cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 15 results. Next

A078840 Table of n-almost-primes T(n,k) (n >= 0, k > 0), read by antidiagonals, starting at T(0,1)=1 followed by T(1,1)=2.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 7, 9, 12, 16, 11, 10, 18, 24, 32, 13, 14, 20, 36, 48, 64, 17, 15, 27, 40, 72, 96, 128, 19, 21, 28, 54, 80, 144, 192, 256, 23, 22, 30, 56, 108, 160, 288, 384, 512, 29, 25, 42, 60, 112, 216, 320, 576, 768, 1024, 31, 26, 44, 81, 120, 224, 432, 640, 1152
Offset: 0

Views

Author

Benoit Cloitre and Paul D. Hanna, Dec 10 2002

Keywords

Comments

An n-almost-prime is a positive integer that has exactly n prime factors.
This sequence is a rearrangement of the natural numbers. - Robert G. Wilson v, Feb 11 2006
Each antidiagonal begins with the n-th prime and ends with 2^n.
From Eric Desbiaux, Jun 27 2009: (Start)
(A001222 gives this sequence)
A001221 gives another table:
1
- 2 3 4 5 7 8 9 11 ... A000961
- 6 10 12 14 15 18 20 21 ... A007774
- 30 42 60 66 70 78 84 90 ... A033992
- 210 330 390 420 462 510 546 570 ... A033993
- 2310 2730 3570 3990 4290 4620 4830 5460 ... A051270
Antidiagonals begin with A000961 and end with A002110.
Diagonal is A073329 which is last term in n-th row of A048692. (End)

Examples

			Table begins:
  1
  -  2  3   5   7  11  13  17  19  23  29 ...
  -  4  6   9  10  14  15  21  22  25  26 ...
  -  8 12  18  20  27  28  30  42  44  45 ...
  - 16 24  36  40  54  56  60  81  84  88 ...
  - 32 48  72  80 108 112 120 162 168 176 ...
  - 64 96 144 160 216 224 240 324 336 352 ...

Links

Crossrefs

Cf. A078840, A078841, A078842, A078843, A078844, A078445, A078846, A109636.
T(1, k)=A000040(k), T(2, k)=A001358(k), T(3, k)=A014612(k), T(4, k)=A014613(k), T(5, k)=A014614(k), T(6, k)=A046306(k), T(7, k)=A046308(k), T(8, k)=A046310(k), T(9, k)=A046312(k), T(10, k)=A046314(k).
T(11, k)=A069272(k), T(12, k)=A069273(k), T(13, k)=A069274(k), T(14, k)=A069275(k), T(15, k)=A069276(k), T(16, k)=A069277(k), T(17, k)=A069278(k), T(18, k)=A069279(k), T(19, k)=A069280(k), T(20, k)=A069281(k).
T(k, 1)=A000079(k), T(k, 2)=A007283(k), T(k, 3)=A116453(k), T(k, k)=A101695(k), T(k, k+1)=A078841(k).
A091538 is this sequence with zeros inserted, making a square array.

Programs

  • Mathematica
    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 *)
AlmostPrime[k_, n_] := Block[{e = Floor[Log[2, n]+k], a, b}, a = 2^e; Do[b = 2^p; While[ AlmostPrimePi[k, a] < n, a = a + b]; a = a - b/2, {p, e, 0, -1}]; a + b/2]; Table[ AlmostPrime[k, n - k + 1], {n, 11}, {k, n}] // Flatten (* Robert G. Wilson v *)
mx = 11; arr = NestList[Take[Union@Flatten@Outer[Times, #, primes], mx] &, primes = Prime@Range@mx, mx]; Prepend[Flatten@Table[arr[[k, n - k + 1]], {n, mx}, {k, n}], 1] (* Ivan Neretin, Apr 30 2016 *)
(* The next code skips the initial 1. *)
width = 15; (seq = Table[
  Rest[NestList[1 + NestWhile[# + 1 &, #, ! PrimeOmega[#] == z &] &,
  2^z, width - z + 1]] - 1, {z, width}]) // TableForm
Flatten[Map[Reverse[Diagonal[Reverse[seq], -width + #]] &, Range[width]]]
(* Peter J. C. Moses, Jun 05 2019 *)
Grid[Table[Select[Range[200], PrimeOmega[#] == n &], {n, 0, 7}]]
(* Clark Kimberling, Nov 17 2024 *)
  • PARI
    T(n,k)=if(k<0,0,s=1; while(sum(i=1,s,if(bigomega(i)-n,0,1))
  • Python
    from math import prod, isqrt
from sympy import primerange, integer_nthroot, primepi, prime
def A078840_T(n,k):
    if n == 1: return prime(k)
    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(k-1+x-sum(primepi(x//prod(c[1] for c in a))-a[-1][0] for a in g(x,0,1,1,n)))
    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

