A014613 -id:A014613 - cp's OEIS frontend

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

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

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

Robert G. Wilson v, Feb 07 2006

nonn

			There are 14 seven-almost primes up to 1000: 128, 192, 288, 320, 432, 448, 480, 648, 672, 704, 720, 800, 832 & 972.

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[7, 10^n], {n, 11}]

More terms from Robert G. Wilson v, Jan 07 2007
Example corrected by Harvey P. Dale, Jan 25 2013
a(15)-a(18) from Henri Lifchitz, Mar 18 2025

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

Robert G. Wilson v, Feb 07 2006

nonn

			There are 2 nine-almost primes up to 1000: 512 & 768.

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[9, 10^n], {n, 12}]

a(13) and a(14) from Robert G. Wilson v, Jan 07 2007

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

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

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

A109287 4-almost primes equal to p*q + 1, where p and q are (not necessarily distinct) primes.

Original entry on oeis.org

16, 36, 40, 56, 88, 135, 156, 184, 204, 210, 220, 248, 250, 260, 296, 306, 315, 328, 330, 340, 342, 372, 414, 459, 472, 490, 516, 536, 546, 580, 584, 636, 650, 686, 690, 708, 714, 732, 735, 738, 804, 808, 819, 836, 850, 852, 870, 872, 940, 950, 966, 975, 996

Offset: 1

Giovanni Teofilatto and Jonathan Vos Post, Aug 20 2005

4-almost primes of the form semiprime + 1.

			a(1) = 16 because (3*5+1)=(2^4) = 16.
a(2) = 36 because (5*7+1)=((2^2)*(3^2)) = 36.
a(3) = 40 because (3*13+1)=((2^3)*5) = 40.
a(4) = 56 because (5*11+1)=((2^3)*7) = 56.
a(5) = 88 because (3*29+1)=((2^3)*11) = 88.
a(6) = 135 because (2*67+1)=((3^3)*5) = 135.
a(7) = 156 because (5*31+1)=((2^2)*3*13) = 156.
a(8) = 184 because (3*61+1)=((2^3)*23) = 184.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Primes are in A000040. Semiprimes are in A001358. 4-almost primes are in A014613.
Primes of the form semiprime + 1 are in A005385 (safe primes).
Semiprimes of the form semiprime + 1 are in A109373.
3-almost primes of the form semiprime + 1 are in A109067.
4-almost primes of the form semiprime + 1 are in this sequence.
5-almost primes of the form semiprime + 1 are in A109383.
Least n-almost prime of the form semiprime + 1 are in A128665.
Similar to A076153; after A076153(0)=3 next difference is A076153(100)=1792 whereas A109287(100)=1794.
Cf. A077065, A079148, A092307, A109288, A109289, A109290.

bo[n_] := Plus @@ Last /@ FactorInteger[n]; Select[Range[1000], bo[ # ] == 4 && bo[ # - 1] == 2 &] (* Ray Chandler, Aug 27 2005 *)

is(n)=bigomega(n)==4 && bigomega(n-1)==2 \\ Charles R Greathouse IV, Sep 16 2015

a(n) is in this sequence iff a(n) is in A014613 and (a(n)-1) is in A001358.

Extended by Ray Chandler, Aug 27 2005

Edited by Ray Chandler, Mar 20 2007

A111176 Sophie Germain 4-almost primes.

Original entry on oeis.org

40, 220, 580, 712, 808, 812, 904, 940, 1062, 1192, 1444, 1592, 1612, 1690, 1812, 1876, 2002, 2152, 2212, 2236, 2254, 2488, 2502, 2562, 2650, 2662, 2788, 3010, 3052, 3064, 3112, 3162, 3208, 3258, 3272, 3352, 3448, 3550, 3580, 3820, 3832, 3892, 3910, 4012

Offset: 1

Jonathan Vos Post, Oct 22 2005

4-almost primes P such that 2*P + 1 are also 4-almost primes. There should also be 4-almost prime chains of length k analogous to Cunningham chains of the first kind and Tomaszewski chains of the first kind. A 4-almost prime chain of length k is a sequence of 4-almost primes a(1) < a(2) < ... < a(k) such that a(i+1) = 2*a(i) + 1 for i = 1, ..., k-1. There are no such chains beginning with integers under 1200.

