A066265 -id:A066265 - cp's OEIS frontend

A120051 Number of 10-almost primes less than or equal to 10^n.

0, 0, 0, 0, 22, 306, 4016, 49163, 578154, 6618221, 74342563, 823164388, 9011965866, 97765974368, 1052666075366, 11263041623194, 119864659464824, 1269754732725522, 13396817167474205, 140847445420555406

Offset: 0

Robert G. Wilson v, Feb 07 2006

			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. A066265, A014613, A116382.

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

More terms from Robert G. Wilson v, Jan 07 2007

a(15)-a(19) from Henri Lifchitz, Mar 20 2025

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

Robert G. Wilson v, Feb 07 2006

			There are 3 twelve-almost primes up to 10000: 4096, 6144, and 9216.

Cf. A066265, A014613, A116382.

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

a(13) and a(14) from Robert G. Wilson v, Jan 07 2007
a(15) from Chai Wah Wu, Aug 24 2024
a(16)-a(19) from Henri Lifchitz, Mar 18 2025

A125527 Number of semiprimes <= 2^n.

Original entry on oeis.org

0, 1, 2, 6, 10, 22, 42, 82, 157, 304, 589, 1124, 2186, 4192, 8110, 15658, 30253, 58546, 113307, 219759, 426180, 827702, 1608668, 3129211, 6091437, 11868599, 23140878, 45150717, 88157689, 172235073, 336717854, 658662065, 1289149627, 2524532330

Offset: 1

Robert G. Wilson v, Dec 29 2006

nonn

Robert G. Wilson v, Table of n, a(n) for n = 1..63 (using data from A120033, terms n=48, 50..57 from Dana Jacobsen)
Eric Weisstein's World of Mathematics, Semiprime.

Cf. A126279, A066265, A007053, A120033, A127396.

SemiPrimePi[n_] := Sum[ PrimePi[ n/Prime@i] - i + 1, {i, PrimePi@ Sqrt@n}]; Table[ SemiPrimePi[2^n], {n, 47}]

a(n)=my(s,i,N=2^n); forprime(p=2, sqrtint(N), s+=primepi(N\p); i++); s - i * (i-1)/2 \\ Charles R Greathouse IV, May 12 2013

use ntheory ":all"; print "$ ",semiprime_count(1 << $),"\n" for 1..48; # Dana Jacobsen, Sep 10 2018

a(n) = A072000(2^n). - R. J. Mathar, Aug 26 2011

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

Robert G. Wilson v, Feb 07 2006

			There are 7 eleven-almost primes up to 10000: 2048, 3072, 4608, 5120, 6912, 7168, and 7680.

Cf. A066265, A014613, A116382.

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}]

a(14) from Robert G. Wilson v, Jan 07 2007

a(15)-a(19) from Henri Lifchitz, Mar 18 2025

A036351 Number of numbers <= 10^n that are products of two distinct primes.

Original entry on oeis.org

2, 30, 288, 2600, 23313, 209867, 1903878, 17426029, 160785135, 1493766851, 13959963049, 131125938680, 1237087821006, 11715901643501, 111329815346924, 1061057287065814, 10139482896634686, 97123037634329553, 932300026078297246, 8966605849186166511, 86389956292394285653, 833671466547121873095, 8056846659972421004731

Offset: 1

Shyam Sunder Gupta

nonn

Index entries for sequences related to numbers of primes in various ranges

Cf. A066265.

Cf. A036352, A122121.

f[n_] := Sum[ PrimePi[n/Prime[i]] - i, {i, PrimePi[ Sqrt[ n]] }]; Table[ f[10^n], {n, 14}] (* Robert G. Wilson v, Feb 07 2012 and modified Dec 28 2016 *)

a(n)=my(s);forprime(p=2,sqrt(10^n),s+=primepi(10^n\p)); s-binomial(primepi(sqrt(10^n))+1,2) \\ Charles R Greathouse IV, Apr 23 2012

from math import isqrt
from sympy import primepi, primerange
def A036351(n): return -(t:=primepi(s:=isqrt(m:=10**n)))-(t*(t-1)>>1)+sum(primepi(m//k) for k in primerange(1, s+1)) # Chai Wah Wu, Aug 15 2024

