A281222 -id:A281222 - cp's OEIS frontend

A030229 Numbers that are the product of an even number of distinct primes.

1, 6, 10, 14, 15, 21, 22, 26, 33, 34, 35, 38, 39, 46, 51, 55, 57, 58, 62, 65, 69, 74, 77, 82, 85, 86, 87, 91, 93, 94, 95, 106, 111, 115, 118, 119, 122, 123, 129, 133, 134, 141, 142, 143, 145, 146, 155, 158, 159, 161, 166, 177, 178, 183, 185, 187, 194, 201, 202, 203, 205, 206, 209, 210, 213, 214

Offset: 1

David W. Wilson

These are the positive integers k with moebius(k) = 1 (cf. A008683). - N. J. A. Sloane, May 18 2021
From Enrique Pérez Herrero, Jul 06 2012: (Start)
This sequence and A030059 form a partition of the squarefree numbers set: A005117.
Also solutions to equation mu(n)=1.
Sum_{n>=1} 1/a(n)^s = (Zeta(s)^2 + Zeta(2*s))/(2*Zeta(s)*Zeta(2*s)).
(End)
A008683(a(n)) = 1; a(A220969(n)) mod 2 = 0; a(A220968(n)) mod 2 = 1. - Reinhard Zumkeller, Dec 27 2012
Characteristic function for values of a(n) = (mu(n)+1)! - 1, where mu(n) is the Mobius function (A008683). - Wesley Ivan Hurt, Oct 11 2013
Conjecture: For the matrix M(i,j) = 1 if j|i and 0 otherwise, Inverse(M)(a,1) = -1, for any a in this sequence. - Benedict W. J. Irwin, Jul 26 2016
Solutions to the equation Sum_{d|n} mu(d)*d = Sum_{d|n} mu(n/d)*d. - Torlach Rush, Jan 13 2018
Solutions to the equation Sum_{d|n} mu(d)*sigma(d) = n, where sigma(n) is the sum of divisors function (A000203). - Robert D. Rosales, May 20 2024
From Peter Munn, Oct 04 2019: (Start)
Numbers n such that omega(n) = bigomega(n) = 2*k for some integer k.
The squarefree numbers in A000379.
The squarefree numbers in A028260.
This sequence is closed with respect to the commutative binary operation A059897(.,.), thus it forms a subgroup of the positive integers under A059897(.,.). A006094 lists a minimal set of generators for this subgroup. The lexicographically earliest ordered minimal set of generators is A100484 with its initial 4 removed.
(End)
The asymptotic density of this sequence is 3/Pi^2 (cf. A104141). - Amiram Eldar, May 22 2020

			(empty product), 2*3, 2*5, 2*7, 3*5, 3*7, 2*11, 2*13, 3*11, 2*17, 5*7, 2*19, 3*13, 2*23,...

B. C. Berndt and R. A. Rankin, Ramanujan: Letters and Commentary, see p. 23; AMS Providence RI 1995
S. Ramanujan, Collected Papers, pp. xxiv, 21.

T. D. Noe, Table of n, a(n) for n = 1..1000
Debmalya Basak, Nicolas Robles, and Alexandru Zaharescu, Exponential sums over Möbius convolutions with applications to partitions, arXiv:2312.17435 [math.NT], 2023. Mentions this sequence.
S. Ramanujan, Irregular numbers, J. Indian Math. Soc. 5 (1913) 105-106.
Eric Weisstein's World of Mathematics, Prime Factor
Eric Weisstein's World of Mathematics, Moebius Function
Eric Weisstein's World of Mathematics, Prime Sums
H. S. Wilf, A Greeting; and a view of Riemann's Hypothesis, Amer. Math. Monthly, 94:1 (1987), 3-6.

A006881, A046386, A067885, A123322, A281222 are subsequences.

Cf. A000379, A005117, A008683, A028260, A030059, A104141, A151797, A245630.

import Data.List (elemIndices)
a030229 n = a030229_list !! (n-1)
a030229_list = map (+ 1) $ elemIndices 1 a008683_list
-- Reinhard Zumkeller, Dec 27 2012

a := n -> `if`(numtheory[mobius](n)=1,n,NULL); seq(a(i),i=1..214); # Peter Luschny, May 04 2009
with(numtheory); t := [ ]: f := [ ]: for n from 1 to 250 do if mobius(n) = 1 then t := [ op(t), n ] else f := [ op(f), n ]; fi; od: t; # Wesley Ivan Hurt, Oct 11 2013
# alternative
A030229 := proc(n)
    option remember;
    local a;
    if n = 1 then
        1;
    else
        for a from procname(n-1)+1 do
            if numtheory[mobius](a) = 1 then
                return a;
            end if;
        end do:
    end if;
end proc:
seq(A030229(n),n=1..40) ; # R. J. Mathar, Sep 22 2020