			n p 2*p+1
1 40 = 2^3 * 5 81 = 3^4
2 220 = 2^2 * 5 * 11 441 = 3^2 * 7^2
3 580 = 2^2 * 5 * 29 1161 = 3^3 * 43
4 712 = 2^3 * 89 1425 = 3 * 5^2 * 19
5 808 = 2^3 * 101 1617 = 3 * 7^2 * 11
6 812 = 2^2 * 7 * 29 1625 = 5^3 * 13

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A005384, A014613, A111153, A111168, A111170, A111171, A111173.

Select[Range[5000],PrimeOmega[#]==PrimeOmega[2#+1]==4&] (* Harvey P. Dale, Nov 09 2011 *)

{a(n)} = a(n) is an element of A014613 and 2*a(n)+1 is an element of A014613.

Extended by Ray Chandler, Oct 22 2005

A111496 4-almost primes with semiprime digits (digits 4, 6, 9 only).

Original entry on oeis.org

444, 644, 664, 666, 966, 996, 999, 4444, 4466, 4494, 4696, 4964, 6644, 6666, 6699, 9444, 9496, 9646, 9964, 9966, 9999, 44444, 44499, 44646, 44996, 46444, 46449, 46964, 46996, 49444, 49494, 49946, 49966, 49996, 64446, 64494, 64644, 64666

Offset: 1

Jonathan Vos Post, Nov 16 2005

			a(1) = 444 = 2^2 * 3 * 37, a(2) = 644 = 2^2 * 7 * 23, a(3) = 664 = 2^3 * 83, a(4) = 666 = 2 * 3^2 * 37, a(5) = 966 = 2 * 3 * 7 * 23, a(6) = 996 = 2^2 * 3 * 83, a(7) = 999 = 3^3 * 37, a(8) = 4466 = 2 * 7 * 11 * 29, a(9) = 4494 = 2 * 3 * 7 * 107, a(10) = 4696 = 2^3 * 587.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A001358, A014613, A107665, A107666, A107342, A111494, A111697.

do(N)=my(v=List(), a=[4, 6, 9]); for(d=1, N, forvec(u=vector(d, i, [1, 3]), t=fromdigits(apply(n->a[n], u)); if(bigomega(t)==4, listput(v, t)))); Set(v) \\ Charles R Greathouse IV, Feb 01 2017

A116453 Third smallest number with exactly n prime factors.

Original entry on oeis.org

5, 9, 18, 36, 72, 144, 288, 576, 1152, 2304, 4608, 9216, 18432, 36864, 73728, 147456, 294912, 589824, 1179648, 2359296, 4718592, 9437184, 18874368, 37748736, 75497472, 150994944, 301989888, 603979776, 1207959552, 2415919104

Offset: 1

Reinhard Zumkeller, Feb 16 2006

Smallest term in A116451 having exactly n prime factors;
a(n) = A116451(A116454(n)) = A116454(n) + 2 * n + 3;
a(n) = A005010(n-2) = 9*2^(n - 2) for n > 1.

			a(1) =  5 = A000040(3) > A000040(2) =  3 > A000040(1) =  2;
a(2) =  9 = A001358(3) > A001358(2) =  6 > A001358(1) =  4;
a(3) = 18 = A014612(3) > A014612(2) = 12 > A014612(1) =  8;
a(4) = 36 = A014613(3) > A014613(2) = 24 > A014613(1) = 16.

Index entries for linear recurrences with constant coefficients, signature (2).

def A116453(n): return 9<1 else 5 # 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

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

cp's OEIS Frontend

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

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A120048 Number of 7-almost primes less than or equal to 10^n.

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A120050 Number of 9-almost primes less than or equal to 10^n.

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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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A109287 4-almost primes equal to p*q + 1, where p and q are (not necessarily distinct) primes.

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A111176 Sophie Germain 4-almost primes.

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A111496 4-almost primes with semiprime digits (digits 4, 6, 9 only).

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