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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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A377505 a(n) is the number of positive integers that have Omega(n) prime factors and these are all <= n.

Original entry on oeis.org

1, 1, 2, 3, 3, 6, 4, 20, 10, 10, 5, 35, 6, 21, 21, 126, 7, 84, 8, 120, 36, 36, 9, 495, 45, 45, 165, 165, 10, 220, 11, 3003, 66, 66, 66, 1001, 12, 78, 78, 1365, 13, 455, 14, 560, 560, 105, 15, 11628, 120, 680, 120, 680, 16, 3876, 136, 3876, 136, 136, 17, 4845, 18
Offset: 1

Views

Author

Felix Huber, Dec 20 2024

Keywords

Comments

If drawing at random with replacement from the primes <= n as many as n has prime factors, 1/a(n) is the probability that the product of the prime numbers drawn is equal to n.

Examples

			a(4) = 3 because 3 positive integers have Omega(4) = 2 prime factors <= 4: 4 = 2*2, 6 = 2*3, 9 = 3*3.
a(6) = 6 because 6 positive integers have Omega(6) = 2 prime factors <= 6: 4 = 2*2, 6 = 2*3, 9 = 3*3, 10 = 2*5, 15 = 3*5, 25 = 5*5.
a(7) = 4 because 4 positive integers have Omega(7) = 1 prime factor <= 7: 2, 3, 5, 7.

Links

Crossrefs

Cf. A000040, A000720, A001222, A007318, A037031, A377537.

Programs

  • Maple
    with(NumberTheory):
A377505:=n->binomial(pi(n)+Omega(n)-1,Omega(n));
seq(A377505(n),n=1..61);
  • Mathematica
    Table[Binomial[PrimePi[n]+PrimeOmega[n]-1,PrimeOmega[n]],{n,61}] (* James C. McMahon, Dec 24 2024 *)

Formula

a(n) = binomial(pi(n) + Omega(n) - 1, Omega(n)) where pi = A000720 and Omega = A001222.
a(p) = pi(p) for prime p.

A382330 a(n) is the number of positive integers k for which Sum_{i=1..j} (p_i+e_i) = n, where p_1^e_1*...*p_j^e_j is the prime factorization of k.

Original entry on oeis.org

0, 0, 1, 2, 2, 3, 4, 6, 8, 11, 15, 21, 27, 36, 47, 61, 79, 104, 133, 170, 215, 272, 343, 433, 542, 678, 845, 1050, 1300, 1608, 1981, 2437, 2988, 3655, 4460, 5433, 6603, 8014, 9705, 11731, 14155, 17055, 20509, 24624, 29512, 35313, 42184, 50315, 59916, 71248, 84598
Offset: 1

Views

Author

Felix Huber, Mar 23 2025

Keywords

Comments

a(n) is the number of positive integers k for A008474(k) = n.

Examples

			The a(7) = 4 positive integers k are 32 = 2^5, 81 = 3^4, 25 = 5^2, 6 = 2^1*3^1 because 2 + 5 = 3 + 4 = 5 + 2 = 2 + 1 + 3 + 1 = 7 and there is no further positive integer with that property.
The a(11) = 15 positive integers k are 512 = 2^9, 6561 = 3^8, 15625 = 5^6, 2401 = 7^4, 96 = 2^5*3^1, 144 = 2^4*3^2, 216 = 2^3*3^3, 324 = 2^2*3^4, 486 = 2^1*3^5, 40 = 2^3*5^1, 100 = 2^2*5^2, 250 = 2^1*5^3, 14 = 2^1*7^1, 45 = 3^2*5^1, 75 = 3^1*5^2 because 2 + 9 = 3 + 8 = 5 + 6 = 7 + 4 = 2 + 5 + 3 + 1 = 2 + 4 + 3 + 2 = 2 + 3 + 3 + 3 = 2 + 2 + 3 + 4 = 2 + 1 + 3 + 5 = 2 + 3 + 5 + 1 = 2 + 2 + 5 + 2 = 2 + 1 + 5 + 3 = 2 + 1 + 7 + 1 = 3 + 2 + 5 + 1 = 3 + 1 + 5 + 2 = 11 and there is no further positive integer with that property.

Links

Crossrefs

Cf. A008474, A219180, A377505, A377537.

Programs

  • Maple
    # processes b and T from Alois P. Heinz (A219180).
b:= proc(n,i) option remember;
      `if`(n=0,[1],`if`(i<1,[],zip((x,y)->x+y,b(n,i-1),
       [0,`if`(ithprime(i)>n,[],b(n-ithprime(i),i-1))[]],0)))
    end:
T:= proc(n) local l;l:=b(n,NumberTheory:-pi(n));
       while nops(l)>0 and l[-1]=0 do l:=subsop(-1=NULL,l) od; l[]
    end:
A382330:=proc(n)
    local a,k,s,i,j,L;
    a:=0;k:=1;s:=0;
    while s+k<=n do
        s:=s+ithprime(k);k:=k+1
    od;
    for i to k-1 do
        for j to n-i do
            L:=[T(j)];
            if nops(L)>=i+1 then
                a:=a+L[i+1]*binomial(n-j-1,n-j-i);
            fi
        od
    od;
    return a
end proc;
seq(A382330(n),n=1..51);
Showing 1-2 of 2 results.

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