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.

A058933 Let k be bigomega(n) (i.e., n is a k-almost-prime). a(n) = number of k-almost-primes <= n.

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 4, 1, 3, 4, 5, 2, 6, 5, 6, 1, 7, 3, 8, 4, 7, 8, 9, 2, 9, 10, 5, 6, 10, 7, 11, 1, 11, 12, 13, 3, 12, 14, 15, 4, 13, 8, 14, 9, 10, 16, 15, 2, 17, 11, 18, 12, 16, 5, 19, 6, 20, 21, 17, 7, 18, 22, 13, 1, 23, 14, 19, 15, 24, 16, 20, 3, 21, 25, 17, 18, 26, 19, 22, 4, 8, 27, 23
Offset: 1

Views

Author

Naohiro Nomoto, Jan 11 2001

Keywords

Comments

Equivalently, the number of positive integers less than or equal to n with the same number of prime factors as n, counted with multiplicity. - Gus Wiseman, Dec 28 2018
There is a close relationship between a(n) and a(n^2). See A209934 for an exploratory quantification. - Peter Munn, Aug 04 2019

Examples

			3 is prime, so a(3)=2. 10 is 2-almost prime (semiprime), so a(10)=4.
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 prime factors as n, counted with multiplicity:
  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     6  9   7   8   11  10  14      13  12  17  18
              2     3     4  6   5       7   9   10      11  8   13  12
                    2        4   3       5   6   9       7       11  8
                                 2       3   4   6       5       7
                                         2       4       3       5
                                                         2       3
                                                                 2
(End)

Links

Crossrefs

Positions of 1's are A000079.
Cf. A001358, A014612, A014613, A014614.
Cf. A000010, A000961, A001222, A006049, A045920, A061142, A067003, A078843, A209934, A302242, A322838, A322839, A322840.
Equivalent sequence restricted to squarefree numbers: A340313.

Programs

  • Maple
    p:= proc() 0 end:
a:= proc(n) option remember; local t;
      t:= numtheory[bigomega](n);
      p(t):= p(t)+1
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Oct 09 2015
  • Mathematica
    p[] = 0; a[n] := a[n] = Module[{t}, t = PrimeOmega[n]; p[t] = p[t]+1]; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Jan 24 2017, after Alois P. Heinz *)
  • PARI
    a(n) = my(k=bigomega(n)); sum(i=1, n, bigomega(i)==k); \\ Michel Marcus, Jun 27 2024
  • Python
    from math import prod, isqrt
from sympy import isprime, primepi, primerange, integer_nthroot, primeomega
def A058933(n):
    if n==1: return 1
    if isprime(n): return primepi(n)
    def g(x,a,b,c,m): yield from (((d,) for d in enumerate(primerange(b,isqrt(x//c)+1),a)) if m==2 else (((a2,b2),)+d for a2,b2 in enumerate(primerange(b,integer_nthroot(x//c,m)[0]+1),a) for d in g(x,a2,b2,c*b2,m-1)))
    return int(sum(primepi(n//prod(c[1] for c in a))-a[-1][0] for a in g(n,0,1,1,primeomega(n)))) # Chai Wah Wu, Aug 28 2024

Formula

Ordinal transform of A001222 (bigomega). - Franklin T. Adams-Watters, Aug 28 2006
If a(n) < a(3^A001222(2n)) = A078843(A001222(2n)) then a(2n) = a(n), otherwise a(2n) > a(n). - Peter Munn, Aug 05 2019

Extensions

Name edited by Peter Munn, Dec 30 2022

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}]

A339910 After 1, numbers k > 1 such that k has fewer prime divisors than k-1, when they are counted with multiplicity.

Original entry on oeis.org

1, 5, 7, 9, 11, 13, 17, 19, 21, 23, 25, 29, 31, 33, 37, 41, 43, 46, 47, 49, 51, 53, 55, 57, 59, 61, 65, 67, 69, 71, 73, 77, 79, 81, 82, 83, 85, 89, 91, 93, 97, 101, 103, 105, 106, 107, 109, 111, 113, 115, 118, 121, 127, 129, 131, 133, 137, 139, 141, 145, 149, 151, 153, 155, 157, 161, 163, 166, 167, 169, 173, 177, 179
Offset: 1

Views

Author

Antti Karttunen, Dec 22 2020

Keywords

Comments

List of starts of nondecreasing runs of values of A001222 (= bigomega, number of prime factors, when counted with multiplicity).
Differs from A303578 (a similar sequence computed for tau, the number of divisors) for the first time at n=44, where a(44) = 105, a term missing from A303578.

Links

Crossrefs

Cf. A001222.
One more than A322839.
Cf. A339911, A339912 for subsequences.
Cf. also A303578.

Programs

  • Mathematica
    1 + {0}~Join~Position[Sign@ Differences[Array[PrimeOmega, 180]], -1][[All, 1]] (* Michael De Vlieger, Dec 22 2020 *)
  • PARI
    isA339910(n) = ((1==n)||(bigomega(n)

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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