Extensions

Edited by Robert G. Wilson v, Feb 11 2006

A078843 Where 3^n occurs in n-almost primes, starting at a(0)=1.

Original entry on oeis.org

1, 2, 3, 5, 8, 14, 23, 39, 64, 103, 169, 269, 427, 676, 1065, 1669, 2628, 4104, 6414, 10023, 15608, 24281, 37733, 58503, 90616, 140187, 216625, 334527, 516126, 795632, 1225641, 1886570, 2901796, 4460359, 6851532, 10518476, 16138642, 24748319
Offset: 0

Views

Author

Benoit Cloitre and Paul D. Hanna, Dec 10 2002

Keywords

Examples

			a(3) = 5 since 3^3 is the 5th 3-almost-prime: 8,12,18,20,27,....., A014612.

Links

Crossrefs

Cf. A078840, A078841, A078842, A078844, A078845, A078846.

Programs

  • Mathematica
    AlmostPrimePi[k_Integer /; k > 1, n_] := Module[{a, i}, a[0] = 1; 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[n, 3^n], {n, 2, 37}] (* Robert G. Wilson v, Feb 09 2006 *)
  • PARI
    a(n)=sum(i=1,3^n,if(bigomega(i)-n,0,1))
  • PARI
    { appi(k,n,m=2) = local(r=0);
if(k==0,return(1));
if(k==1,return(primepi(n)));
forprime(p=m, floor(sqrtn(n,k)+1e-20),
r+=appi(k-1,n\p,p)-(k==2)*(primepi(p)-1));
r }
{ appi3(k,n) = appi(k,n) - if(k>=1,appi(k-1,n\3)) }
a=1; for(n=1,50, k=ceil(n*log(5/3)/log(5/2)); a+=appi3(n-k,3^n\2^k); print1(a,", "))
\\ Max Alekseyev, Jan 06 2008
  • Python
    from math import isqrt, prod
from sympy import primerange, integer_nthroot, primepi
def A078843(n):
    def almostprimepi(n,k):
        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))
    return almostprimepi(3**n,n) if n else 1 # Chai Wah Wu, Sep 01 2024

Formula

a(n) = a(n-1) + appi3(n-k, floor(3^n/2^k)), where k = ceiling(n*c) with c = log(5/3)/log(5/2) = 0.55749295065024006729857073190835923443... and appi3(k,n) is the number of k-almost primes not divisible by 3 and not exceeding n. - Max Alekseyev, Jan 06 2008

Extensions

a(14)-a(37) from Robert G. Wilson v, Feb 09 2006

A078841 Main diagonal of the table of k-almost primes (A078840): a(n) = (n+1)-st integer that is an n-almost prime.

Original entry on oeis.org

1, 3, 9, 20, 54, 112, 240, 648, 1344, 2816, 5760, 12800, 26624, 62208, 129024, 270336, 552960, 1114112, 2293760, 4915200, 9961472, 20447232, 47775744, 96468992, 198180864, 411041792, 830472192, 1698693120, 3422552064, 7046430720
Offset: 0

Views

Author

Benoit Cloitre and Paul D. Hanna, Dec 10 2002

Keywords

Comments

A k-almost prime is a positive integer that has exactly k prime factors counted with multiplicity.

Examples

			a(0) = 1 since one is the multiplicative identity,
a(1) = 2nd 1-almost prime is the second prime number = A000040(2) = 3,
a(2) = 3rd 2-almost prime = 3rd semiprime = A001358(3) = 9 = {3*3}.
a(3) = 4th 3-almost prime = A014612(4) = 20 = {2*2*5}.
a(4) = 5th 4-almost prime = A014613(5) = 54 = {2*3*3*3},
a(5) = 6th 5-almost prime = A014614(6) = 112 = {2*2*2*2*7}, ....

