A072047 -id:A072047 - cp's OEIS frontend

A340316 Square array A(n,k), n>=1, k>=1, read by antidiagonals, where row n is the increasing list of all squarefree numbers with n primes.

2, 3, 6, 5, 10, 30, 7, 14, 42, 210, 11, 15, 66, 330, 2310, 13, 21, 70, 390, 2730, 30030, 17, 22, 78, 462, 3570, 39270, 510510, 19, 26, 102, 510, 3990, 43890, 570570, 9699690, 23, 33, 105, 546, 4290, 46410, 690690, 11741730, 223092870

Offset: 1

Peter Dolland, Jan 04 2021

This is a permutation of all squarefree numbers > 1.

			First six rows and columns:
      2     3     5     7    11    13
      6    10    14    15    21    22
     30    42    66    70    78   102
    210   330   390   462   510   546
   2310  2730  3570  3990  4290  4830
  30030 39270 43890 46410 51870 53130

Cf. A005117 (squarefree numbers), A072047 (number of prime factors), A340313 (indexing), A078840 (all natural numbers, not only squarefree).
Rows n=1..10: A000040, A006881, A007304, A046386, A046387, A067885, A123321, A123322, A115343, A281222.
Columns k=1..2: A002110, A306237.
Main diagonal gives A340467.
Cf. A358677.

a340316 n k = a340316_row n !! (k-1)
a340316_row n = [a005117_list !! k | k <- [0..], a072047_list !! k == n]

from math import prod, isqrt
from sympy import prime, primerange, integer_nthroot, primepi
def A340316_T(n,k):
    if n == 1: return prime(k)
    def g(x,a,b,c,m): yield from (((d,) for d in enumerate(primerange(b+1,isqrt(x//c)+1),a+1)) if m==2 else (((a2,b2),)+d for a2,b2 in enumerate(primerange(b+1,integer_nthroot(x//c,m)[0]+1),a+1) for d in g(x,a2,b2,c*b2,m-1)))
    def f(x): return int(k+x-sum(primepi(x//prod(c[1] for c in a))-a[-1][0] for a in g(x,0,1,1,n)))
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    return bisection(f) # Chai Wah Wu, Aug 31 2024

A(A072047(n), A340313(n)) = A005117(n) for n > 1.

A376365 The number of distinct prime factors of the cubefree numbers.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 1, 2, 1, 2, 1, 2, 2, 1, 2, 1, 2, 2, 2, 1, 1, 2, 2, 1, 3, 1, 2, 2, 2, 2, 1, 2, 2, 1, 3, 1, 2, 2, 2, 1, 1, 2, 2, 2, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 3, 1, 2, 2, 3, 1, 1, 2, 2, 2, 2, 3, 1, 2, 1, 3, 2, 2, 2, 1, 3, 2, 2, 2, 2, 2, 1, 2, 2, 2, 1, 3, 1, 3, 2, 1, 1, 3, 2, 1, 3, 2, 2, 2, 2, 2, 1, 2, 2, 2

Offset: 1

Amiram Eldar, Sep 21 2024

Amiram Eldar, Table of n, a(n) for n = 1..10000
Sourabhashis Das, Wentang Kuo, and Yu-Ru Liu, Distribution of omega(n) over h-free and h-full numbers, arXiv:2409.10430 [math.NT], 2024. See Theorem 1.1.

Cf. A001221, A004709, A059956, A072047, A077761, A376361, A376363, A376366.

f[k_] := Module[{e = If[k == 1, {}, FactorInteger[k][[;; , 2]]]}, If[AllTrue[e, # < 3 &], Length[e], Nothing]]; Array[f, 150]

lista(kmax) = {my(e, is); for(k = 1, kmax, e = factor(k)[, 2]; is = 1; for(i = 1, #e, if(e[i] > 2, is = 0; break)); if(is, print1(#e, ", ")));}

a(n) = A001221(A004709(n)).

Sum_{A004709(k) <= x} a(k) = (6/Pi^2) * x * (log(log(x)) + B - C) + O(x/log(x)), where B is Mertens's constant (A077761) and C = Sum_{p prime} (p-1)/(p*(p^3-1)) = 0.10770743252352371604... (Das et al., 2024).

A319246 Sum of prime indices of the n-th squarefree number.

