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 1-4 of 4 results.

A036336 Smallest positive integer with n digits and exactly n prime factors (counted with multiplicity).

Original entry on oeis.org

2, 10, 102, 1012, 10010, 100040, 1000125, 10000096, 100000032, 1000000080, 10000000080, 100000000512, 1000000001280, 10000000014336, 100000000004096, 1000000000010880, 10000000000008192, 100000000000008192, 1000000000000010240, 10000000000000045056
Offset: 1

Views

Author

Patrick De Geest, Dec 15 1998

Keywords

Links

Crossrefs

Cf. A036335, A036337, A036338.

Programs

  • Maple
    f:= proc(n) local k;
  for k from 10^(n-1) do
    if numtheory:-bigomega(k) = n then return k fi
  od
end proc:
map(f, [$1..20]); # Robert Israel, May 31 2018
  • Mathematica
    npf[n_]:=Module[{k=1,st=10^(n-1)-1},While[PrimeOmega[st+k]!=n,k++];st+k]; Array[npf,20] (* Harvey P. Dale, Mar 25 2012 *)
  • Python
    from sympy import factorint
def a(n):
  for m in range(10**(n-1), 10**n):
    if sum(factorint(m).values()) == n: return m
print([a(n) for n in range(1, 13)]) # Michael S. Branicky, Feb 10 2021

Extensions

More terms from Matthew Conroy, May 27 2001
Offset corrected, and a(19)-a(20) from Robert Israel, May 31 2018

A036337 Largest integer with n digits and exactly n prime factors (counted with multiplicity).

Original entry on oeis.org

7, 95, 994, 9999, 99996, 999992, 9999968, 99999840, 999999968, 9999999900, 99999999840, 999999999744, 9999999998720, 99999999998400, 999999999999000, 9999999999999744, 99999999999995904, 999999999999967232, 9999999999999989760, 99999999999999995904
Offset: 1

Views

Author

Patrick De Geest, Dec 15 1998

Keywords

Comments

If all prime factors are distinct then a(n) >= A002110(n) which might give a contradiction for large enough n and so some primes have a multiplicity > k for some nonnegative k. - David A. Corneth, Oct 30 2018

Examples

			95 = 5 * 19, while 96, 97, 98, 99 and 100 have, respectively, 6,1,3,3 and 4 prime factors; thus 95 is the largest two digit number with exactly two prime factors.

Links

Crossrefs

Cf. A002110, A036335, A036336, A036338.

Programs

  • Mathematica
    Table[Module[{k=10^n-1},While[PrimeOmega[k]!=n,k--];k],{n,20}] (* Harvey P. Dale, Sep 02 2022 *)
  • PARI
    a(n) = forstep(i = 10^n-1,10^(n-1),-1,if(bigomega(i) == n, return(i))) \\ David A. Corneth, Oct 30 2018

Extensions

More terms and better description from Matthew Conroy, May 25 2001
a(19) and a(20) from Zak Seidov, Oct 30 2018

A036338 Composites whose digit length is equal to their number of prime factors (counted with multiplicity).

Original entry on oeis.org

10, 14, 15, 21, 22, 25, 26, 33, 34, 35, 38, 39, 46, 49, 51, 55, 57, 58, 62, 65, 69, 74, 77, 82, 85, 86, 87, 91, 93, 94, 95, 102, 105, 110, 114, 116, 117, 124, 125, 130, 138, 147, 148, 153, 154, 164, 165, 170, 171, 172, 174, 175, 182, 186, 188, 190, 195, 207
Offset: 1

Views

Author

Patrick De Geest, Dec 15 1998

Keywords

Links

  • Carlos Rivera, Composed primes, The Prime Puzzles and Problems Connection.

Crossrefs

Cf. A036335, A036336, A036337.

Extensions

1 removed by Sean A. Irvine, Oct 26 2020

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
Showing 1-4 of 4 results.

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