Links

Crossrefs

Cf. A078840, A078842, A078843, A078844, A078445, A078846, A101695.

Programs

  • Mathematica
    f[n_] := Plus @@ Last /@ FactorInteger@n; t = Table[{}, {40}]; Do[a = f[n]; AppendTo[ t[[a]], n]; t[[a]] = Take[t[[a]], 10], {n, 2, 148*10^8}]; Table[ t[[n, n + 1]], {n, 30}] (* Robert G. Wilson v, Feb 11 2006 *)
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 *)
AlmostPrime[k_, n_] := Block[{e = Floor[ Log[2, n] + k], a, b}, a = 2^e; Do[b = 2^p; While[ AlmostPrimePi[k, a] < n, a = a + b]; a = a - b/2, {p, e, 0, -1}]; a + b/2]; AlmostPrime[1, 1] = 2; lst = {}; Do[ AppendTo[lst, AlmostPrime[n-1, n]], {n, 30}]; lst (* Robert G. Wilson v, Nov 13 2007 *)
  • Python
    from math import prod, isqrt
from sympy import primerange, integer_nthroot, primepi
def A078841(n):
    if n <= 1: return (n<<1)+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)))
    def f(x): return int(n+x-sum(primepi(x//prod(c[1] for c in a))-a[-1][0] for a in g(x,0,1,1,n)))
    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

Formula

Conjecture: Lim as n->inf. of a(n+1)/a(n) = 2. - Robert G. Wilson v, Nov 13 2007

Extensions

a(14)-a(29) from Robert G. Wilson v, Feb 11 2006

A078846 Where 11^n occurs in n-almost-primes, starting at a(0)=1.

Original entry on oeis.org

1, 5, 40, 328, 2556, 18452, 126096, 827901, 5276913, 32887213, 201443165, 1217389949, 7279826998, 43168558912, 254258462459, 1489291941733, 8683388113017, 50433408838966
Offset: 0

Views

Author

Benoit Cloitre and Paul D. Hanna, Dec 10 2002

Keywords

Comments

A k-almost-prime is a positive integer that has exactly k prime factors, counted with multiplicity.

Examples

			a(2) = 40 since 11^2 is the 40th 2-almost-prime: A001358(40) = 121.

Links

Crossrefs

Cf. A078840, A078841, A078842, A078843, A078844, A078845.

Programs

  • Mathematica
    AlmostPrimePi[k_Integer /; k > 1, n_] := Module[{a, i}, a[0] = 1; 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[n, 11^n], {n, 2, 11}] (* Robert G. Wilson v, Feb 09 2006 *)
  • PARI
    almost_prime_count(N, k) = if(k==1, return(primepi(N))); (f(m, p, k, j=0) = my(c=0, s=sqrtnint(N\m, k)); if(k==2, forprime(q=p, s, c += primepi(N\(m*q))-j; j += 1), forprime(q=p, s, c += f(m*q, q, k-1, j); j += 1)); c); f(1, 2, k);
a(n) = if(n == 0, 1, almost_prime_count(11^n, n)); \\ Daniel Suteu, Jul 10 2023
  • Python
    from math import isqrt, prod
from sympy import primerange, integer_nthroot, primepi
def A078846(n):
    def almostprimepi(n, k):
        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))
    return almostprimepi(11**n, n) if n else 1 # Chai Wah Wu, Sep 01 2024

Extensions

a(6)-a(11) from Robert G. Wilson v, Feb 09 2006
a(12)-a(15) from Donovan Johnson, Sep 27 2010
a(16)-a(17) from Daniel Suteu, Jul 10 2023

A116430 The number of n-almost primes less than or equal to 10^n, starting with a(0)=1.

Original entry on oeis.org

1, 4, 34, 247, 1712, 11185, 68963, 409849, 2367507, 13377156, 74342563, 407818620, 2214357712, 11926066887, 63809981451, 339576381990, 1799025041767, 9494920297227, 49950199374227, 262036734664892
Offset: 0

Views

Author

Robert G. Wilson v, Feb 10 2006, Jun 01 2006

Keywords

Comments

If instead we asked for those less than or equal to 2^n, then the sequence is A000012.

