A109251 -id:A109251 - cp's OEIS frontend

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

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

A120034 Number of 3-almost primes t such that 2^n < t <= 2^(n+1).

Original entry on oeis.org

0, 0, 1, 1, 5, 6, 17, 30, 65, 131, 257, 536, 1033, 2132, 4187, 8370, 16656, 33123, 65855, 130460, 259431, 513737, 1019223, 2019783, 4003071, 7930375, 15712418, 31126184, 61654062, 122137206, 241920724, 479226157, 949313939, 1880589368, 3725662783

Offset: 0

Jonathan Vos Post and Robert G. Wilson v, Mar 20 2006

nonn

The partial sum equals the number of Pi_3(2^n) = A127396(n).

			(2^3, 2^4] there is one semiprime, namely 12. 8 was counted in the previous entry.

Cf. A014612, A072114, A109251, A036378, A120033 - A120043.

ThreeAlmostPrimePi[n_] := Sum[PrimePi[n/(Prime@i*Prime@j)] - j + 1, {i, PrimePi[n^(1/3)]}, {j, i, PrimePi@Sqrt[n/Prime@i]}]; t = Table[ ThreePrimePi[2^n], {n, 0, 35}]; Rest@t - Most@t

A072114 Number of 3-almost primes (A014612) <= n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 4, 4, 4, 4, 4, 4, 4, 5, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 9, 10, 10, 10, 10, 10, 11, 11, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 13, 13, 13, 14, 14, 15, 15, 16, 16, 16, 16, 16, 17, 18, 18, 19, 19, 19, 19, 19, 19, 19

Offset: 0

Benoit Cloitre, Jun 19 2002

Number of k <= n such that bigomega(k) = 3.
Let A be a positive integer then card{ x <= n : bigomega(x) = A } ~ (n/Log(n))*Log(Log(n))^(A-1)/(A-1)!. For which n, card{ x <= n : bigomega(x) = 3 } >= card{ x <= n : bigomega(x) = 2 } ?
15530 is the first number for which there are more 3-almost primes than 2-almost primes. See A125149.

E. Landau, Handbuch der Lehre von der Verteilung der Primzahlen, vol. 1, Teubner, Leipzig; third edition : Chelsea, New York (1974).
G. Tenenbaum, Introduction à la théorie analytique et probabiliste des nombres, p. 203, Publications de l'Institut Cartan, 1990.

Charles R Greathouse IV, Table of n, a(n) for n = 0..10000
E. Landau, Handbuch der Lehre von der Verteilung der Primzahlen, vol. 1 and vol. 2, Leipzig, Berlin, B. G. Teubner, 1909.

Cf. A014612, A109251, A001358, A072000.
Partial sums of A101605.
Cf. A125149.

Table[Sum[KroneckerDelta[PrimeOmega[i], 3], {i, n}], {n, 0, 50}] (* Wesley Ivan Hurt, Oct 07 2014 *)

for(n=1,100,print1(sum(i=1,n,bigomega(i)==3),","))

a(n)=my(j,s);forprime(p=2,(n+.5)^(1/3),j=primepi(p)-2;forprime(q=p,sqrtint(n\p),s+=primepi(n\(p*q))-j++));s \\ Charles R Greathouse IV, Mar 21 2012

from math import isqrt
from sympy import primepi, primerange, integer_nthroot
def A072114(n): return int(sum(primepi(n//(k*m))-b for a,k in enumerate(primerange(integer_nthroot(n,3)[0]+1)) for b,m in enumerate(primerange(k,isqrt(n//k)+1),a))) # Chai Wah Wu, Aug 17 2024

a(n) = card{ x <= n : bigomega(x) = 3 }, asymptotically : a(n) ~ (n/log(n))*log(log(n))^2/2 [Landau, p. 211].

A127396 Number of 3-almostprimes <= 2^n.

Original entry on oeis.org

0, 0, 1, 2, 7, 13, 30, 60, 125, 256, 513, 1049, 2082, 4214, 8401, 16771, 33427, 66550, 132405, 262865, 522296, 1036033, 2055256, 4075039, 8078110, 16008485, 31720903, 62847087, 124501149, 246638355, 488559079, 967785236, 1917099175, 3797688543

Offset: 1

Robert G. Wilson v, Dec 29 2006

Robert G. Wilson v, Table of n, a(n) for n = 1..53
Eric Weisstein's World of Mathematics, Semiprime.

Cf. A126279, A109251, A125527.

ThreeAlmostPrimePi[n_] := Sum[PrimePi[n/(Prime@i*Prime@j)] - j + 1, {i, PrimePi[n^(1/3)]}, {j, i, PrimePi[Sqrt[n/Prime@i]]}]; Table[ ThreeAlmostPrimePi[2^n], {n, 30}]

a(n) = A072114(2^n). - R. J. Mathar, Aug 26 2011

A126280 Triangle read by rows: T(k,n) is number of numbers <= 10^n that are products of k primes.

Original entry on oeis.org

4, 4, 1, 25, 34, 22, 12, 4, 2, 168, 299, 247, 149, 76, 37, 14, 7, 2, 1229, 2625, 2569, 1712, 963, 485, 231, 105, 47, 22, 7, 3, 1, 9592, 23378, 25556, 18744, 11185, 5933, 2973, 1418, 671, 306, 138, 63, 25, 11, 4, 2, 78498, 210035, 250853, 198062, 124465, 68963

Offset: 1

Jonathan Vos Post & Robert G. Wilson v, Dec 22 2006

The n-th row's sum is 10^n - 1.