Original entry on oeis.org

0, 1, 2, 3, 3, 4, 4, 5, 6, 5, 5, 7, 8, 6, 6, 9, 7, 10, 6, 11, 7, 8, 7, 12, 9, 8, 13, 7, 14, 10, 15, 9, 16, 8, 10, 11, 17, 18, 12, 9, 8, 19, 11, 8, 20, 21, 13, 9, 9, 22, 14, 23, 10, 15, 12, 24, 10, 13, 16, 11, 25, 26, 10, 27, 9, 17, 28, 29, 9, 14, 30, 11, 12

Offset: 1

Gus Wiseman, Sep 15 2018

nonn

A prime index of n is a number m such that prime(m) divides n.

			The 19th squarefree number is 30 with prime indices (3,2,1), so a(19) = 6.

Rows sums of A319247.

Cf. A000720, A001222, A005117, A056239, A072047, A265668, A299755, A299757, A319241, A319242.

Table[Total[Cases[FactorInteger[n],{p_,k_}:>k*PrimePi[p]]],{n,Select[Range[100],SquareFreeQ]}]

A376361 The number of distinct prime factors of the powerful numbers.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 2, 2, 1, 1, 1, 2, 1, 2, 2, 2, 2, 1, 1, 2, 1, 2, 1, 1, 2, 2, 2, 2, 2, 2, 1, 1, 2, 1, 2, 2, 2, 1, 2, 2, 1, 2, 3, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 1, 2, 1, 2, 2, 2, 2, 2, 1, 2, 3, 3, 1, 2, 2, 2, 2, 1, 2, 1, 1, 1, 2, 2, 1, 2, 2, 2, 3, 2, 2, 1, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2

Offset: 1

Amiram Eldar, Sep 21 2024

Amiram Eldar, Table of n, a(n) for n = 1..10000
Sourabhashis Das, Wentang Kuo, and Yu-Ru Liu, Distribution of omega(n) over h-free and h-full numbers, arXiv:2409.10430 [math.NT], 2024. See Theorem 1.2.
Index entries for sequences related to powerful numbers.

Cf. A001221, A001694, A072047, A077761, A090699, A376362, A376363, A376365.

f[k_] := Module[{e = If[k == 1, {}, FactorInteger[k][[;; , 2]]]}, If[AllTrue[e, # > 1 &], Length[e], Nothing]]; Array[f, 3500]

lista(kmax) = {my(e, is); for(k = 1, kmax, e = factor(k)[, 2]; is = 1; for(i = 1, #e, if(e[i] == 1, is = 0; break)); if(is, print1(#e, ", ")));}

a(n) = A001221(A001694(n)).

Sum_{A001694(k) <= x} a(k) = c * sqrt(x) * (log(log(x)) + B - log(2) + L(2, 3) - L(2, 4)) + O(sqrt(x)/log(x)), where c = zeta(3/2)/zeta(3) (A090699), B is Mertens's constant (A077761), L(h, r) = Sum_{p prime} 1/(p^(r/h - 1) * (p - p^(1 - 1/h) + 1)), L(2, 3) = 1.07848461669337535407..., and L(2, 4) = 0.57937575954505652569... (Das et al., 2024).

A376363 The number of distinct prime factors of the cubefull numbers.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 1, 2, 2, 1, 2, 1, 2, 2, 2, 1, 1, 1, 1, 2, 2, 1, 2, 2, 2, 2, 1, 1, 2, 2, 2, 2, 1, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 1, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 2, 3, 2, 2, 1, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 3, 2, 2, 1, 2

Offset: 1

Amiram Eldar, Sep 21 2024

Amiram Eldar, Table of n, a(n) for n = 1..10000
Sourabhashis Das, Wentang Kuo, and Yu-Ru Liu, Distribution of omega(n) over h-free and h-full numbers, arXiv:2409.10430 [math.NT], 2024. See Theorem 1.2.
Index entries for sequences related to powerful numbers.

Cf. A001221, A036966, A072047, A077761, A362974, A376361, A376364, A376365.

f[k_] := Module[{e = If[k == 1, {}, FactorInteger[k][[;; , 2]]]}, If[AllTrue[e, # > 2 &], Length[e], Nothing]]; Array[f, 60000]

lista(kmax) = {my(e, is); for(k = 1, kmax, e = factor(k)[, 2]; is = 1; for(i = 1, #e, if(e[i] < 3, is = 0; break)); if(is, print1(#e, ", ")));}

a(n) = A001221(A036966(n)).

