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 1-7 of 7 results.

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

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

A116429 The number of n-almost primes less than or equal to 9^n, starting with a(0)=1.

Original entry on oeis.org

1, 4, 26, 181, 1095, 6416, 35285, 187929, 973404, 4934952, 24628655, 121375817, 592337729, 2868086641, 13798982719, 66043675287, 314715355786, 1494166794434, 7071357084444, 33374079939405
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 *)
Table[ AlmostPrimePi[n, 9^n], {n, 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(9^n, n)); \\ Daniel Suteu, Jul 10 2023

Extensions

a(14)-a(16) from Donovan Johnson, Oct 01 2010
a(16) corrected and a(17)-a(19) from Daniel Suteu, Jul 10 2023

A116431 The number of n-almost primes less than or equal to 12^n, starting with a(0)=1.

Original entry on oeis.org

1, 5, 48, 434, 3695, 29165, 218283, 1569995, 10950776, 74621972, 499495257, 3297443264, 21533211312, 139411685398, 896352197825, 5730605551626, 36465861350230
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 *)
Table[ AlmostPrimePi[n, 12^n], {n, 12}]
  • 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(12^n, n)); \\ Daniel Suteu, Jul 10 2023
  • Python
    from math import isqrt, prod
from sympy import primerange, integer_nthroot, primepi
def A116431(n):
    if n<=1: return 4*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(12**n//prod(c[1] for c in a))-a[-1][0] for a in g(12**n,0,1,1,n))) # Chai Wah Wu, Sep 28 2024

Extensions

a(13)-a(14) from Donovan Johnson, Oct 01 2010
a(15)-a(16) from Daniel Suteu, Jul 10 2023

A116432 The number of n-almost primes less than or equal to e^n, starting with a(0)=1.

Original entry on oeis.org

1, 1, 2, 4, 5, 7, 12, 18, 24, 37, 54, 74, 107, 159, 218, 315, 450, 634, 888, 1269, 1782, 2496, 3520, 4933, 6899, 9681, 13555, 18888, 26407, 36855, 51352, 71526, 99654, 138608, 192708, 267833, 372107, 516420, 716816, 994191, 1378195, 1909694
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 *)
Table[ AlmostPrimePi[n, E^n], {n, 42}]

A116433 Consider the array T(r,c) where is the number of c-almost primes less than or equal to r^c, r >= 1, c >= 0. Read the array by antidiagonals.

Original entry on oeis.org

0, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 2, 3, 1, 0, 1, 3, 6, 5, 1, 0, 1, 3, 9, 13, 8, 1, 0, 1, 4, 13, 30, 34, 14, 1, 0, 1, 4, 17, 50, 90, 77, 23, 1, 0, 1, 4, 22, 82, 200, 269, 177, 39, 1, 0, 1, 4, 26, 125, 385, 726, 788, 406, 64, 1, 0, 1, 5, 34, 181, 669, 1688, 2613, 2249, 887, 103, 1, 0, 1, 5
Offset: 0

Views

Author

Paul D. Hanna and Robert G. Wilson v, Feb 15 2006

Keywords

Examples

			The array begins:
  0 0 0 0 0 0 0 0 0 0 0
  1 1 1 1 1 1 1 1 1 1 1
  1 2 3 5 8 14 23 39 64 103 169
  1 2 6 13 34 77 177 406 887 1962 4225
  1 3 9 30 90 269 788 2249 6340 17526 47911
T(3,2)=3 because there are 3 2-almost primes <= 3^2 = 9, namely 4, 6, and 9 (see A001358).

Crossrefs

Cf. The rows are: A000004, A000012, A078843, A116426, A078844, A116427, A078845, A116428, A116429, A116430, A078846, A116431.
The columns are: A057427, A000720.

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[ If[k == 0, 1, AlmostPrimePi[n - k + 1, k^(n - k + 1)]], {n, 0, 7}, {k, n, 0, -1}] // Flatten

Extensions

NAME corrected by R. J. Mathar, Jun 20 2021
Showing 1-7 of 7 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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