			4 4 1
25 34 22 12 4 2
168 299 247 149 76 37 14
7 2
1229 2625 2569 1712 963 485 231
105 47 22 7 3 1
9592 23378 25556 18744 11185 5933 2973
1418 671 306 138 63 25 11 4 2
78498 210035 250853 198062 124465 68963 35585 17572
8491 4016 1878 865 400 179 79 35 14 7 2
664579 1904324 2444359 2050696 1349779 774078 409849 207207
101787 49163 23448 11068 5210 2406 1124 510 233 102 45 21 7 3 1

Robert G. Wilson v, Table of n, a(n) for n = 1..342
Eric Weisstein's World of Mathematics, Semiprime.
Eric Weisstein's World of Mathematics, Almost Prime.

By columns: A006880, A036352, A109251, A114106, A114453, A120047, A120048, A120049, A120050, A120051, A120052, A120053.

The n-th row's sum: A002283 = 10^n -1, A116430, A126279: same array but for powers of two.

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[m, 10^n], {n, 6}, {m, Floor[Log[2, 10^n]] }] // Flatten

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

A120044 The 10^n-th 3-almost prime.

Original entry on oeis.org

8, 45, 412, 3918, 38991, 395085, 4046429, 41657362, 429891626, 4439956573, 45851698382, 473238120286, 4880292241955, 50280826966354

Offset: 0

Robert G. Wilson v, Feb 15 2006

Cf. A109251, A006988, A114125, A120045, A120046.

ThreeAlmostPrimePi[n_] := Sum[ PrimePi[n/(Prime@i*Prime@j)] - j + 1, {i, PrimePi[n^(1/3)]}, {j, i, PrimePi@ Sqrt[n/Prime@i]}]; ThreeAlmostPrime[n_] := Block[{e = Floor[Log[2, n]], a, b}, a = 2^e; Do[b = 2^p; While[ThreeAlmostPrimePi[a] < n, a = a + b]; a = a - b/2, {p, e, 0, -1}]; a + b/2]; Do[ Print@ThreeAlmostPrime[10^n], {n, 0, 13}]
ThreePrime[n_] := Block[{e = Floor[ Log[2, n] +2], a, b}, a = 2^e; Do[b = 2^p; While[ ThreePrimePi[a] < n, a = a + b]; a = a - b/2, {p, e, 0, -1}]; a + b/2]; Table[ ThreePrime[n], {n, 0, 13}]

A124033 Number of n-digit numbers having exactly n prime factors (counted with multiplicity).

Original entry on oeis.org

4, 31, 225, 1563, 10222, 63030, 374264, 2160300, 12196405, 67724342, 371233523, 2014305995, 10841722966, 57974736592, 308361428628, 1632877406997

Offset: 1

J. M. Bergot, Apr 08 2011

Essentially the same as A036335.

What would be the ratio between a(n) and all possible numbers with n digits for each n?

			a(1) = A006880(1) = 4.
a(2) = A066265(2) - A066265(1) = 34 - 3 = 31.
a(3) = A109251(3) - A109251(2) = 247 - 22 = 225.
a(4) = A114106(4) - A114106(3) = 1712 - 149 = 1563.
a(5) = A114453(5) - A114453(4) = 11185 - 963 = 10222.
a(6) = A120047(6) - A120047(5) = 68963 - 5933 = 63030.
a(7) = A120048(7) - A120048(6) = 409849 - 35585 = 374264.
a(8) = A120049(8) - A120049(7) = 2367507 - 207207 = 2160300.
a(9) = A120050(9) - A120050(8) = 13377156 - 1180751 = 12196405.
a(10) = A120051(10) - A120051(9) = 74342563 - 6618221 = 67724342.
a(11) = A120052(11) - A120052(10) = 407818620 - 36585097 = 371233523.
a(12) = A120053(12) - A120053(11) = 2214357712 - 200051717 = 2014305995.

Table[Count[Range[10^(n-1),10^n-1],?(PrimeOmega[#]==n&)],{n,8}]  (* _Harvey P. Dale, Apr 22 2011 *)
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 *)
f[n_] := AlmostPrimePi[n, 10^n - 1] - AlmostPrimePi[n, 10^(n - 1) - 1]; Array[f, 12] (* Robert G. Wilson v, Jul 06 2012 *)

Corrected and extended by Ray Chandler, Apr 11 2011
a(9)-a(12) from Ray Chandler, Apr 12 2011
a(13)-a(16) from Robert G. Wilson v, Jul 06 2012

cp's OEIS Frontend

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

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A120034 Number of 3-almost primes t such that 2^n < t <= 2^(n+1).

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A072114 Number of 3-almost primes (A014612) <= n.

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A127396 Number of 3-almostprimes <= 2^n.

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A126280 Triangle read by rows: T(k,n) is number of numbers <= 10^n that are products of k primes.

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

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A120044 The 10^n-th 3-almost prime.

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A124033 Number of n-digit numbers having exactly n prime factors (counted with multiplicity).

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