A073514
Number of primes less than 10^n with initial digit 4.
Original entry on oeis.org
0, 3, 20, 139, 1069, 8747, 74114, 641594, 5661135, 50653546, 458352691, 4185483176, 38510936699, 356622729564, 3320632228693, 31067060521057, 291869049531878, 2752144407792176, 26035873192178041, 247025281876786013, 2349914303292170310, 22407593754131275705
Offset: 1
a(2)=3 because there are 3 primes up to 10^2 whose initial digit is 4 (namely 41, 43 and 47).
For primes with initial digit d (1 <= d <= 9) see
A045707,
A045708,
A045709,
A045710,
A045711,
A045712,
A045713,
A045714,
A045715;
A073517,
A073516,
A073515,
A073514,
A073513,
A073512,
A073511,
A073510,
A073509
A073515
Number of primes less than 10^n with initial digit 3.
Original entry on oeis.org
1, 3, 19, 139, 1097, 8960, 75290, 651085, 5735086, 51247361, 463196868, 4225763390, 38851672813, 359541975662, 3345924530873, 31288310624754, 293820812588401, 2769490109678920, 26191046215879444, 248421640738371325, 2362546444095790527, 22522418647770393663
Offset: 1
a(2)=3 because there are 3 primes up to 10^2 whose initial digit is 3 (namely 3, 31 and 37).
For primes with initial digit d (1 <= d <= 9) see
A045707,
A045708,
A045709,
A045710,
A045711,
A045712,
A045713,
A045714,
A045715;
A073517,
A073516,
A073515,
A073514,
A073513,
A073512,
A073511,
A073510,
A073509
A073516
Number of primes less than 10^n with initial digit 2.
Original entry on oeis.org
1, 3, 19, 146, 1129, 9142, 77025, 664277, 5837665, 52064915, 469864125, 4281198201, 39319600765, 363545360347, 3380562309312, 31590949437540, 296487794277035, 2793170342851930, 26402713858800478, 250324979315879678, 2379753569255122805, 22678735843184786383
Offset: 1
a(2)=3 because there are 3 primes up to 10^2 whose initial digit is 2 (namely 2, 23 and 29).
For primes with initial digit d (1 <= d <= 9) see
A045707,
A045708,
A045709,
A045710,
A045711,
A045712,
A045713,
A045714,
A045715;
A073517,
A073516,
A073515,
A073514,
A073513,
A073512,
A073511,
A073510,
A073509
A145540
Number of numbers removed in each step of Eratosthenes's sieve for 10^4.
Original entry on oeis.org
4999, 1666, 666, 380, 207, 159, 110, 94, 76, 59, 56, 46, 41, 37, 33, 27, 23, 21, 17, 15, 12, 9, 8, 6, 3
Offset: 1
Artur Jasinski with assistance from Bob Hanlon (hanlonr(AT)cox.net), Oct 14 2008
-
A145540:=Array([seq(0,j=1..25)]): lim:=10^4: p:=Array([seq(ithprime(j),j=1..25)]): for n from 4 to lim do if(isprime(n))then n:=n+1: fi: for k from 1 to 25 do if(n mod p[k] = 0)then A145540[k]:=A145540[k]+1: break: fi: od: od: seq(A145540[j],j=1..25); # Nathaniel Johnston, Jun 23 2011
-
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 = 4; kk = PrimePi[Sqrt[10^nn]]; t3 = f3[10^nn, kk] (* Bob Hanlon (hanlonr(AT)cox.net) *)
A119290
a(n) is the total number of digits in the first 10^n primes.
Original entry on oeis.org
1, 16, 271, 3803, 48982, 610484, 7245905, 83484450, 942636916, 10487584405, 115369529592, 1257761617574, 13611696080735, 146406754329933, 1566562183907264, 16687323842873339, 177063766685219106, 1872323812397478246, 19738266145121133639, 207517446542560214799, 2176390177056541482871, 22774922890367225576581
Offset: 0
At a(1) there are 10^1 primes, 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, and the total number of digits is 16.
