A046391 -id:A046391 - cp's OEIS frontend

A112643 Odd squarefree abundant numbers.

15015, 19635, 21945, 23205, 25935, 26565, 31395, 33495, 33915, 35805, 39585, 41055, 42315, 42735, 45885, 47355, 49665, 50505, 51765, 54285, 55965, 58695, 61215, 64155, 68145, 70455, 72345, 77385, 80535, 82005, 83265, 84315, 91245

Offset: 1

Labos Elemer, Sep 20 2005

nonn

Deviates from A046391 (does not contain 36465, 40755 for example).
The numbers of terms not exceeding 10^k, for k = 5, 6, ..., are 34, 134, 1663, 16328, 175630, 1694621, 16726454, ... . Apparently, the asymptotic density of this sequence exists and equals 0.00016... . - Amiram Eldar, Sep 02 2022
From Amiram Eldar, Jan 15 2025: (Start)
The least term that is not divisible by 5 is a(3696) = 22309287.
The least term that is not divisible by 3 is a(5607800) = 33426748355.
The least term that is coprime to 15 is 1357656019974967471687377449. (End)

			199815 = 3 * 5 * 7 * 11 * 173, with 32 divisors adding up to 400896 = 2 * 199815 + 1266.

Donovan Johnson, Table of n, a(n) for n = 1..10000

Cf. A001221, A005101, A005231, A005408, A046391, A087248.

# see A087248 for the additional code
isA112643 := proc(n)
    isA087248(n) and type(n,'odd') ;
end proc:
for n from 1 do
    if isA112643(n) then
        print(n);
    end if;
end do: # R. J. Mathar, Nov 10 2014

ta = {{0}}; Do[g = n; s = DivisorSigma[1, n] - 2 * n; If[Greater[s, 0] && Equal[Abs[MoebiusMu[n]], 1] && !Equal[Mod[n, 2], 0], Print[n, PrimeFactorList[n], s]; ta = Append[ta, n]], {n, 1, 200000}];{ta = Delete[ta, 1], g}(* Elemer *)
Select[Range[1, 99999, 2], MoebiusMu[#] != 0 && DivisorSigma[1, #] > 2 # &] (* Alonso del Arte, Nov 11 2017 *)

is(n)=if(n%2==0, return(0)); my(f=factor(n)); sigma(f)>2*n && vecmax(f[,2])==1 \\ Charles R Greathouse IV, Feb 21 2017

A087248 INTERSECT A005408.

omega(a(n)) >= 5, where omega(n) = A001221(n) is the number of distinct primes dividing n. - Amiram Eldar, Jan 15 2025

A069158 a(n) = Product{d|n} mu(d), product over positive divisors, d, of n, where mu(d) = Moebius function (A008683).

Original entry on oeis.org

1, -1, -1, 0, -1, 1, -1, 0, 0, 1, -1, 0, -1, 1, 1, 0, -1, 0, -1, 0, 1, 1, -1, 0, 0, 1, 0, 0, -1, 1, -1, 0, 1, 1, 1, 0, -1, 1, 1, 0, -1, 1, -1, 0, 0, 1, -1, 0, 0, 0, 1, 0, -1, 0, 1, 0, 1, 1, -1, 0, -1, 1, 0, 0, 1, 1, -1, 0, 1, 1, -1, 0, -1, 1, 0, 0, 1, 1, -1, 0, 0, 1, -1, 0, 1, 1, 1, 0, -1, 0, 1, 0, 1, 1, 1, 0, -1, 0, 0, 0, -1, 1, -1, 0, 1, 1

Offset: 1

Leroy Quet, Apr 08 2002

sign

Absolute value of a(n) = absolute value of mu(n).
Differs from A080323 at n=2, 105, 165, 195, 231, ..., 15015,..., 19635,.. (cf. A046389, A046391, ...) - R. J. Mathar, Dec 15 2008
Not multiplicative: For example a(2)*a(15) <> a(30). - R. J. Mathar, Mar 31 2012
Row products of table A225817. - Reinhard Zumkeller, Jul 30 2013

			a(6) = mu(1)*mu(2)*mu(3)*mu(6) = 1*(-1)*(-1)*1 = 1.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A008683, A008966, A010051, A013929, A046389, A046391, A080323, A225817.

a069158 = product . a225817_row  -- Reinhard Zumkeller, Jul 30 2013

f := function(n); t1 := &*[MoebiusMu(d) : d in Divisors(n) ]; return t1; end function;

A069158 := proc(n)
    mul(numtheory[mobius](d),d=numtheory[divisors](n)) ;
end proc: # R. J. Mathar, May 28 2016

a[n_] := Product[MoebiusMu[d], {d, Divisors[n]}]; Array[a, 106] (* Jean-François Alcover, Feb 22 2018 *)

a(n) = vecprod(apply(moebius, divisors(n))); \\ Amiram Eldar, Feb 10 2025

a(n) = 0 if mu(n) = 0 (A013929); a(n) = -1 if n = prime; a(n) = 1 if n = squarefree composite (A120944) or 1.

a(n) = A008966(n) - 2*A010051(n). - Amiram Eldar, Feb 10 2025

A168352 Products of 6 distinct odd primes.

