A120049 -id:A120049 - cp's OEIS frontend

A130051 A bisection of A120049.

0, 6, 17, 32, 51, 72, 97, 127, 161, 197, 236, 279, 327, 378, 432, 489, 550, 614, 681, 751, 826, 906, 989, 1074, 1162, 1252, 1347, 1445, 1546, 1650, 1759, 1870, 1986, 2105, 2227, 2352, 2480, 2611, 2745, 2884, 3026, 3171, 3319, 3470, 3624, 3781, 3943, 4108

Offset: 1

Clark Kimberling, May 03 2007

nonn

The other bisection is A130050; both bisections are strictly increasing.

			A130049 = (0,3,6,7,17,12,32,20,51,29,72,...), so that
A130051 = (0,6,17,32,51,72,...).

Cf. A130048, A130049, A130050.

a(n)=b(2n-1), for n>=1, where b=A130049.

A130050 A bisection of A120049.

Original entry on oeis.org

3, 7, 12, 20, 29, 39, 50, 63, 77, 92, 108, 126, 145, 166, 188, 211, 235, 260, 286, 313, 341, 371, 402, 435, 469, 504, 540, 577, 615, 655, 696, 738, 781, 825, 870, 916, 963, 1011, 1060, 1112, 1165, 1219, 1274, 1330, 1387, 1445, 1504, 1564, 1625, 1687, 1751

Offset: 1

Clark Kimberling, May 03 2007

nonn

The other bisection is A130051; both bisections are strictly increasing.

Cf. A130048, A130049, A130051.

a(n)=b(2n), for n>=1, where b=A130049.

A046310 Numbers that are divisible by exactly 8 primes counting multiplicity.

Original entry on oeis.org

256, 384, 576, 640, 864, 896, 960, 1296, 1344, 1408, 1440, 1600, 1664, 1944, 2016, 2112, 2160, 2176, 2240, 2400, 2432, 2496, 2916, 2944, 3024, 3136, 3168, 3240, 3264, 3360, 3520, 3600, 3648, 3712, 3744, 3968, 4000, 4160, 4374, 4416, 4536, 4704, 4736

Offset: 1

Patrick De Geest, Jun 15 1998

nonn

Also called 8-almost primes. Products of exactly 8 primes (not necessarily distinct). Any 8-almost prime can be represented in several ways as a product of two 4-almost primes A014613 and in several ways as a product of four semiprimes A001358. - Jonathan Vos Post, Dec 11 2004

Odd terms are in A046321; first odd term is a(64)=6561=3^8. - Zak Seidov, Feb 08 2016

T. D. Noe, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Reference

Cf. A046309, A120049 (number of 8-almost primes <= 10^n).
Cf. A101637, A101638, A101605, A101606.
Sequences listing r-almost primes, that is, the n such that A001222(n) = r: A000040 (r = 1), A001358 (r = 2), A014612 (r = 3), A014613 (r = 4), A014614 (r = 5), A046306 (r = 6), A046308 (r = 7), this sequence (r = 8), A046312 (r = 9), A046314 (r = 10), A069272 (r = 11), A069273 (r = 12), A069274 (r = 13), A069275 (r = 14), A069276 (r = 15), A069277 (r = 16), A069278 (r = 17), A069279 (r = 18), A069280 (r = 19), A069281 (r = 20).
Cf. A046321.

A046310 := proc(n)
    option remember;
    if n = 1 then
        2^8 ;
    else
        for a from procname(n-1)+1 do
            if numtheory[bigomega](a) = 8 then
                return a;
            end if;
        end do:
    end if;
end proc:
seq(A046310(n),n=1..30) ; # R. J. Mathar, Dec 21 2018

Select[Range[1600], Plus @@ Last /@ FactorInteger[ # ] == 8 &] (* Vladimir Joseph Stephan Orlovsky, Apr 23 2008 *)
Select[Range[5000],PrimeOmega[#]==8&]  (* Harvey P. Dale, Apr 19 2011 *)

is(n)=bigomega(n)==8 \\ Charles R Greathouse IV, Mar 21 2013

from math import prod, isqrt
from sympy import primerange, integer_nthroot, primepi
def A046310(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 f(x): return int(n-1+x-sum(primepi(x//prod(c[1] for c in a))-a[-1][0] for a in g(x,0,1,1,8)))
    kmin, kmax = 1,2
    while f(kmax) >= kmax:
        kmax <<= 1
    while True:
        kmid = kmax+kmin>>1
        if f(kmid) < kmid:
            kmax = kmid
        else:
            kmin = kmid
        if kmax-kmin <= 1:
            break
    return kmax # Chai Wah Wu, Aug 23 2024

Product p_i^e_i with Sum e_i = 8.
a(n) ~ 5040n log n / (log log n)^7. - Charles R Greathouse IV, May 06 2013
a(n) = A078840(8,n). - R. J. Mathar, Jan 30 2019

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

Robert G. Wilson v, Feb 07 2006

			There are 2 six-almost primes up to 100: 64 and 96, so a(2) = 2.

Cf. A066265, A014613, A116382.
Cf. A006880, A036352, A109251, A114106, A114453.
Cf. 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[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

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

a(15)-a(18) from Henri Lifchitz, Feb 03 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

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

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

cp's OEIS Frontend

A130051 A bisection of A120049.

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A130050 A bisection of A120049.

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A046310 Numbers that are divisible by exactly 8 primes counting multiplicity.

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A120047 Number of 6-almost primes less than or equal to 10^n.

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

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