Sum_{A036966(k) <= x} a(k) = c * x^(1/3) * (log(log(x)) + B - log(3) + L(3, 4) - L(3, 6)) + O(x^(1/3)/log(x)), where c = A362974, B is Mertens's constant (A077761), L(h, r) = Sum_{p prime} 1/(p^(r/h - 1) * (p - p^(1 - 1/h) + 1)), L(3, 4) = 1.65235055631578303808..., and L(3, 6) = 0.67060646664392140547... (Das et al., 2024).

A340313 The n-th squarefree number is the a(n)-th squarefree number having its number of primes.

Original entry on oeis.org

1, 1, 2, 3, 1, 4, 2, 5, 6, 3, 4, 7, 8, 5, 6, 9, 7, 10, 1, 11, 8, 9, 10, 12, 11, 12, 13, 2, 14, 13, 15, 14, 16, 15, 16, 17, 17, 18, 18, 19, 3, 19, 20, 4, 20, 21, 21, 22, 5, 22, 23, 23, 24, 25, 26, 24, 27, 28, 29, 30, 25, 26, 6, 27, 7, 31, 28, 29, 8, 32, 30, 9

Offset: 1

Peter Dolland, Jan 04 2021

nonn

The sequence gives the column index of A005117(n) in the array A340316 and may be understood as a complementary addition to A072047 giving the row index.

			{x|x <= 6, A072047(x) = A072047(6) = 1} = {2,3,4,6}, therefore a(6) = 4.
{x|x <= 28, A072047(x) = A072047(28) = 3} = {19,28}, therefore a(28) = 2.

David A. Corneth, Table of n, a(n) for n = 1..10000
Alois P. Heinz, Plot of n, a(n) for n = 1..1000000

Cf. A001221, A001222, A005117 (squarefree numbers), A058933, A067003, A072047 (number of prime factors), A340316 (squarefree numbers array).

a340313 n = a340313_list !! (n-1)
a340313_list = repetitions a072047_list
    where
    repetitions [] = []
    repetitions (a:as) = 1 : h a as (repetitions as)
    h  []  = []
    h b (c:cs) (r:rs) = (if c == b then succ else id) r : h b cs rs

with(numtheory):
b:= proc(n) option remember; local k; if n=1 then 1 else
      for k from 1+b(n-1) while not issqrfree(k) do od; k fi
    end:
p:= proc() 0 end:
a:= proc(n) option remember; local h; a(n-1);
      h:= bigomega(b(n)); p(h):= p(h)+1;
    end: a(0):=0:
seq(a(n), n=1..100);  # Alois P. Heinz, Jan 06 2021

b[n_] := b[n] = Module[{k}, If[n == 1, 1,
     For[k = 1 + b[n - 1], !SquareFreeQ[k], k++]; k]];
p[_] = 0;
a[n_] := a[n] = Module[{h}, a[n - 1];
     h = PrimeOmega[b[n]]; p[h] = p[h]+1];
a[0] = 0;
Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Mar 28 2022, after Alois P. Heinz *)

first(n) = {v = vector(5); n--; res = vector(n); t = 0; for(i = 2, oo, f = factor(i)[,2]; if(vecmax(f) == 1, if(#f > #v, v = concat(v, vector(#f - #v)) ); t++; v[#f]++; res[t] = v[#f]; if(t >= n, return(concat(1, res)) ) ) ) } \\ David A. Corneth, Jan 07 2021

from math import isqrt, prod
from sympy import primerange, integer_nthroot, mobius, primenu, primepi
def A340313(n):
    if n == 1: return 1
    def g(x,a,b,c,m): yield from (((d,) for d in enumerate(primerange(b+1,isqrt(x//c)+1),a+1)) if m==2 else (((a2,b2),)+d for a2,b2 in enumerate(primerange(b+1,integer_nthroot(x//c,m)[0]+1),a+1) for d in g(x,a2,b2,c*b2,m-1)))
    def f(x): return n+x-sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    kmax = bisection(f)
    return int(sum(primepi(kmax//prod(c[1] for c in a))-a[-1][0] for a in g(kmax,0,1,1,m)) if (m:=primenu(kmax)) > 1 else primepi(kmax)) # Chai Wah Wu, Aug 31 2024

a(n) = #{x|x <= n, A072047(x) = A072047(n)}.

