A120043 -id:A120043 - cp's OEIS frontend

A120033 Number of semiprimes s such that 2^n < s <= 2^(n+1).

0, 1, 1, 4, 4, 12, 20, 40, 75, 147, 285, 535, 1062, 2006, 3918, 7548, 14595, 28293, 54761, 106452, 206421, 401522, 780966, 1520543, 2962226, 5777162, 11272279, 22009839, 43006972, 84077384, 164482781, 321944211, 630487562, 1235382703

Offset: 0

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

nonn

The partial sum equals the number of Pi_2(2^n) = A125527(n).

			(2^2, 2^3] there is one semiprime, namely 6. 4 was counted in the previous entry.

Dana Jacobsen, Table of n, a(n) for n = 0..62 (first 48 terms from Charles R Greathouse IV, corrected a(47)-a(48))

Cf. A001358, A072000, A066265, A036378, A120033-A120043.

SemiPrimePi[n_] := Sum[PrimePi[n/Prime[i]] - i + 1, {i, PrimePi[Sqrt[n]]}]; t = Table[SemiPrimePi[2^n], {n, 0, 35}]; Rest@t - Most@t

pi2(n)=my(s,i); forprime(p=2, sqrt(n), s+=primepi(n\p); i++); s - i * (i-1)/2
a(n)=pi2(2^(n+1))-pi2(2^n) \\ Charles R Greathouse IV, May 15 2012

use ntheory ":all"; print "$ ",semiprime_count(1+(1<<$), 1<<($+1)),"\n" for 0..48; # _Dana Jacobsen, Mar 04 2019

use ntheory ":all"; my $l=0; for (0..48) { my $c=semiprime_count(1<<($+1)); print "$ ",$c-$l,"\n"; $l=$c; } # Dana Jacobsen, Mar 04 2019

A120035 Number of 4-almost primes f such that 2^n < f <= 2^(n+1).

Original entry on oeis.org

0, 0, 0, 1, 1, 5, 7, 20, 37, 81, 173, 344, 736, 1461, 3065, 6208, 12643, 25662, 52014, 105487, 212566, 430007, 865650, 1744136, 3508335, 7053390, 14167804, 28441899, 57065447, 114418462, 229341261, 459442819, 920097130, 1841946718, 3686197728

Offset: 0

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

nonn

The partial sum equals the number of Pi_4(2^n) = A334069(n).

			(2^4, 2^5] there is one semiprime, namely 24. 16 was counted in the previous entry.

Cf. A014613, A082996, A114106, A036378, A120033 - A120043, A334069.

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)]}]; t = Table[ FourAlmostPrimePi[2^n], {n, 0, 37}]; Rest@t - Most@t

from math import isqrt
from sympy import primepi, primerange, integer_nthroot
def A120035(n):
    x = 1<Chai Wah Wu, Mar 28 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

A120036 Number of 5-almost primes 5ap such that 2^n < 5ap <= 2^(n+1).

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 5, 8, 21, 41, 91, 199, 403, 873, 1767, 3740, 7709, 15910, 32759, 67185, 138063, 281566, 576165, 1173435, 2390366, 4860357, 9873071, 20033969, 40612221, 82266433, 166483857, 336713632, 680482316, 1374413154, 2774347425

Offset: 0

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

nonn

The partial sum equals the number of Pi_5(2^n) = 0, 0, 0, 0, 1, 2, 7, 15, 36, 77, 168, 367, 770, 1643,..

			(2^5, 2^6] there is one semiprime, namely 48. 32 was counted in the previous entry.

Cf. A014614, A114453, A036378, A120033, A120034, A120035, A120036, A120037, A120038, A120039, A120040, A120041, A120042, A120043.

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)]}]; t = Table[ FiveAlmostPrimePi[2^n], {n, 0, 37}]; Rest@t - Most@t

A120037 Number of 6-almost primes 6ap such that 2^n < 6ap <= 2^(n+1).

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 5, 8, 22, 44, 96, 215, 439, 959, 1967, 4185, 8735, 18143, 37695, 77939, 161479, 332008, 684502, 1404867, 2882712, 5904454, 12078654, 24682057, 50375102, 102724466, 209250102, 425921989, 866187909, 1760280404, 3574740094

Offset: 0

Jonathan Vos Post and Robert G. Wilson v, Mar 21 2006

nonn

The partial sum equals the number of Pi_6(2^n).

			(2^6, 2^7] there is one semiprime, namely 96. 64 was counted in the previous entry.

Cf. A046306, A036378, A120033, A120034, A120035, A120036, A120037, A120038, A120039, A120040, A120041, A120042, A120043.

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 *)
t = Table[AlmostPrimePi[6, 2^n], {n, 0, 30}]; Rest@t - Most@t

A120038 Number of 7-almost primes 7ap such that 2^n < 7ap <= 2^(n+1).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 1, 5, 8, 22, 46, 99, 224, 461, 1013, 2093, 4459, 9388, 19603, 40946, 85087, 177200, 366248, 758686, 1565038, 3226717, 6641105, 13648299, 28018956, 57445770, 117667693, 240751326, 492172466, 1005221914, 2051468099

Offset: 0

Jonathan Vos Post and Robert G. Wilson v, Mar 21 2006

nonn

The partial sum equals the number of Pi_7(2^n).

