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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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

A216593 Number of semiprimes among n-th million natural numbers.

Original entry on oeis.org

210035, 197249, 193162, 190540, 188288, 187308, 185657, 184788, 183856, 183441, 182123, 181556, 181125, 180878, 180234, 179649, 179055, 178710, 178652, 178034, 178015, 177307, 177391, 177003, 176568, 176419, 176021, 175788, 175655, 175189, 174915, 175357
Offset: 1

Views

Author

Zak Seidov, Sep 09 2012

Keywords

Comments

Let f(m) = number of semiprimes
a(n) = 0 for almost all n. It seems infeasible to find the first such n. - Charles R Greathouse IV, Sep 09 2012

Links

Crossrefs

Cf. A001358, A036352, A064911, A066265, A072000.

Programs

  • Mathematica
    f[m_] := Sum[ PrimePi[(m - 1)/Prime[i]], {i, PrimePi[ Sqrt[m]]}] - Binomial[ PrimePi[ Sqrt[m]], 2]; ta=Table[f[n*10^6],{n,0,1000}];s=Rest[ta]-Most[ta] (* for first 1000 terms *)
(* using Mmca code by Robert G. Wilson v in A066265 - Zak Seidov, Sep 09 2012 *)
  • PARI
    a(n)=sum(k=10^6*(n-1),10^6*n,bigomega(k)==2) \\ Charles R Greathouse IV, Sep 09 2012

Formula

a(n) ~ 1000000 n log log n / log n. - Charles R Greathouse IV, Sep 23 2012

A108216 Number of semiprimes between 10n and 10n + 9.

Original entry on oeis.org

3, 3, 4, 5, 2, 4, 3, 2, 4, 4, 1, 4, 4, 2, 5, 3, 3, 2, 3, 1, 6, 6, 2, 2, 2, 3, 3, 2, 2, 4, 5, 2, 5, 3, 2, 2, 3, 2, 3, 5, 2, 4, 2, 1, 3, 4, 2, 3, 4, 2, 3, 5, 3, 4, 3, 4, 3, 2, 4, 2, 0, 2, 4, 3, 1, 1, 3, 3, 4, 5, 3, 3, 2, 3, 3, 3, 3, 3, 3, 4, 3, 4, 0, 3, 4, 1, 4, 3, 2, 4, 2, 3, 4, 3, 2, 4, 2, 3, 3, 3, 3, 2, 1, 1, 5
Offset: 0

Views

Author

Giovanni Teofilatto, Jun 16 2005

Keywords

Comments

a(60) = a(82) = a(142) = 0. - Jonathan Vos Post, Jun 16 2005

Examples

			a(0) = 3 because between 0 and 9 there are three semiprimes: 4, 6 and 9.
a(1) = 3 because between 10 and 19 there are three semiprimes: 10, 14 and 15.

Crossrefs

Cf. A038800 number of primes between 10n and 10n+9.
Cf. A064911, A066265, A072000.

Programs

  • Mathematica
    f[n_] := Sum[ PrimePi[n/Prime[i]] - (i - 1), {i, PrimePi[ Sqrt[n]]}]; Table[f[10n + 9] - f[Max[10n - 1, 0]], {n, 0, 104}] (* Robert G. Wilson v, Ray Chandler *)

Extensions

Edited and extended by Ray Chandler, Jul 07 2005

A115854 Difference between number of semiprimes <= 10^n and the asymptotic approximation round(10^n*loglog(10^n)/log(10^n)).

Original entry on oeis.org

0, 0, 0, 19, 214, 2154, 19974, 179590, 1610937, 14515403, 131560754, 1199914216, 11009605949, 101581094033, 942018562525
Offset: 0

Views

Author

Jonathan Vos Post, Mar 14 2006

Keywords

Examples

			a(5) = A036352(5) - round(...10^5...) = 23378 - 21224 = 2154.

Crossrefs

Cf. A001358, A036352, A066265, A072000.

Extensions

Edited by Don Reble, Mar 29 2006

A194895 Difference between the number of semiprimes less than 10^n and the number of primes less than 10^n.

Original entry on oeis.org

0, -1, 9, 131, 1396, 13786, 131537, 1239745, 11665803, 109941002, 1038723932, 9841935529, 93518105160, 891022511814, 8510960557278, 81485246876212, 781818951793344
Offset: 0

Views

Author

Harvey P. Dale, Sep 04 2011

Keywords

Comments

A066265 minus A006880

Examples

			There are 2625 semiprimes less than 10^4 and there are 1229 primes less than 10^4, so a(4) = 2625 - 1229 = 1396.

Crossrefs

Cf. A006880, A066265

Programs

  • Mathematica
    SemiPrimePi[n_] := Sum[ PrimePi[n/Prime@ i] - i + 1, {i, PrimePi@ Sqrt@ n}]; f[n_] := SemiPrimePi[10^n - 1] - PrimePi[10^n - 1]; Array[f, 15, 0] (* Robert G. Wilson v, Jul 05 2012 *)

A242134 Number of semiprimes with n digits.

Original entry on oeis.org

3, 31, 265, 2326, 20753, 186657, 1694289, 15522934, 143361278, 1332987907, 12466213899, 117166026836, 1105962031475, 10478814259427, 99613914990801, 949727475528388, 9078425620890083, 86983554771459735, 835176988544997091, 8034305823411044844, 77423350444120266442
Offset: 1

Views

Author

Derek Orr, Aug 15 2014

Keywords

Examples

			4, 6, and 9 are semiprimes with 1 digit. So a(1) = 3.

Crossrefs

Cf. A001358, A066265.

Programs

  • PARI
    vector(8,n,sum(k=10^(n-1),10^n-1,bigomega(k)==2))

Formula

a(n) = A066265(n) - A066265(n-1). - Jens Kruse Andersen, Aug 16 2014

Extensions

a(9)-a(18) from A066265, added by Jens Kruse Andersen, Aug 16 2014
a(19)-a(21) from A066265, added by Jon E. Schoenfield, Feb 10 2022

A296653 a(n) is the smallest k > 15 such that the density of semiprimes in 1..k is 1/n.

Original entry on oeis.org

18, 26432, 3066830, 348933114, 44690978122, 6553736049264
Offset: 3

Views

Author

Jon E. Schoenfield, Dec 17 2017

Keywords

Comments

The condition that k > 15 is included in the definition because the ratio (number of semiprimes in 1..k)/k is 0 for k < 4 and reaches its maximum value (2/5) only at k = 10 (the 4th semiprime) and at k = 15 (the 6th semiprime), and decreases (although not monotonically) beyond that.

Examples

			For k > 15, the ratio (number of semiprimes in 1..k)/k first decreases to --
1/3 at k = 18 (the 6th semiprime), so a(3) = 18;
1/4 at k = 26432 (the 6608th semiprime), so a(4) = 26432.

Crossrefs

Cf. A001358, A066265, A106130.

Formula

a(n) = exp(n log n + n log log n + O(n)). - Charles R Greathouse IV, Dec 14 2022

Extensions

a(7)-a(8) from Giovanni Resta, Aug 18 2018
Previous Showing 21-27 of 27 results.

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