a(n) = (1/2)*(pi(10^(n/2)) + Sum_{i=1..pi(10^n)} pi((10^n-1)/P_i)) -1 = Sum_{i=1..pi(sqrt(10^n))} (pi((10^n-1)/P_i) -1) - binomial(pi(sqrt(10^n)), 2). - Robert G. Wilson v, May 19 2005

a(n) = A036352(n) - A122121(n). - Robert G. Wilson v, Feb 07 2012

a(14) from Robert G. Wilson v, May 19 2005
a(15)-a(16) from Donovan Johnson, Oct 16 2010
Corrected a(15) and a(16) by Henri Lifchitz, Nov 11 2012
a(17)-a(19) from Henri Lifchitz, Nov 11 2012
a(20)-a(21) from Henri Lifchitz, Jul 03 2015
a(22)-a(23) from Henri Lifchitz, Nov 09 2024

A114125 a(n) is the 10^n-th semiprime.

Original entry on oeis.org

4, 26, 314, 3595, 40882, 459577, 5109839, 56168169, 611720495, 6609454805, 70937808071, 757060825018, 8040423200947, 85037651263063, 896113850117314, 9413000361625346, 98597629032410971, 1030179406403917981, 10739422018595513973, 111729397883168684917, 1160260967837159869621

Offset: 0

Robert G. Wilson v, Feb 11 2006

nonn

Eric Weisstein's World of Mathematics, Semiprime.

Cf. A001358, A006880, A006988, A066265, A078972.

fQ[n_] := Plus @@ Last /@ FactorInteger@n == 2; c = 0; k = 2; Do[While[c < 10^n, If[fQ@k, c++ ]; k++ ]; Print[k - 1], {n, 0, 8}]
(* checked by *) SemiPrimePi[n_] := Sum[ PrimePi[n/Prime@i] - i + 1, {i, PrimePi@ Sqrt@n}]

use ntheory ":all"; print "$ ",nth_semiprime(10**$),"\n" for 0..15; # Dana Jacobsen, Oct 08 2018

a(14) from Donovan Johnson, Sep 27 2010
Corrected a(14), added a(15)-a(18) from Dana Jacobsen, Oct 10 2018
a(19)-a(20) from Henri Lifchitz, Nov 08 2024

A085770 Number of odd semiprimes < 10^n. Number of terms of A046315 < 10^n.

Original entry on oeis.org

0, 1, 19, 204, 1956, 18245, 168497, 1555811, 14426124, 134432669, 1258822220, 11840335764, 111817881036, 1059796387004, 10076978543513, 96091983644261, 918679875630905, 8803388145953381, 84537081118605467, 813340036541900706, 7838825925851034479, 75669246175972479567

Offset: 0

Hugo Pfoertner, Jul 22 2003

nonn

			a(1)=1 because A046315(1)=9=3*3 is the only odd semiprime < 10^1,
a(2)=19 because there are 19 terms of A046315 < 10^2.

Cf. A046315 (odd numbers divisible by exactly 2 primes), A066265 (number of semiprimes < 10^n), A220262, A292785.

OddSemiPrimePi[n_] := Sum[ PrimePi[n/Prime@i] - i + 1, {i, 2, PrimePi@ Sqrt@ n}]; Table[ OddSemiPrimePi[10^n], {n, 14}] (* Robert G. Wilson v, Feb 02 2006 *)

from math import isqrt
from sympy import primepi, primerange
def A085770(n): return int((-(t:=primepi(s:=isqrt(m:=10**n)))*(t-1)>>1)+sum(primepi(m//k) for k in primerange(3, s+1))) if n>1 else n # Chai Wah Wu, Oct 17 2024

a(n) = A066265(n) - A220262(n) for n > 1. - Jinyuan Wang, Jul 30 2021

a(10)-a(14) from Robert G. Wilson v, Feb 02 2006
a(15)-a(16) from Donovan Johnson, Mar 18 2010
a(0) inserted by and a(17)-a(21) from Jinyuan Wang, Jul 30 2021

A220262 Number of even semiprimes < 10^n. Number of terms of A100484 < 10^n.

Original entry on oeis.org

0, 3, 15, 95, 669, 5133, 41538, 348513, 3001134, 26355867, 234954223, 2119654578, 19308136142, 177291661649, 1638923764567, 15237833654620, 142377417196364, 1336094767763971, 12585956566571620, 118959989688273472, 1127779923790184543, 10720710117789005897

Offset: 0

Robert G. Wilson v, Dec 08 2012

nonn

All such semiprimes have the form 2*p, where p is prime. - T. D. Noe, Dec 09 2012

Kim Walisch, Fast C++ prime counting function implementation.

