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.

A067003 Number of numbers <= n with same number of distinct prime factors as n.

Original entry on oeis.org

1, 1, 2, 3, 4, 1, 5, 6, 7, 2, 8, 3, 9, 4, 5, 10, 11, 6, 12, 7, 8, 9, 13, 10, 14, 11, 15, 12, 16, 1, 17, 18, 13, 14, 15, 16, 19, 17, 18, 19, 20, 2, 21, 20, 21, 22, 22, 23, 23, 24, 25, 26, 24, 27, 28, 29, 30, 31, 25, 3, 26, 32, 33, 27, 34, 4, 28, 35, 36, 5, 29, 37, 30, 38, 39, 40, 41
Offset: 1

Views

Author

Henry Bottomley, Dec 21 2001

Keywords

Examples

			a(11)=8 since 2,3,4,5,7,8,9,11 each have one distinct prime factor. a(12)=3 since 6,10,12 each have two distinct prime factors.
From _Gus Wiseman_, Dec 28 2018: (Start)
Column n lists the a(n) positive integers less than or equal to n with the same number of distinct prime factors as n:
  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20
  ---------------------------------------------------------------------
  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20
        2  3  4     5  7  8  6   9   10  11  12  14  13  16  15  17  18
           2  3     4  5  7      8   6   9   10  12  11  13  14  16  15
              2     3  4  5      7       8   6   10  9   11  12  13  14
                    2  3  4      5       7       6   8   9   10  11  12
                       2  3      4       5           7   8   6   9   10
                          2      3       4           5   7       8   6
                                 2       3           4   5       7
                                         2           3   4       5
                                                     2   3       4
                                                         2       3
                                                                 2
(End)

Links

Crossrefs

Positions of 1's are A002110.
Cf. A001221, A008479, A058933, A067004. Inverse of A000961, A007774, A033992, A033993, A051270 etc.
Cf. A000010, A006049, A061142, A294277, A294278, A302242, A322837, A322841.

Programs

  • Mathematica
    Table[Length[Select[Range[n],PrimeNu[#]==PrimeNu[n]&]],{n,100}] (* Gus Wiseman, Dec 28 2018 *)
  • PARI
    a(n) = my(nb = #factor(n)~); sum(k=1, n, #factor(k)~ == nb); \\ Michel Marcus, Jul 13 2019

Formula

a(A002110(n)) = 1.

A322838 Number of positive integers less than n with more prime factors than n, counted with multiplicity.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 2, 0, 1, 1, 5, 0, 6, 2, 2, 0, 9, 1, 10, 1, 5, 5, 13, 0, 6, 6, 2, 2, 18, 2, 19, 0, 10, 10, 10, 1, 24, 11, 11, 1, 27, 5, 28, 5, 5, 15, 31, 0, 16, 6, 17, 6, 36, 2, 19, 2, 20, 20, 41, 2, 42, 21, 9, 0, 23, 10, 47, 10, 25, 10, 50, 1, 51, 27, 11, 11
Offset: 1

Views

Author

Gus Wiseman, Dec 28 2018

Keywords

Examples

			Column n lists the a(n) positive integers less than n with more prime factors than n:
  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20
  ---------------------------------------------------------------------
              4     6     8  8   10      12  12  12      16  16  18  16
                    4            9       10  8   8       15      16
                                 8       9               14      15
                                 6       8               12      14
                                 4       6               10      12
                                         4               9       10
                                                         8       9
                                                         6       8
                                                         4       6
                                                                 4

Crossrefs

Positions of zeros appear to be A029744.
Cf. A000010, A000961, A001221, A001222, A045920, A058933, A061142, A067003, A302242, A322839, A322841.

Programs

  • Mathematica
    Table[Length[Select[Range[n],PrimeOmega[#]>PrimeOmega[n]&]],{n,100}]

A334655 Number of integers less than n with the same number of distinct prime factors as n.

Original entry on oeis.org

0, 0, 1, 2, 3, 0, 4, 5, 6, 1, 7, 2, 8, 3, 4, 9, 10, 5, 11, 6, 7, 8, 12, 9, 13, 10, 14, 11, 15, 0, 16, 17, 12, 13, 14, 15, 18, 16, 17, 18, 19, 1, 20, 19, 20, 21, 21, 22, 22, 23, 24, 25, 23, 26, 27, 28, 29, 30, 24, 2, 25, 31, 32, 26, 33, 3, 27, 34, 35, 4, 28, 36, 29, 37, 38, 39, 40, 5, 30, 41
Offset: 1

Views

Author

Ilya Gutkovskiy, Oct 31 2020

Keywords

Examples

			a(12) = 2 because omega(12) = 2 and also omega(6) = omega(10) = 2.

Links

Crossrefs

Cf. A001221, A002110 (positions of 0's), A047983, A067003, A067004, A322837, A322841, A335097.

Programs

  • Maple
    R:= NULL:
for n from 1 to 100 do
  w:= nops(numtheory:-factorset(n));
  if assigned(V[w]) then V[w]:= V[w]+1 else V[w]:= 1 fi;
  R:= R, V[w]-1
od:
R; # Robert Israel, Feb 25 2024
  • Mathematica
    Table[Length[Select[Range[n - 1], PrimeNu[#] == PrimeNu[n] &]], {n, 80}]
  • PARI
    a(n)={my(t=omega(n)); sum(k=1, n-1, omega(k)==t)} \\ Andrew Howroyd, Oct 31 2020

Formula

a(n) = |{j < n : omega(j) = omega(n)}|.
a(n) = A067003(n) - 1.

A322837 Number of positive integers less than n with fewer distinct prime factors than n.

Original entry on oeis.org

0, 1, 1, 1, 1, 5, 1, 1, 1, 8, 1, 9, 1, 10, 10, 1, 1, 12, 1, 13, 13, 13, 1, 14, 1, 15, 1, 16, 1, 29, 1, 1, 19, 19, 19, 19, 1, 20, 20, 20, 1, 40, 1, 22, 22, 22, 1, 23, 1, 24, 24, 24, 1, 25, 25, 25, 25, 25, 1, 57, 1, 27, 27, 1, 28, 62, 1, 29, 29, 65, 1, 30, 1, 31
Offset: 1

Views

Author

Gus Wiseman, Dec 28 2018

Keywords

Examples

			Column n lists the a(n) positive integers less than n with fewer distinct prime factors than n:
  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20
  ---------------------------------------------------------------------
     1  1  1  1  5  1  1  1  9   1   11  1   13  13  1   1   17  1   19
                 4           8       9       11  11          16      17
                 3           7       8       9   9           13      16
                 2           5       7       8   8           11      13
                 1           4       5       7   7           9       11
                             3       4       5   5           8       9
                             2       3       4   4           7       8
                             1       2       3   3           5       7
                                     1       2   2           4       5
                                             1   1           3       4
                                                             2       3
                                                             1       2
                                                                     1

Links

Crossrefs

Positions of 1's are A246655.
Cf. A000010, A001221, A006049, A067003, A294277, A294278, A302242, A322838, A322841.

Programs

  • Mathematica
    Table[Length[Select[Range[n],PrimeNu[#]
  • PARI
    \\ See Corneth link
Showing 1-4 of 4 results.

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