-
Accumulate@Table[c = 0; i0 = If[n == 0, 1, 10^(n - 1) + 1]; For[i = i0, i <= 10^n, i++, c += IntegerLength[Prime[i]]]; c, {n, 0, 6}] (* Robert Price, Jun 09 2019 *)
A122121
Number of primes <= 10^(n/2).
Original entry on oeis.org
0, 2, 4, 11, 25, 65, 168, 446, 1229, 3401, 9592, 27293, 78498, 227647, 664579, 1951957, 5761455, 17082666, 50847534, 151876932, 455052511, 1367199811, 4118054813, 12431880460, 37607912018, 113983535775, 346065536839, 1052370166553, 3204941750802, 9773865306521
Offset: 0
N. J. A. Sloane, based on a suggestion from Klaus Kastberg (Kastberg(AT)aapt.net.au), Oct 17 2006
a(3) = 11: sqrt(1000) = 31.62277660..., pi(31) = 11.
-
a={}; For[n=0, n<=27, n++, AppendTo[a,PrimePi[10^(n/2)]]]; Print[a]; (* John W. Layman, Mar 12 2010 *)
-
{ a= 0; n= 1; p=2 ; while(1, a++ ; pnext =nextprime(p+1) ; if( p^2 <= 10^n && pnext^2>10^n, print(a) ; n++ ; ) ; p=pnext ; ) ; } \\ R. J. Mathar, Jan 13 2007
-
from math import isqrt
from sympy import primepi
def A122121(n): return primepi(isqrt(10**n)) # Chai Wah Wu, Oct 17 2024
a(0)-a(17) confirmed, and a(18)-a(26) added using Mathematica, by
John W. Layman, Mar 12 2010
a(27) and a(28) added using Mathematica, by
David Baugh, Oct 06 2011
a(30)-a(46) added using Kim Walisch's primecount program, by
David Baugh, Feb 10 2015
a(47)-a(52) from
David Baugh using Kim Walisch's primecount program, Jun 19 2016
A114106
Number of 4-almost primes less than or equal to 10^n.
Original entry on oeis.org
0, 0, 12, 149, 1712, 18744, 198062, 2050696, 20959322, 212385942, 2139236881, 21454599814, 214499908019, 2139634739326, 21306682904040, 211905511283590, 2105504493045818, 20905484578206982, 207458897819329031, 2057931819347474462
Offset: 0
There are 12 primes with four almost primes up to 100: 16, 24, 36, 40, 54, 56, 60, 81, 84, 88, 90 and 100, so a(2) = 12.
-
FourAlmostPrimePi[n_] := Sum[ PrimePi[n/(Prime@i*Prime@j*Prime@k)] - k + 1, {i, PrimePi[n^(1/4)]}, {j, i, PrimePi[(n/Prime@i)^(1/3)]}, {k, j, PrimePi@Sqrt[n/(Prime@i*Prime@j)]}]; Table[ FourAlmostPrimePi[n], {n, 0, 13}]
-
from math import isqrt
from sympy import primepi, primerange, integer_nthroot
def A114106(n): return sum(primepi(10**n//(k*m*r))-c for a,k in enumerate(primerange(integer_nthroot(10**n,4)[0]+1)) for b,m in enumerate(primerange(k,integer_nthroot(10**n//k,3)[0]+1),a) for c,r in enumerate(primerange(m,isqrt(10**n//(k*m))+1),b)) # Chai Wah Wu, Aug 17 2024
A116430
The number of n-almost primes less than or equal to 10^n, starting with a(0)=1.
Original entry on oeis.org
1, 4, 34, 247, 1712, 11185, 68963, 409849, 2367507, 13377156, 74342563, 407818620, 2214357712, 11926066887, 63809981451, 339576381990, 1799025041767, 9494920297227, 49950199374227, 262036734664892
Offset: 0
Cf.
A078840,
A078841,
A078842,
A116432,
A078843,
A116426,
A078844,
A116427,
A078845,
A116428,
A116429,
A116430,
A078846,
A116431.