A368779 The number of prime factors of the cubefree numbers, counted with multiplicity.

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 1, 2, 2, 1, 3, 1, 2, 2, 1, 3, 1, 3, 2, 2, 1, 2, 2, 3, 1, 3, 1, 2, 2, 2, 4, 1, 2, 2, 1, 3, 1, 3, 3, 2, 1, 2, 3, 2, 3, 1, 2, 2, 2, 1, 4, 1, 2, 3, 2, 3, 1, 3, 2, 3, 1, 1, 2, 3, 3, 2, 3, 1, 2, 1, 4, 2, 2, 2, 1, 4, 2, 3, 2, 2, 2, 1, 3, 3, 4, 1, 3

Offset: 1

Amiram Eldar, Jan 05 2024

Amiram Eldar, Table of n, a(n) for n = 1..10000
Rafael Jakimczuk and Matilde Lalín, The Number of Prime Factors on Average in Certain Integer Sequences, Journal of Integer Sequences, Vol. 25 (2022), Article 22.2.3.

Cf. A001222, A004709, A072047.

f[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, If[AllTrue[e, # < 3 &], Total[e], Nothing]]; f[1] = 0; Array[f, 100]

lista(max) = {my(e); for(k = 1, max, e = factor(k)[,2]; if(k == 1 || vecmax(e) < 3, print1(vecsum(e), ", ")));}

from sympy import mobius, integer_nthroot, primeomega
def A368779(n):
    def f(x): return n+x-sum(mobius(k)*(x//k**3) for k in range(1, integer_nthroot(x,3)[0]+1))
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    return primeomega(m) # Chai Wah Wu, Aug 06 2024

a(n) = A001222(A004709(n)).

Sum_{A004709(k) <= x} a(k) = (1/zeta(3)) * x * log(log(x)) + O(x) (Jakimczuk and Lalín, 2022). [corrected Sep 21 2024]

A340467 a(n) is the n-th squarefree number having n prime factors.

Original entry on oeis.org

2, 10, 66, 462, 4290, 53130, 903210, 17687670, 406816410, 11125544430, 338431883790, 11833068917670, 457077357006270, 20384767656323070, 955041577211912190, 49230430891074322890, 2740956243836856315270, 168909608387276001835590, 11054926927790884163355330

Offset: 1

Alois P. Heinz, Jan 08 2021

nonn

a(n) is the n-th product of n distinct primes.
All terms are even.
This sequence differs from A073329 which has also nonsquarefree terms.

			a(1) = A000040(1) = 2.
a(2) = A006881(2) = 10.
a(3) = A007304(3) = 66.
a(4) = A046386(4) = 462.
a(5) = A046387(5) = 4290.
a(6) = A067885(6) = 53130.
a(7) = A123321(7) = 903210.
a(8) = A123322(8) = 17687670.
a(9) = A115343(9) = 406816410.
a(10) = A281222(10) = 11125544430.

Alois P. Heinz, Table of n, a(n) for n = 1..350

Main diagonal of A340316.
Cf. A001221, A001222, A005117, A070826, A072047, A073329, A101695, A340313.
Cf. A000040, A006881, A007304, A046386, A046387, A067885, A123321, A123322, A115343, A281222.

from math import isqrt, prod
from sympy import primerange, integer_nthroot, primepi
def A340467(n):
    if n == 1: return 2
    def g(x,a,b,c,m): yield from (((d,) for d in enumerate(primerange(b+1,isqrt(x//c)+1),a+1)) if m==2 else (((a2,b2),)+d for a2,b2 in enumerate(primerange(b+1,integer_nthroot(x//c,m)[0]+1),a+1) for d in g(x,a2,b2,c*b2,m-1)))
    def f(x): return int(n+x-sum(primepi(x//prod(c[1] for c in a))-a[-1][0] for a in g(x,0,1,1,n)))
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    return bisection(f) # Chai Wah Wu, Aug 31 2024

a(n) = A340316(n,n).
a(n) = A005117(m) <=> A072047(m) = n = A340313(m).
A001221(a(n)) = A001222(a(n)) = n.
a(n) < A070826(n+1), the least odd number with exactly n distinct prime divisors.

