A145535 -id:A145535

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A145583

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

+10
11

49, 16, 6, 3

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

OFFSET

1,1

COMMENTS

Number of steps in Eratosthenes's sieve for 10^n is A122121(n).

Number of primes less than 10^2 is equal to 10^2 - (sum all of numbers in this sequence) - 1 = A006880(2).

LINKS

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

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 = 2; kk = PrimePi[Sqrt[10^nn]]; t3 = f3[10^nn, kk] (*Bob Hanlon (hanlonr(AT)cox.net) *)

CROSSREFS

Cf. A006880, A122121, A145532, A145533, A145534, A145535, A145536, A145537, A145538, A145539, A145540, A145583, A145584, A145585, A145586, A145587, A145588, A145589, A145590, A145591, A145592.

KEYWORD

fini,nonn

AUTHOR

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

STATUS

approved

A139172

Natural numbers of the form (n!-2)/2.

+10
10

0, 2, 11, 59, 359, 2519, 20159, 181439, 1814399, 19958399, 239500799, 3113510399, 43589145599, 653837183999, 10461394943999, 177843714047999, 3201186852863999, 60822550204415999, 1216451004088319999

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

OFFSET

2,2

COMMENTS

Natural numbers of the form (n!-m)/m:

for m=1 n!-1 see A033312;

for m=2 (n!-2)/2 see A139172;

for m=3 (n!-3)/3 see A139173;

for m=4 (n!-4)/4 see A139174;

for m=5 (n!-5)/5 see A139175;

for m=6 (n!-6)/6 see A139176;

for m=7 (n!-7)/7 see A139177;

for m=8 (n!-8)/8 see A139183;

for m=9 (n!-9)/9 see A139184;

for m=10 (n!-10)/10 see A139185.

From Artur Jasinski, Oct 14 2008: (Start)

a(n) = Number of numbers removed in first step of Eratosthenes's sieve for n!

a(5)=A145532(1), a(6)=A145533(1), a(7)=A145534(1), a(8)=A145535(1), a(9)=A145536(1), a(10)=A145537(1). (End)

Generally, for n >= m, the formula a(n) = n*(a(n-1) + 1) - 1 applies to all natural numbers of the form (n!-m)/m, m >= 2. - Bob Selcoe, Mar 28 2015

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 2..300

FORMULA

a(n) = Sum_{k=1..floor(n/2)} s(n,n-2*k), where s(n,k) are Stirling numbers of the first kind, A048994. - Mircea Merca, Apr 07 2012

a(n) = n*(a(n-1) + 1) - 1. - Bob Selcoe, Mar 28 2015

MATHEMATICA

Table[(n! - 2)/2, {n, 2, 20}]

PROG

(Magma) [(Factorial(n)-2)/2: n in [2..25]]; // Vincenzo Librandi, Jul 20 2011

(PARI) a(n)=n!/2-1 \\ Charles R Greathouse IV, Apr 07 2012

CROSSREFS

Cf. A033312, A139172, A139173, A139174, A139175, A139176, A139177, A139183, A139184.

KEYWORD

nonn,easy

AUTHOR

Artur Jasinski, Apr 11 2008

STATUS

approved

A227799

Number of composites removed in each step of the Sieve of Eratosthenes for 10^10.

+10
3

4999999999, 1666666666, 666666666, 380952380, 207792207, 159840159, 112828348, 95013343, 74358271, 56409724, 50950713, 41311372, 36273411, 33742734, 30153115, 26170720, 23065826, 21931483, 19640105, 18256894, 17506397, 15954848, 14993294, 13813524, 12531256

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

OFFSET

1,1

COMMENTS

a(n) = the number of composites <= 10^10 for which the n-th prime is the least prime factor.

pi(sqrt(10^10)) = the number of terms of this sequence.

The sum of a(n) for n = 1..3401 = A000720(10^10) + A065855(10^10).

LINKS

Eric F. O'Brien, Table of n, a(n) for n = 1..9592

EXAMPLE

a(1) = 10^10 \ 2 - 1.

a(2) = 10^10 \ 3 - 10^10 \ (2*3) - 1.

a(3) = 10^10 \ 5 - 10^10 \ (2*5) - 10^10 \ (3*5) + 10^10 \ (2*3*5) - 1.

a(4) = 10^10 \ 7 - 10^10 \ (2*7) - 10^10 \ (3*7) - 10^10 \ (5*7) + 10^10 \ (2*3*7) + 10^10 \ (2*5*7) + 10^10 \ (3*5*7) - 10^10 \ (2*3*5*7) - 1.

CROSSREFS

Cf. A133228, A145538, A145539, A145540, A145583, A227155, A227797, A227798, A145532, A145533, A145534, A145535, A145536, A145537.

KEYWORD

nonn,fini

AUTHOR

Eric F. O'Brien, Jul 31 2013

STATUS

approved

A145584

a(n) = number of numbers removed in step n of Eratosthenes's sieve for 2^6.

+10
2

31, 10, 3, 1

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

OFFSET

1,1

COMMENTS

Number of steps in Eratosthenes's sieve for 2^n is A060967(n).

Number of primes less than 2^6 is equal to 2^6 - (sum all of numbers in this sequence) - 1 = A007053(6).

LINKS

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

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[2^nn]]; t3 = f3[2^nn, kk] (* Bob Hanlon (hanlonr(AT)cox.net) *)

CROSSREFS

Cf. A006880, A122121.

Cf. A145532, A145533, A145534, A145535, A145536, A145537, A145538, A145539, A145540.

Cf. A145583, A145584, A145585, A145586, A145587, A145588, A145589, A145590, A145591, A145592.

KEYWORD

fini,full,nonn

AUTHOR

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

STATUS

approved

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