Crossrefs

Cf. A036352, A114106, A114453.
Cf. A078840, A078841, A078842, A116432, A078843, A116426, A078844, A116427, A078845, A116428, A116429, A116430, A078846, A116431.
Cf. A006880, A036352, A109251, A114106, A114453, A120047 - A120053.

Programs

  • Mathematica
    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[n, 10^n], {n, 0, 13}]
  • PARI
    almost_prime_count(N, k) = if(k==1, return(primepi(N))); (f(m, p, k, j=0) = my(c=0, s=sqrtnint(N\m, k)); if(k==2, forprime(q=p, s, c += primepi(N\(m*q))-j; j += 1), forprime(q=p, s, c += f(m*q, q, k-1, j); j += 1)); c); f(1, 2, k);
a(n) = if(n == 0, 1, almost_prime_count(10^n, n)); \\ Daniel Suteu, Jul 10 2023
  • Python
    from math import prod, isqrt
from sympy import primerange, integer_nthroot, primepi
def A116430(n):
    if n<=1: return 3*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(10**n//prod(c[1] for c in a))-a[-1][0] for a in g(10**n,0,1,1,n))) # Chai Wah Wu, Aug 23 2024

Extensions

Edited by N. J. A. Sloane, Aug 08 2008 at the suggestion of R. J. Mathar
a(15)-a(16) from Donovan Johnson, Oct 01 2010
a(17)-a(19) from Daniel Suteu, Jul 10 2023

A078842 Sums of the antidiagonals of the table of k-almost primes (A078840).

Original entry on oeis.org

1, 2, 7, 19, 44, 95, 195, 395, 794, 1583, 3172, 6334, 12665, 25313, 50596, 101180, 202326, 404635, 809227, 1618410, 3236766, 6473474, 12946903, 25893723, 51787365, 103574668, 207149213, 414298342, 828596584, 1657193052, 3314385970
Offset: 0

Views

Author

Benoit Cloitre and Paul D. Hanna, Dec 11 2002

Keywords

Comments

A k-almost prime is a positive integer that has exactly k prime factors counted with multiplicity.

Examples

			a(3) = 19 = 5 (3rd prime) + 6 (2nd 2-almost prime) + 8 (first 3-almost prime).

Links

Crossrefs

Cf. A078840, A078841, A078843, A078844, A078845, A078846.

Programs

  • Mathematica
    f[n_] := Plus @@ Last /@ FactorInteger@n; t = Table[{}, {40}]; Do[a = f[n]; AppendTo[t[[a]], n]; t[[a]] = Take[t[[a]], 10], {n, 2, 148*10^8}]; Plus @@@ Table[t[[n - k + 1, k]], {n, 30}, {k, n, 1, -1}] (* Or *)
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 *)
AlmostPrime[k_, n_] := Block[{e = Floor[Log[2, n]+k], a, b}, a = 2^e; Do[b = 2^p; While[ AlmostPrimePi[k, a] < n, a = a + b]; a = a - b/2, {p, e, 0, -1}]; a + b/2]; Table[ Sum[ AlmostPrime[k, n - k + 1], {k, n}], {n, 150}] (* Robert G. Wilson v, Feb 11 2006 *)

Formula

a(n) = Sum_{i=0..n-1} A078840(i+1, n-i).

Extensions

a(12)-a(30) from Robert G. Wilson v, Feb 11 2006

A078845 Where 7^n occurs in n-almost-primes, starting at a(0)=1.

Original entry on oeis.org

1, 4, 17, 82, 385, 1688, 7089, 28893, 115180, 450906, 1740244, 6640747, 25115604, 94312569, 352110321, 1308256678, 4841115048, 17852264639, 65636109307, 240689877440, 880582139867
Offset: 0

Views

Author

Benoit Cloitre and Paul D. Hanna, Dec 10 2002

Keywords

Comments

A k-almost-prime is a positive integer that has exactly k prime factors, counted with multiplicity.

Examples

			a(2) = 17 since 7^2 is the 17th 2-almost-prime: {4,6,9,10,14,15,21,22,25,26,33,34,35,38,39,46,49,...}.

Links

Crossrefs

Cf. A078840, A078841, A078842, A078843, A078844, A078846.

