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.

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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

Views

Author

Jonathan Vos Post & Robert G. Wilson v, Dec 22 2006

Keywords

Comments

The n-th row's sum is 10^n - 1.

Examples

			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

Links

Crossrefs

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.

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[m, 10^n], {n, 6}, {m, Floor[Log[2, 10^n]] }] // Flatten

A117526 Least number a(n) which is a product of n primes and such that Pi_n(a(n))/a(n) is maximum.

Original entry on oeis.org

3, 10, 9837, 259441550133
Offset: 1

Views

Author

Martin Raab and Robert G. Wilson v, Mar 25 2006

Keywords

Comments

Pi_n(a(n))/a(n): 0.66667, 0.40000, 0.25801, 0.2145967653
3=3, 10=2*5, 9837=3*3*1093 & 259441550133=3*89*311*3124409.
3 is the second prime, 10 is the fourth semiprime, 9837 is the 3-almost prime, and 259441550133 is the 4-almost prime.

Examples

			a(1)=3 because Pi(2)/2=1/2 < Pi(3)/3=2/3 > Pi(5)/5=3/5.
a(2)=10 because Pi_2(9)/9=1/3 < Pi_2(10)/10=2/5 > Pi_2(14)/14=5/14; Pi_2(10)/10 = Pi_2(15)/15 but 10 < 15.

Links

Crossrefs

Cf. A006880, A066265, A109251, A114106, A114453.

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 *)
fQ[n_] := Plus @@ Last /@ FactorInteger@n == 4; c = r = 0; Do[If[fQ@n, c++ ]; If[c/n > r, Print[n]; r = c/n], {n, 10^6}]

Extensions

Comment edited and a(4) added by Donovan Johnson, Mar 10 2010

A120046 The 10^n-th 5-almost prime.

Original entry on oeis.org

32, 176, 1272, 10374, 89896, 810220, 7475818, 70185558, 667561977, 6411296283, 62037096770, 603813941738
Offset: 0

Views

Author

Robert G. Wilson v, Feb 15 2006

Keywords

Crossrefs

Cf. A014614, A114453, A006988, A114125, A120044, A120045.

Programs

  • Mathematica
    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@Sqrt[(n/(Prime@i*Prime@j*Prime@k))]}];
FiveAlmostPrime[n_] := Block[{e = Floor[Log[2, n] +4], a, b}, a = 2^e; Do[b = 2^p; While[FiveAlmostPrimePi[a] < n, a = a + b]; a = a - b/2, {p, e, 0, -1}]; a + b/2]; Do[ Print@FiveAlmostPrime[10^n], {n, 0, 13}]
  • PARI
    lista(nmax) = {my(pow = 1, c = 0, n = 0); for(k = 1, oo, if(bigomega(k) == 5, c++; if(c == pow, print1(k, ", "); if(n == nmax, break); pow *= 10; n++)));} \\ Amiram Eldar, Apr 29 2024

Formula

a(n) = A014614(10^n). - Amiram Eldar, Apr 29 2024

Extensions

a(6) corrected and a(7)-a(9) added by Amiram Eldar, Apr 29 2024
a(10)-a(11) from David A. Corneth, Apr 29 2024

A124033 Number of n-digit numbers having exactly n prime factors (counted with multiplicity).

Original entry on oeis.org

4, 31, 225, 1563, 10222, 63030, 374264, 2160300, 12196405, 67724342, 371233523, 2014305995, 10841722966, 57974736592, 308361428628, 1632877406997
Offset: 1

Views

Author

J. M. Bergot, Apr 08 2011

Keywords

Comments

Essentially the same as A036335.
What would be the ratio between a(n) and all possible numbers with n digits for each n?

Examples

			a(1) = A006880(1) = 4.
a(2) = A066265(2) - A066265(1) = 34 - 3 = 31.
a(3) = A109251(3) - A109251(2) = 247 - 22 = 225.
a(4) = A114106(4) - A114106(3) = 1712 - 149 = 1563.
a(5) = A114453(5) - A114453(4) = 11185 - 963 = 10222.
a(6) = A120047(6) - A120047(5) = 68963 - 5933 = 63030.
a(7) = A120048(7) - A120048(6) = 409849 - 35585 = 374264.
a(8) = A120049(8) - A120049(7) = 2367507 - 207207 = 2160300.
a(9) = A120050(9) - A120050(8) = 13377156 - 1180751 = 12196405.
a(10) = A120051(10) - A120051(9) = 74342563 - 6618221 = 67724342.
a(11) = A120052(11) - A120052(10) = 407818620 - 36585097 = 371233523.
a(12) = A120053(12) - A120053(11) = 2214357712 - 200051717 = 2014305995.

Programs

  • Mathematica
    Table[Count[Range[10^(n-1),10^n-1],?(PrimeOmega[#]==n&)],{n,8}]  (* _Harvey P. Dale, Apr 22 2011 *)
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 *)
f[n_] := AlmostPrimePi[n, 10^n - 1] - AlmostPrimePi[n, 10^(n - 1) - 1]; Array[f, 12] (* Robert G. Wilson v, Jul 06 2012 *)

Extensions

Corrected and extended by Ray Chandler, Apr 11 2011
a(9)-a(12) from Ray Chandler, Apr 12 2011
a(13)-a(16) from Robert G. Wilson v, Jul 06 2012
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