Select[Range[214], MoebiusMu[#] == 1 &] (* Jean-François Alcover, Oct 04 2011 *)

isA030229(n)= #(n=factor(n)[,2]) % 2 == 0 && (!n || vecmax(n)==1 )

is(n)=moebius(n)==1 \\ Charles R Greathouse IV, Jan 31 2017
for(n=1,500, isA030229(n)&print1(n",")) \\ M. F. Hasler

from math import isqrt, prod
from sympy import primerange, integer_nthroot, primepi
def A030229(n):
    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-1+x-sum(sum(primepi(x//prod(c[1] for c in a))-a[-1][0] for a in g(x,0,1,1,i)) for i in range(2,x.bit_length(),2)))
    kmin, kmax = 0,1
    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 # Chai Wah Wu, Aug 29 2024

a(n) < n*Pi^2/3 infinitely often; a(n) > n*Pi^2/3 infinitely often. - Charles R Greathouse IV, Oct 04 2011; corrected Sep 07 2017

{a(n)} = {m : m = A059897(A030059(k), p), k >= 1} for prime p, where {a(n)} denotes the set of integers in the sequence. - Peter Munn, Oct 04 2019

A115343 Products of 9 distinct primes.

Original entry on oeis.org

223092870, 281291010, 300690390, 340510170, 358888530, 363993630, 380570190, 397687290, 406816410, 417086670, 434444010, 455885430, 458948490, 481410930, 485555070, 497668710, 504894390, 512942430, 514083570, 531990690, 538047510, 547777230, 551861310

Offset: 1

Jonathan Vos Post, Mar 06 2006

			514083570 is in the sequence as it is equal to 2*3*5*7*11*13*17*19*53.

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 1045 terms from Vincenzo Librandi and Chai Wah Wu)

Cf. A000040, A006881, A007304, A046386, A046387, A067885, A123321, A123322, A115343, A281222.

N:= 10^9: # to get all terms < N
n0:= mul(ithprime(i),i=1..8):
Primes:= select(isprime,[$1..floor(N/n0)]):
nPrimes:= nops(Primes):
for i from 1 to 9 do
  for j from 1 to nPrimes do
    M[i,j]:= convert(Primes[1..min(j,i)],`*`);
od od:
A:= {}:
for i9 from 9 to nPrimes do
  m9:= Primes[i9];
for i8 in select(t -> M[7,t-1]*Primes[t]*m9 <= N, [$8..i9-1]) do
  m8:= m9*Primes[i8];
for i7 in select(t -> M[6,t-1]*Primes[t]*m8 <= N, [$7..i8-1]) do
  m7:= m8*Primes[i7];
for i6 in select(t -> M[5,t-1]*Primes[t]*m7 <= N, [$6..i7-1]) do
  m6:= m7*Primes[i6];
for i5 in select(t -> M[4,t-1]*Primes[t]*m6 <= N, [$5..i6-1]) do
  m5:= m6*Primes[i5];
for i4 in select(t -> M[3,t-1]*Primes[t]*m5 <= N, [$4..i5-1]) do
  m4:= m5*Primes[i4];
for i3 in select(t -> M[2,t-1]*Primes[t]*m4 <= N, [$3..i4-1]) do
  m3:= m4*Primes[i3];
for i2 in select(t -> M[1,t-1]*Primes[t]*m3 <= N, [$2..i3-1]) do
  m2:= m3*Primes[i2];
for i1 in select(t -> Primes[t]*m2 <= N, [$1..i2-1]) do
  A:= A union {m2*Primes[i1]};
od od od od od od od od od:
A; # Robert Israel, Sep 02 2014

Module[{n=6*10^8,k},k=PrimePi[n/Times@@Prime[Range[8]]];Select[ Union[ Times@@@ Subsets[Prime[Range[k]],{9}]],#<=n&]](* Harvey P. Dale with suggestions from Jean-François Alcover, Sep 03 2014 *)
n = 10^9; n0 = Times @@ Prime[Range[8]]; primes = Select[Range[Floor[n/n0]], PrimeQ]; nPrimes = Length[primes]; Do[M[i, j] = Times @@ primes[[1 ;; Min[j, i]]], {i, 1, 9}, {j, 1, nPrimes}]; A = {};
Do[m9 = primes[[i9]];
Do[m8 = m9*primes[[i8]];
Do[m7 = m8*primes[[i7]];
Do[m6 = m7*primes[[i6]];
Do[m5 = m6*primes[[i5]];
Do[m4 = m5*primes[[i4]];
Do[m3 = m4*primes[[i3]];
Do[m2 = m3*primes[[i2]];
Do[A = A ~Union~ {m2*primes[[i1]]},
{i1, Select[Range[1, i2-1], primes[[#]]*m2 <= n &]}],
{i2, Select[Range[2, i3-1], M[1, #-1]*primes[[#]]*m3 <= n &]}],
{i3, Select[Range[3, i4-1], M[2, #-1]*primes[[#]]*m4 <= n &]}],
{i4, Select[Range[4, i5-1], M[3, #-1]*primes[[#]]*m5 <= n &]}],
{i5, Select[Range[5, i6-1], M[4, #-1]*primes[[#]]*m6 <= n &]}],
{i6, Select[Range[6, i7-1], M[5, #-1]*primes[[#]]*m7 <= n &]}],
{i7, Select[Range[7, i8-1], M[6, #-1]*primes[[#]]*m8 <= n &]}],
{i8, Select[Range[8, i9-1], M[7, #-1]*primes[[#]]*m9 <= n &]}],
{i9, 9, nPrimes}];
A (* Jean-François Alcover, Sep 03 2014, translated and adapted from Robert Israel's Maple program *)