Cf. A066265, A085770, A100484.

Mathematica
```
Table[PrimePi[10^n/2], {n, 0, 14}]
```

a(n)=primepi(10^n\2) \\ Charles R Greathouse IV, Sep 08 2015

from sympy import primepi
def A220262(n): return primepi(10**n>>1) # Chai Wah Wu, Oct 17 2024

a(n) = A066265(n) - A085770(n) for n > 1.

a(15)-a(20) from Hugo Pfoertner, Oct 14 2017

a(21) from Jinyuan Wang, Jul 30 2021

A292785 Number of odd squarefree semiprimes < 10^n.

Original entry on oeis.org

0, 16, 194, 1932, 18181, 168330, 1555366, 14424896, 134429269, 1258812629, 11840308472, 111817802539, 1059796159358, 10076977878935, 96091981692305, 918679869869451, 8803388128870716, 84537081067757934, 813340036390023775, 7838825925395981969, 75669246174605279757

Offset: 1

Hugo Pfoertner, Oct 10 2017

nonn

			a(2)=16 because there are 16 squarefree odd semiprimes < 10^2: 15=3*5, 21=3*7, 33=3*11, 35=5*7, 39=3*13, 51=3*17, 55=5*11, 57=3*19, 65=5*13, 69=3*23, 77=7*11, 85=5*17, 87=3*29, 91=7*13, 93=3*31, 95=5*19.

Cf. A046388, A066265, A122121, A220262.

-1 + Accumulate@ Table[Count[Range[10^n + 1, 10^(n + 1) - 1, 2], ?(And[ SquareFreeQ@ #, PrimeNu[#] == 2] &)], {n, 0, 5}] (* _Michael De Vlieger, Oct 10 2017 *)

from math import isqrt
from sympy import primepi, primerange
def A292785(n): return int((-(t:=primepi(s:=isqrt(m:=10**n)))*(t+1)>>1)+1+sum(primepi(m//k) for k in primerange(3, s+1))) if n>1 else 0 # Chai Wah Wu, Oct 17 2024

a(n) = A066265(n) - A122121(n) - A220262(n) + 1 for n > 1.

a(21) from Jinyuan Wang, Jul 30 2021

A117526 Least number a(n) which is a product of n primes and such that Pi_n(a(n))/a(n) is maximum.

Original entry on oeis.org

3, 10, 9837, 259441550133

Offset: 1

Martin Raab and Robert G. Wilson v, Mar 25 2006

nonn

Pi_n(a(n))/a(n): 0.66667, 0.40000, 0.25801, 0.2145967653
3=3, 10=2*5, 9837=3*3*1093 & 259441550133=3*89*311*3124409.
3 is the second prime, 10 is the fourth semiprime, 9837 is the 3-almost prime, and 259441550133 is the 4-almost prime.

			a(1)=3 because Pi(2)/2=1/2 < Pi(3)/3=2/3 > Pi(5)/5=3/5.
a(2)=10 because Pi_2(9)/9=1/3 < Pi_2(10)/10=2/5 > Pi_2(14)/14=5/14; Pi_2(10)/10 = Pi_2(15)/15 but 10 < 15.

Eric Weisstein's World of Mathematics, Almost Prime.
Eric Weisstein's World of Mathematics, Prime Factor.

Cf. A006880, A066265, A109251, A114106, A114453.

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 *)
fQ[n_] := Plus @@ Last /@ FactorInteger@n == 4; c = r = 0; Do[If[fQ@n, c++ ]; If[c/n > r, Print[n]; r = c/n], {n, 10^6}]

Comment edited and a(4) added by Donovan Johnson, Mar 10 2010

cp's OEIS Frontend

A120051 Number of 10-almost primes less than or equal to 10^n.

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A120053 Number of 12-almost primes less than or equal to 10^n.

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A125527 Number of semiprimes <= 2^n.

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A120052 Number of 11-almost primes less than or equal to 10^n.

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A036351 Number of numbers <= 10^n that are products of two distinct primes.

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A114125 a(n) is the 10^n-th semiprime.

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A085770 Number of odd semiprimes < 10^n. Number of terms of A046315 < 10^n.

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A220262 Number of even semiprimes < 10^n. Number of terms of A100484 < 10^n.

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