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.

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.

A322841 Number of positive integers less than n with more distinct prime factors than n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 2, 0, 3, 0, 0, 5, 5, 0, 6, 0, 0, 0, 9, 0, 10, 0, 11, 0, 12, 0, 13, 13, 1, 1, 1, 1, 17, 1, 1, 1, 20, 0, 21, 2, 2, 2, 24, 2, 25, 2, 2, 2, 28, 2, 2, 2, 2, 2, 33, 0, 34, 3, 3, 36, 3, 0, 38, 4, 4, 0, 41, 5, 42, 5, 5, 5, 5, 0, 47, 6, 48
Offset: 1

Views

Author

Gus Wiseman, Dec 28 2018

Keywords

Examples

			Column n lists the a(n) positive integers less than n with more 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
  ---------------------------------------------------------------------
                    6  6  6      10      12          15  15      18
                                  6      10          14  14      15
                                          6          12  12      14
                                                     10  10      12
                                                      6   6      10
                                                                  6

Links

Crossrefs

Positions of zeros are A116998.
Cf. A000010, A000961, A001221, A006049, A058933, A067003, A294277, A302242, A322837, A322838.

Programs

  • Maple
    b:= proc(n) option remember; nops(numtheory[factorset](n)) end:
a:= proc(n) option remember;
      (t-> add(`if`(b(i)>t, 1, 0), i=1..n-1))(b(n))
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Dec 28 2018
  • Mathematica
    Table[Length[Select[Range[n],PrimeNu[#]>PrimeNu[n]&]],{n,100}]
  • PARI
    a(n) = my(omegan=omega(n)); sum(k=1, n-1, omega(k) > omegan); \\ Michel Marcus, Dec 29 2018
  • PARI
    first(n) = {my(t = 1, pp = 1, res = vector(n)); forprime(p = 2, oo, pp*=p; if(pp > n, v = vector(t); break); t++); for(i = 1, n, o = omega(i); res[i] = v[o+1]; for(j = 1, o, v[j]++)); res} \\ David A. Corneth, Dec 29 2018

A322840 Positive integers n with fewer prime factors (counted with multiplicity) than n + 1.

Original entry on oeis.org

1, 3, 5, 7, 11, 13, 15, 17, 19, 23, 26, 29, 31, 35, 37, 39, 41, 43, 47, 49, 51, 53, 55, 59, 61, 62, 63, 65, 67, 69, 71, 73, 74, 77, 79, 83, 87, 89, 91, 95, 97, 99, 101, 103, 107, 109, 111, 113, 115, 119, 123, 125, 127, 129, 131, 134, 137, 139, 143, 146, 149
Offset: 1

Views

Author

Gus Wiseman, Dec 28 2018

Keywords

Examples

			49 = 7*7 has two prime factors, while 50 = 2*5*5 has three, so 49 belongs to the sequence.

Links

Crossrefs

Cf. A001222, A006049, A045920, A294277, A294278, A302242, A322837, A322838, A322839.

Programs

  • Mathematica
    Select[Range[100],PrimeOmega[#]?(#[[1]]< #[[2]]&),1,Heads->False]//Flatten (* _Harvey P. Dale, Sep 23 2021 *)
  • PARI
    isok(n) = bigomega(n) < bigomega(n+1); \\ Michel Marcus, Dec 29 2018
Showing 1-4 of 4 results.

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