *)

CROSSREFS

Cf. A003604, A133228, A145532-A145540.

KEYWORD

fini,full,nonn

AUTHOR

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

STATUS

approved

A145535

a(n) is the number of numbers removed in each step of Eratosthenes's sieve for 8!.

+10
5

20159, 6719, 2687, 1535, 836, 642, 454, 381, 297, 223, 204, 170, 154, 146, 134, 119, 108, 103, 92, 84, 81, 76, 70, 64, 56, 53, 51, 47, 45, 42, 36, 32, 30, 28, 23, 21, 18, 16, 15, 12, 8, 6, 5, 3, 2, 1

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

Number of steps in Eratosthenes's sieve for n! is A133228(n).

Number of primes less than 8! is 8! - (sum all numbers in this sequence) - 1 = A003604(8).

LINKS

Table of n, a(n) for n=1..46.

MAPLE

A145535 := {$(1..8!)}: for n from 1 do p:=ithprime(n): r:=0: lim:=8!/p: for k from 2 to lim do if(member(k*p, A145535))then r:=r+1: fi: A145535 := A145535 minus {k*p}: od: printf("%d, ", r): if(r=0)then break: fi: od: # 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 = 8; kk = PrimePi[Sqrt[nn! ]]; t3 = f3[nn!, kk] (* Bob Hanlon (hanlonr(AT)cox.net) *)

CROSSREFS

Cf. A003604, A133228, A145532-A145540.

KEYWORD

fini,full,nonn

AUTHOR

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

STATUS

approved

A145536

a(n) is the number of numbers removed in each step of Eratosthenes's sieve for 9!.

+10
5

181439, 60479, 24191, 13823, 7540, 5800, 4092, 3446, 2701, 2046, 1842, 1487, 1296, 1200, 1070, 927, 817, 782, 703, 665, 645, 600, 574, 538, 498, 477, 465, 451, 441, 425, 385, 372, 351, 346, 326, 322, 308, 294, 288, 277, 267, 263, 248, 246, 238, 236, 221, 211

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

Number of steps in Eratosthenes's sieve for n! is A133228(n).

Number of primes less than 9! is 9! - (sum all numbers in this sequence) - 1 = A003604(9).

LINKS

Nathaniel Johnston, Table of n, a(n) for n = 1..110 (full sequence)

MAPLE

A145536:=Array([seq(0, j=1..110)]): lim:=9!: p:=Array([seq(ithprime(j), j=1..110)]): for n from 4 to lim do if(isprime(n))then n:=n+1: fi: for k from 1 to 110 do if(n mod p[k] = 0)then A145536[k]:=A145536[k]+1: break: fi: od: od: seq(A145536[j], j=1..110); # 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 = 9; kk = PrimePi[Sqrt[nn! ]]; t3 = f3[nn!, kk] (* Bob Hanlon (hanlonr(AT)cox.net) *)

CROSSREFS

Cf. A003604, A133228, A145532-A145540.

KEYWORD

fini,nonn

AUTHOR

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

STATUS

approved

A062274

Number of prime divisors (with repetition) of (n!)!, A000197.

+10
1

0, 0, 1, 7, 45, 291, 2030, 15695, 135045, 1287243, 13495669, 154516663, 1919455487, 25721712601, 369942275033

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,4

LINKS

Table of n, a(n) for n=0..14.

FORMULA

a(n) = A001222(A000197(n)). - Michel Marcus, Oct 20 2019

MATHEMATICA

Table[PrimeOmega[(n!)!], {n, 0, 10}] (* Harvey P. Dale, Apr 29 2015 *)

PROG

(PARI) for(n=0, 10, print1(bigomega(n!!), ", "))

(PARI) a(n) = { my(res = 0, nf = n!); forprime(p = 2, nf, res+=val(nf, p) ); res }

val(n, p) = my(r=0); while(n, r+=n\=p); r \\ David A. Corneth, Apr 10 2021

(Python)

from sympy import factorial, factorint

def A062274(n): return sum(sum(factorint(i).values()) for i in range(2, factorial(n)+1)) # Chai Wah Wu, Apr 10 2021

CROSSREFS

Cf. A000197, A001222, A003604, A022559.

KEYWORD

nonn,more

AUTHOR

Jason Earls, Jul 04 2001

EXTENSIONS

More terms from David W. Wilson, Jul 06 2001

a(11)-a(13) from Jinyuan Wang, Apr 01 2020

a(14) from David A. Corneth, Apr 10 2021

STATUS

approved