Programs

  • Mathematica
    l = Table[0, {30}]; e = 0; Do[f = Plus @@ Last /@ FactorInteger[n]; l[[f+1]]++; If[n == 7^e, Print[l[[f+1]]]; e++ ], {n, 1, 7^10}] (* Ryan Propper, Aug 08 2005 *)
AlmostPrimePi[k_Integer /; k > 1, n_] := Module[{a, i}, a[0] = 1; 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[n, 7^n], {n, 2, 15}] (* Robert G. Wilson v, Feb 09 2006 *)
  • Python
    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 A078845(n): return almostprimepi(7**n,n) if n else 1 # Chai Wah Wu, Oct 02 2024

Extensions

a(7)-a(10) from Ryan Propper, Aug 08 2005
a(11)-a(15) from Robert G. Wilson v, Feb 09 2006
a(16)-a(20) from Donovan Johnson, Sep 27 2010

A116426 The number of n-almost primes less than or equal to 4^n, starting with a(0)=1.

Original entry on oeis.org

1, 2, 6, 13, 34, 77, 177, 406, 887, 1962, 4225, 9094, 19482, 41414, 87706, 184976, 389357, 816193, 1708412, 3566209, 7431153, 15457234, 32098652, 66560309, 137830562, 285062028, 588871107, 1215176367, 2505048537, 5159228725
Offset: 0

Views

Author

Robert G. Wilson v, Feb 10 2006

Keywords

Crossrefs

Cf. A078840, A078841, A078842, A116432, A078843, A116426, A078844, A116427, A078845, A116428, A116429, A116430, A078846, A116431.

Programs

  • Mathematica
    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 *)
Join[{1},Table[AlmostPrimePi[n, 4^n], {n, 29}]]
  • Python
    from math import isqrt, prod
from sympy import primerange, integer_nthroot, primepi
def A116426(n):
    if n<=1: return 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((1<<(n<<1))//prod(c[1] for c in a))-a[-1][0] for a in g(1<<(n<<1),0,1,1,n))) # Chai Wah Wu, Oct 02 2024

A116427 The number of n-almost primes less than or equal to 6^n, starting with a(0)=1.

Original entry on oeis.org

1, 3, 13, 50, 200, 726, 2613, 9061, 30779, 102637, 338230, 1102674, 3566001, 11455355, 36597558, 116395587, 368749900, 1164407829, 3666312894, 11515047829, 36085395700, 112857846859, 352329509934, 1098136237818
Offset: 0

Views

Author

Robert G. Wilson v, Feb 10 2006

Keywords

Crossrefs

Cf. A078840, A078841, A078842, A116432, A078843, A116426, A078844, A116427, A078845, A116428, A116429, A116430, A078846, A116431.

Programs

  • Mathematica
    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 *)
Join[{1},Table[AlmostPrimePi[n, 6^n], {n, 21}]]

Extensions

a(22)-a(23) from Donovan Johnson, Oct 01 2010

A116428 The number of n-almost primes less than or equal to 8^n, starting with a(0)=1.

Original entry on oeis.org

1, 4, 22, 125, 669, 3410, 16677, 78369, 359110, 1612613, 7133274, 31185350, 135062165, 580556958, 2480278767, 10542976739, 44626102826, 188215850830, 791374442571, 3318478309647, 13882441625034, 57952990683107
Offset: 0

Views

Author

Robert G. Wilson v, Feb 14 2006

Keywords

Crossrefs

Cf. A078840, A078841, A078842, A116432, A078843, A116426, A078844, A116427, A078845, A116428, A116429, A116430, A078846, A116431.

Programs

  • Mathematica
    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}]]]]];
Table[ AlmostPrimePi[n, 8^n], {n, 14}] (* Eric W. Weisstein, Feb 07 2006 *)
  • PARI
    almost_prime_count(N, k) = if(k==1, return(primepi(N))); (f(m, p, k, j=0) = my(c=0, s=sqrtnint(N\m, k)); if(k==2, forprime(q=p, s, c += primepi(N\(m*q))-j; j += 1), forprime(q=p, s, c += f(m*q, q, k-1, j); j += 1)); c); f(1, 2, k);
a(n) = if(n == 0, 1, almost_prime_count(8^n, n)); \\ Daniel Suteu, Jul 10 2023

Extensions

a(15)-a(18) from Donovan Johnson, Oct 01 2010
a(19)-a(21) from Daniel Suteu, Jul 10 2023
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