is(n)=omega(n)==9 && bigomega(n)==9 \\ Hugo Pfoertner, Dec 18 2018

from operator import mul
from functools import reduce
from sympy import nextprime, sieve
from itertools import combinations
n = 190
m = 9699690*nextprime(n-1)
A115343 = []
for x in combinations(sieve.primerange(1,n),9):
    y = reduce(mul,(d for d in x))
    if y < m:
        A115343.append(y)
A115343 = sorted(A115343) # Chai Wah Wu, Sep 02 2014

from math import prod, isqrt
from sympy import primerange, integer_nthroot, primepi
def A115343(n):
    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,0,1,1,9)))
    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

Corrected and extended by Don Reble, Mar 09 2006

More terms and corrected b-file from Chai Wah Wu, Sep 02 2014

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.

Original entry on oeis.org

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.

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.

A356683 a(n) is the smallest positive k such that the count of squarefree numbers <= k that have n prime factors is equal to the count of squarefree numbers <= k that have n-1 prime factors (and the count is positive).

Original entry on oeis.org

2, 39, 1279786, 8377774397163159586

Offset: 1

Jon E. Schoenfield, Nov 22 2022

			The first two squarefree numbers are 1 and 2; 1 has 0 prime factors and 2 has 1 prime factor, so a(1)=2.
At k=39, in the interval [1..k], there are 12 squarefree numbers with 1 prime factor (i.e., 12 primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37), and 12 squarefree numbers with 2 prime factors (i.e., 6, 10, 14, 15, 21, 22, 26, 33, 34, 35, 38, 39). k=39 is the smallest such positive number for which these two counts are the same (and are positive), so a(2)=39.
At k=1279786, the interval [1..k] includes 265549 squarefree numbers with 2 prime factors and the same number of squarefree numbers with 3 prime factors, and there is no smaller positive number k that has this property (where the counts are positive), so a(3)=1279786. There are 75 numbers with this property, the last one being 1281378.
At k=8377774397163159586, the interval [1..k] includes 1356557942402075858 squarefree numbers with 3 prime factors and the same number of squarefree numbers with 4 prime factors, and there is no smaller positive number k that has this property (where the counts are positive), so a(4)=8377774397163159586. There are 14 numbers with this property, the last one being 8377774397163162544. - _Henri Lifchitz_, Jan 31 2025

Cf. A005117, A072047.
Cf. 1 to 5 distinct primes: A000040, A006881, A007304, A046386, A046387.
Cf. 6 to 10 distinct primes: A067885, A123321, A123322, A115343, A281222.
Cf. A340316.

a(n) = my(nbm = 0, nbn = 0); for (k=1, oo, if (issquarefree(k), my(o=omega(k)); if (o==n, nbn++); if (o==n-1, nbm++); if (nbm && (nbn==nbm), return(k)))); \\ Michel Marcus, Nov 25 2022

from itertools import count
from math import prod, isqrt
from sympy import primerange, integer_nthroot, primepi
def A356683(n):
    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(k,n): return sum(primepi(k//prod(c[1] for c in a))-a[-1][0] for a in g(k,0,1,1,n)) if n>1 else primepi(k)
    return 2 if n==1 else next(k for k in count(1) if (x:=f(k,n-1))>0 and x==f(k,n)) # Chai Wah Wu, Aug 31 2024

a(4) from Henri Lifchitz, Jan 31 2025

cp's OEIS Frontend

A030229 Numbers that are the product of an even number of distinct primes.

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A115343 Products of 9 distinct primes.

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

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A356683 a(n) is the smallest positive k such that the count of squarefree numbers <= k that have n prime factors is equal to the count of squarefree numbers <= k that have n-1 prime factors (and the count is positive).

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