-
AlmostPrimePi[k_Integer, n_] := Module[{a, i}, a[0] = 1; If[k == 1, PrimePi[n], Sum[PrimePi[n/Times @@ Prime[Array[a, k - 1]]] - a[k - 1] + 1, Evaluate[ Sequence @@ Table[{a[i], a[i - 1], PrimePi[(n/Times @@ Prime[Array[a, i - 1]])^(1/(k - i + 1))]}, {i, k - 1}]]]]]; (* Eric W. Weisstein, Feb 07 2006 *)
Table[ AlmostPrimePi[n, 10^n], {n, 0, 13}]
-
almost_prime_count(N, k) = if(k==1, return(primepi(N))); (f(m, p, k, j=0) = my(c=0, s=sqrtnint(N\m, k)); if(k==2, forprime(q=p, s, c += primepi(N\(m*q))-j; j += 1), forprime(q=p, s, c += f(m*q, q, k-1, j); j += 1)); c); f(1, 2, k);
a(n) = if(n == 0, 1, almost_prime_count(10^n, n)); \\ Daniel Suteu, Jul 10 2023
-
from math import prod, isqrt
from sympy import primerange, integer_nthroot, primepi
def A116430(n):
if n<=1: return 3*n+1
def g(x,a,b,c,m): yield from (((d,) for d in enumerate(primerange(b,isqrt(x//c)+1),a)) if m==2 else (((a2,b2),)+d for a2,b2 in enumerate(primerange(b,integer_nthroot(x//c,m)[0]+1),a) for d in g(x,a2,b2,c*b2,m-1)))
return int(sum(primepi(10**n//prod(c[1] for c in a))-a[-1][0] for a in g(10**n,0,1,1,n))) # Chai Wah Wu, Aug 23 2024
A114453
Number of 5-almost primes less than or equal to 10^n.
Original entry on oeis.org
0, 0, 4, 76, 963, 11185, 124465, 1349779, 14371023, 150982388, 1570678136, 16218372618, 166497674684, 1701439985694, 17323079621014, 175846040834673, 1780617141307093, 17993699600756449, 181520864946969233
Offset: 0
There are 4 five-almost primes up to 100: 32,48,72 and 80, so a(2) = 4.
-
FiveAlmostPrimePi[n_] := Sum[ PrimePi[n/(Prime@i*Prime@j*Prime@k*Prime@l)] - l + 1, {i, PrimePi[n^(1/5)]}, {j, i, PrimePi[(n/Prime@i)^(1/4)]}, {k, j, PrimePi[(n/(Prime@i*Prime@j))^(1/3)]}, {l, k, PrimePi[(n/(Prime@i*Prime@j*Prime@k))^(1/2)]}]; Table[ FiveAlmostPrimePi[10^n], {n, 0, 12}]
-
from math import prod, isqrt
from sympy import primerange, integer_nthroot, primepi
def A114453(n):
def g(x,a,b,c,m): yield from (((d,) for d in enumerate(primerange(b,isqrt(x//c)+1),a)) if m==2 else (((a2,b2),)+d for a2,b2 in enumerate(primerange(b,integer_nthroot(x//c,m)[0]+1),a) for d in g(x,a2,b2,c*b2,m-1)))
return int(sum(primepi(10**n//prod(c[1] for c in a))-a[-1][0] for a in g(10**n,0,1,1,5))) # Chai Wah Wu, Sep 18 2024
A097952
Sum of the number of digits in the prime numbers less than 10^n.
Original entry on oeis.org
0, 4, 46, 475, 4719, 46534, 459970, 4562537, 45337545, 451112256, 4493162026, 44786187348, 446664473808, 4456613596481, 44480880591963, 444075310669968, 4434375640450064, 44287795522995300, 442382943864554586
Offset: 0
There are 25 primes < 100; 4 of them are 1-digit numbers and 21 are 2-digit numbers. Thus a(2) = 4 + 21*2 = 46.
-
Accumulate[Table[n(PrimePi[10^n]-PrimePi[10^(n-1)]),{n,0,14}]] (* This generates the first 15 terms of the sequence, but if n exceeds 14 the function PrimePi in Mathematica cannot calculate it. *) (* Harvey P. Dale, Jun 13 2014 *)
-
g(n) = for(j=0,n,s=0;forprime(x=2,10^j,y=length(Str(x));s+=y);print1(s","))
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