			(2^7, 2^8] there is one semiprime, namely 192. 128 was counted in the previous entry.

Cf. A046308, A036378, A120033, A120034, A120035, A120036, A120037, A120038, A120039, A120040, A120041, A120042, A120043.

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 *)
t = Table[AlmostPrimePi[7, 2^n], {n, 0, 30}]; Rest@t - Most@t

A120039 Number of 8-almost primes 8ap such that 2^n < 8ap <= 2^(n+1).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 1, 5, 8, 22, 47, 101, 229, 473, 1044, 2171, 4634, 9796, 20513, 43020, 89684, 187361, 388633, 807508, 1671160, 3455934, 7135226, 14708436, 30286472, 62280024, 127944070, 262543635, 538266791, 1102507513, 2256357137

Offset: 0

Jonathan Vos Post and Robert G. Wilson v, Mar 21 2006

nonn

The partial sum equals the number of Pi_8(2^n).

			(2^8, 2^9] there is one semiprime, namely 384. 256 was counted in the previous entry.

Cf. A046310, A036378, A120033, A120034, A120035, A120036, A120037, A120038, A120039, A120040, A120041, A120042, A120043.

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 *)
t = Table[AlmostPrimePi[8, 2^n], {n, 0, 30}]; Rest@t - Most@t

A120040 Number of 9-almost primes 9ap such that 2^n < 9ap <= 2^(n+1).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 5, 8, 22, 47, 102, 232, 482, 1062, 2217, 4738, 10051, 21083, 44315, 92608, 193824, 402936, 838879, 1739794, 3605077, 7457977, 15404202, 31781036, 65481376, 134777594, 277096118, 569173839, 1168002568, 2394834166

Offset: 0

Jonathan Vos Post and Robert G. Wilson v, Mar 21 2006

nonn

The partial sum equals the number of Pi_9(2^n).

			(2^9, 2^10] there is one semiprime, namely 768. 512 was counted in the previous entry.

Cf. A046312, A036378, A120033, A120034, A120035, A120036, A120037, A120038, A120039, A120040, A120041, A120042, A120043.

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 *)
t = Table[AlmostPrimePi[9, 2^n], {n, 0, 30}]; Rest@t - Most@t

A120041 Number of 10-almost primes k such that 2^n < k <= 2^(n+1).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 5, 8, 22, 47, 103, 233, 487, 1072, 2246, 4803, 10202, 21440, 45115, 94434, 197891, 412010, 858846, 1783610, 3700698, 7665755, 15853990, 32750248, 67564405, 139238488, 286625278, 589472979, 1211146741, 2486322304

Offset: 0

Jonathan Vos Post and Robert G. Wilson v, Mar 21 2006

nonn

Chai Wah Wu, Table of n, a(n) for n = 0..47

Cf. A046314, A036378, A120033, A120034, A120035, A120036, A120037, A120038, A120039, A120040, A120041, A120042, A120043.

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 *)
t = Table[AlmostPrimePi[10, 2^n], {n, 0, 39}]; Rest@t - Most@t

from math import isqrt, prod
from sympy import primerange, integer_nthroot, primepi
def A120041(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)))
    def almostprimepi(n,k): 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))
    return -almostprimepi(m:=1<Chai Wah Wu, Aug 31 2024

a(n) ~ 2^n log^9 n/(725760 n log 2). [Charles R Greathouse IV, Dec 28 2011]

A120042 Number of 11-almost primes 11ap such that 2^n < 11ap <= 2^(n+1).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 5, 8, 22, 47, 103, 234, 490, 1078, 2261, 4844, 10294, 21659, 45609, 95580, 200422, 417715, 871452, 1811412, 3761623, 7798409, 16142081, 33373093, 68906782, 142120436, 292797806, 602653984, 1239225631

Offset: 0

Jonathan Vos Post and Robert G. Wilson v, Mar 21 2006

nonn

The partial sum equals the number of Pi_11(2^n).

			(2^11, 2^12] there is one semiprime, namely 3072. 2048 was counted in the previous entry.

Chai Wah Wu, Table of n, a(n) for n = 0..50

Cf. A069272, A036378, A120033, A120034, A120035, A120036, A120037, A120038, A120039, A120040, A120041, A120042, A120043.

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 *)
t = Table[AlmostPrimePi[11, 2^n], {n, 0, 30}]; Rest@t - Most@t

from math import isqrt, prod
from sympy import primerange, integer_nthroot, primepi
def A120042(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)))
    def almostprimepi(n, k): 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))
    return -almostprimepi(m:=1<Chai Wah Wu, Jun 17 2025

cp's OEIS Frontend

A120033 Number of semiprimes s such that 2^n < s <= 2^(n+1).

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A120035 Number of 4-almost primes f such that 2^n < f <= 2^(n+1).

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

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A120036 Number of 5-almost primes 5ap such that 2^n < 5ap <= 2^(n+1).

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A120037 Number of 6-almost primes 6ap such that 2^n < 6ap <= 2^(n+1).

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A120038 Number of 7-almost primes 7ap such that 2^n < 7ap <= 2^(n+1).

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A120039 Number of 8-almost primes 8ap such that 2^n < 8ap <= 2^(n+1).

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A120040 Number of 9-almost primes 9ap such that 2^n < 9ap <= 2^(n+1).

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A120041 Number of 10-almost primes k such that 2^n < k <= 2^(n+1).

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