A120049
Number of 8-almost primes less than or equal to 10^n.
Original entry on oeis.org
0, 0, 0, 7, 105, 1418, 17572, 207207, 2367507, 26483012, 291646797, 3173159326, 34192782745, 365561221293, 3882841742380, 41015564702074, 431227959019552, 4515480975731045, 47115876816676830
Offset: 0
There are 7 eight-almost primes up to 1000: 256, 384, 576, 640, 864, 896 & 960.
Cf.
A006880,
A036352,
A109251,
A114106,
A114453,
A120047,
A120048,
A120049,
A120050,
A120051,
A120052,
A120053,
A116430.
-
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[8, 10^n], {n, 12}]
-
from math import prod, isqrt
from sympy import primerange, integer_nthroot, primepi
def A120049(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,0,1,1,8))) # Chai Wah Wu, Aug 23 2024
A120047
Number of 6-almost primes less than or equal to 10^n.
Original entry on oeis.org
0, 0, 2, 37, 485, 5933, 68963, 774078, 8493366, 91683887, 977694273, 10327249593, 108264085934, 1128049914377, 11694704489580, 120734708167792, 1242063105505230, 12739510126065301, 130330025583399801
Offset: 0
There are 2 six-almost primes up to 100: 64 and 96, so a(2) = 2.
-
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[6, 10^n], {n, 0, 13}]
-
from math import isqrt, prod
from sympy import primerange, integer_nthroot, primepi
def almostprimepi(n,k):
if k==0: return int(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(n//prod(c[1] for c in a))-a[-1][0] for a in g(n,0,1,1,k)) if k>1 else primepi(n))
def A120047(n): return almostprimepi(10**n,6) # Chai Wah Wu, Dec 09 2024
A120048
Number of 7-almost primes less than or equal to 10^n.
Original entry on oeis.org
0, 0, 0, 14, 231, 2973, 35585, 409849, 4600247, 50678212, 550454756, 5913771637, 62981797962, 665997804082, 7001087934965, 73232029374751, 762783057783010, 7916319351632036, 81898808371556517
Offset: 0
There are 14 seven-almost primes up to 1000: 128, 192, 288, 320, 432, 448, 480, 648, 672, 704, 720, 800, 832 & 972.
Cf.
A006880,
A036352,
A109251,
A114106,
A114453,
A120047,
A120048,
A120049,
A120050,
A120051,
A120052,
A120053,
A116430.
-
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[7, 10^n], {n, 11}]
A120050
Number of 9-almost primes less than or equal to 10^n.
Original entry on oeis.org
0, 0, 0, 2, 47, 671, 8491, 101787, 1180751, 13377156, 148930536, 1636170477, 17787688377, 191742524399, 2052389350029, 21838745177567, 231206458686127, 2437121982958248, 25591920108631224, 267840642082525459
Offset: 0
There are 2 nine-almost primes up to 1000: 512 & 768.
Cf.
A006880,
A036352,
A109251,
A114106,
A114453,
A120047,
A120048,
A120049,
A120050,
A120051,
A120052,
A120053,
A116430.
-
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[9, 10^n], {n, 12}]
A120051
Number of 10-almost primes less than or equal to 10^n.
Original entry on oeis.org
0, 0, 0, 0, 22, 306, 4016, 49163, 578154, 6618221, 74342563, 823164388, 9011965866, 97765974368, 1052666075366, 11263041623194, 119864659464824, 1269754732725522, 13396817167474205, 140847445420555406
Offset: 0
There are 22 ten-almost primes up to 10000: 1024, 1536, 2304, 2560, 3456, 3584, 3840, 5184, 5376, 5632, 5760, 6400, 6656, 7776, 8064, 8448, 8640, 8704, 8960, 9600, 9728, and 9984.
Cf.
A006880,
A036352,
A109251,
A114106,
A114453,
A120047,
A120048,
A120049,
A120050,
A120051,
A120052,
A120053,
A116430.
-
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[10, 10^n], {n, 12}]
-
from math import isqrt, prod
from sympy import primerange, integer_nthroot, primepi
def A120051(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,0,1,1,10))) # Chai Wah Wu, Nov 03 2024
A120053
Number of 12-almost primes less than or equal to 10^n.
Original entry on oeis.org
0, 0, 0, 0, 3, 63, 865, 11068, 133862, 1563465, 17836903, 200051717, 2214357712, 24255601105, 263439785143, 2841076717752, 30457549169277, 324855769153426, 3449587218984911, 36489283363168885
Offset: 0
There are 3 twelve-almost primes up to 10000: 4096, 6144, and 9216.
Cf.
A006880,
A036352,
A109251,
A114106,
A114453,
A120047,
A120048,
A120049,
A120050,
A120051,
A120052,
A120053,
A116430.
-
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[12, 10^n], {n, 11}]
-
from math import prod, isqrt
from sympy import primerange, integer_nthroot, primepi
def A120053(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,0,1,1,12))) # Chai Wah Wu, Aug 23 2024
A120052
Number of 11-almost primes less than or equal to 10^n.
Original entry on oeis.org
0, 0, 0, 0, 7, 138, 1878, 23448, 279286, 3230577, 36585097, 407818620, 4490844534, 48972151631, 529781669333, 5693047157230, 60832290450373, 646862625625663, 6849459596884350, 72259172519243461
Offset: 0
There are 7 eleven-almost primes up to 10000: 2048, 3072, 4608, 5120, 6912, 7168, and 7680.
Cf.
A006880,
A036352,
A109251,
A114106,
A114453,
A120047,
A120048,
A120049,
A120050,
A120051,
A120052,
A120053,
A116430.
-
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[11, 10^n], {n, 12}]
A116426
The number of n-almost primes less than or equal to 4^n, starting with a(0)=1.
Original entry on oeis.org
1, 2, 6, 13, 34, 77, 177, 406, 887, 1962, 4225, 9094, 19482, 41414, 87706, 184976, 389357, 816193, 1708412, 3566209, 7431153, 15457234, 32098652, 66560309, 137830562, 285062028, 588871107, 1215176367, 2505048537, 5159228725
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 *)
Join[{1},Table[AlmostPrimePi[n, 4^n], {n, 29}]]
-
from math import isqrt, prod
from sympy import primerange, integer_nthroot, primepi
def A116426(n):
if n<=1: return 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((1<<(n<<1))//prod(c[1] for c in a))-a[-1][0] for a in g(1<<(n<<1),0,1,1,n))) # Chai Wah Wu, Oct 02 2024
A116427
The number of n-almost primes less than or equal to 6^n, starting with a(0)=1.
Original entry on oeis.org
1, 3, 13, 50, 200, 726, 2613, 9061, 30779, 102637, 338230, 1102674, 3566001, 11455355, 36597558, 116395587, 368749900, 1164407829, 3666312894, 11515047829, 36085395700, 112857846859, 352329509934, 1098136237818
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 *)
Join[{1},Table[AlmostPrimePi[n, 6^n], {n, 21}]]
A116428
The number of n-almost primes less than or equal to 8^n, starting with a(0)=1.
Original entry on oeis.org
1, 4, 22, 125, 669, 3410, 16677, 78369, 359110, 1612613, 7133274, 31185350, 135062165, 580556958, 2480278767, 10542976739, 44626102826, 188215850830, 791374442571, 3318478309647, 13882441625034, 57952990683107
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}]]]]];
Table[ AlmostPrimePi[n, 8^n], {n, 14}] (* Eric W. Weisstein, Feb 07 2006 *)
-
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(8^n, n)); \\ Daniel Suteu, Jul 10 2023
Showing 1-10 of 15 results.