A360729 a(n) is the number of prime factors of the n-th powerful number (counted with repetition).

Original entry on oeis.org

0, 2, 3, 2, 4, 2, 3, 5, 4, 2, 6, 5, 4, 4, 5, 2, 3, 7, 6, 2, 4, 5, 6, 4, 5, 8, 7, 2, 6, 3, 2, 5, 6, 7, 4, 4, 5, 9, 2, 8, 4, 7, 5, 4, 6, 6, 7, 2, 8, 6, 2, 5, 7, 6, 10, 4, 5, 9, 4, 4, 8, 5, 3, 5, 2, 5, 4, 4, 7, 8, 2, 9, 6, 7, 2, 6, 8, 7, 6, 11, 4, 7, 3, 2, 10, 5

Offset: 1

Amiram Eldar, Feb 18 2023

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000
Rafael Jakimczuk and Matilde Lalín, The Number of Prime Factors on Average in Certain Integer Sequences, Journal of Integer Sequences, Vol. 25 (2022), Article 22.2.3.

Cf. A001222, A001694, A083342, A090699.

Similar sequences: A072047, A076399.

PrimeOmega[Select[Range[3000], # == 1 || Min[FactorInteger[#][[;; , 2]]] > 1 &]]

apply(bigomega, select(ispowerful, [1..3000]))

a(n) = A001222(A001694(n)).

Sum_{A001694(k) < x} a(k) = (2*zeta(3/2)/zeta(3))*sqrt(x)*log(log(x)) + (2*(B_2 - log(2)) + Sum_{p prime} (3/((p^(3/2)+1))))*(zeta(3/2)/zeta(3))*sqrt(x) + O(sqrt(x)/sqrt(log(x))), where B_2 = A083342 (Jakimczuk and Lalín, 2022). [corrected Sep 21 2024]

A384008 Irregular triangle read by rows where row n lists the first differences of the 0-prepended prime indices of the n-th squarefree number.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, May 23 2025

All rows are different.

			The 28-th squarefree number is 42, with 0-prepended prime indices (0,1,2,4), with differences (1,1,2), so row 28 is (1,1,2).
The squarefree numbers and corresponding rows begin:
    1: ()        23: (9)        47: (15)
    2: (1)       26: (1,5)      51: (2,5)
    3: (2)       29: (10)       53: (16)
    5: (3)       30: (1,1,1)    55: (3,2)
    6: (1,1)     31: (11)       57: (2,6)
    7: (4)       33: (2,3)      58: (1,9)
   10: (1,2)     34: (1,6)      59: (17)
   11: (5)       35: (3,1)      61: (18)
   13: (6)       37: (12)       62: (1,10)
   14: (1,3)     38: (1,7)      65: (3,3)
   15: (2,1)     39: (2,4)      66: (1,1,3)
   17: (7)       41: (13)       67: (19)
   19: (8)       42: (1,1,2)    69: (2,7)
   21: (2,2)     43: (14)       70: (1,2,1)
   22: (1,4)     46: (1,8)      71: (20)

Row-lengths are A072047, sums A243290.
This is the restriction of A383534 (ranked by A383535) to rows of squarefree index.
A000040 lists the primes, differences A001223.
A048767 is the Look-and-Say transform, union A351294, complement A351295.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A320348 counts strict partitions with distinct 0-appended differences, ranks A325388.
A325324 counts partitions with distinct 0-appended differences, ranks A325367.
Cf. A001221, A005117, A055396, A061395, A325325, A355536, A358137, A384009.

sql=Select[Range[100],SquareFreeQ];
prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Differences[Prepend[prix[sql[[n]]],0]],{n,Length[sql]}]

cp's OEIS Frontend

A340316 Square array A(n,k), n>=1, k>=1, read by antidiagonals, where row n is the increasing list of all squarefree numbers with n primes.

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A376365 The number of distinct prime factors of the cubefree numbers.

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A319246 Sum of prime indices of the n-th squarefree number.

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A376361 The number of distinct prime factors of the powerful numbers.

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A376363 The number of distinct prime factors of the cubefull numbers.

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A340313 The n-th squarefree number is the a(n)-th squarefree number having its number of primes.

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A368779 The number of prime factors of the cubefree numbers, counted with multiplicity.

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A340467 a(n) is the n-th squarefree number having n prime factors.

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