Original entry on oeis.org

255255, 285285, 345345, 373065, 435435, 440895, 451605, 465465, 504735, 533715, 555555, 569415, 596505, 608685, 615615, 636405, 645645, 672945, 680295, 692835, 705705, 719355, 726495, 752115, 770385, 780045, 795795, 803985, 805035, 811965, 823515, 838695, 844305, 858585

Offset: 1

Vladimir Joseph Stephan Orlovsky, Nov 23 2009

			255255 = 3*5*7*11*13*17
285285 = 3*5*7*11*13*19
345345 = 3*5*7*11*13*23
435435 = 3*5*7*11*13*29

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A005408, A067885.

Cf. A046391 (5 distinct odd primes).

f[n_]:=Last/@FactorInteger[n]=={1,1,1,1,1,1}&&FactorInteger[n][[1,1]]>2; lst={};Do[If[f[n],AppendTo[lst,n]],{n,6*9!}];lst

is(n) = {n%2 == 1 && factor(n)[,2]~ == [1,1,1,1,1,1]} \\ David A. Corneth, Aug 26 2020

from sympy import primefactors, factorint
print([n for n in range(1, 1000000, 2) if len(primefactors(n)) == 6 and max(list(factorint(n).values())) == 1]) # Karl-Heinz Hofmann, Mar 01 2023

from math import prod, isqrt
from sympy import primerange, integer_nthroot, primepi
def A168352(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,1,2,1,6)))
    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,n,n) # Chai Wah Wu, Sep 10 2024

A067885 INTERSECT A005408. [R. J. Mathar, Nov 24 2009]

Definition corrected by R. J. Mathar, Nov 24 2009

More terms from David A. Corneth, Aug 26 2020

A278569 Numbers of the form p^iq^jr^k where p,q,r are distinct odd primes and i,j,k >= 1.

Original entry on oeis.org

105, 165, 195, 231, 255, 273, 285, 315, 345, 357, 385, 399, 429, 435, 455, 465, 483, 495, 525, 555, 561, 585, 595, 609, 615, 627, 645, 651, 663, 665, 693, 705, 715, 735, 741, 759, 765, 777, 795, 805, 819, 825, 855, 861, 885, 897, 903, 915, 935, 945, 957, 969, 975, 987, 1001, 1005, 1015, 1023, 1035, 1045, 1065, 1071, 1085, 1095, 1105, 1113, 1131, 1173, 1185, 1197, 1209

Offset: 1

N. J. A. Sloane, Nov 27 2016

nonn

More than the usual number of terms are included to show the difference from A216918 (the latter includes 3*5*7*11 = 1155 and all terms of A046390, A046391 etc).

If i,j,k are all equal to 1 we get A046389.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Includes A046389, subsequence of A216918.

Select[Range@ 1500, PrimeNu@ # == 3 && OddQ@ # &] (* Michael De Vlieger, Dec 05 2016 *)

A033992 INTERSECT A005408. - R. J. Mathar, Dec 05 2016

A112644 Odd and squarefree abundant numbers not divisible by 5.

Original entry on oeis.org

22309287, 28129101, 30069039, 34051017, 35888853, 36399363, 38057019, 39768729, 40681641, 41708667, 43444401, 45588543, 45894849, 48141093, 48555507, 50489439, 51294243, 51408357, 53804751, 54777723, 55186131, 56429373, 57228171, 58555497, 59168109

Offset: 1

Labos Elemer, Sep 20 2005

nonn

The least term that is not divisible by 3 is 73#/5# = Product_{k=4..21} prime(k) = 1357656019974967471687377449. - Amiram Eldar, Aug 15 2024

			99906807 = 3*7*11*13*17*19*103 is a term since it is an odd squarefree number that is not divisible by 5, and sigma(99906807) = 201277440 > 2*99906807.

Donovan Johnson, Table of n, a(n) for n = 1..1000

Cf. A087248, A046391, A112643, A112642, A112640.

Cf. A005231, A047802.

ta={{0}};Do[g=n;s=DivisorSigma[1, n]-2*n; If[Greater[s, 0]&&Equal[Abs[MoebiusMu[n]], 1]&& !Equal[Mod[n, 2], 0]&&!Equal[Mod[n, 5], 0], Print[n, PrimeFactorList[n], s];ta=Append[ta, n]], {n, 10000000, 100000000}];{ta=Delete[ta, 1], g}

issfab(k) = my(f = factor(k)); issquarefree(f) && sigma(f, -1) > 2;
is(k) = gcd(k, 10) == 1 && issfab(k); \\ Amiram Eldar, Aug 15 2024

A361075 Products of exactly 7 distinct odd primes.

Original entry on oeis.org

4849845, 5870865, 6561555, 7402395, 7912905, 8273265, 8580495, 8843835, 9444435, 10015005, 10140585, 10465455, 10555545, 10705695, 10818885, 10975965, 11565015, 11696685, 11996985, 12267255, 12777765, 12785955, 13096545, 13408395, 13498485, 13528515, 13667745, 13803405

Offset: 1

Karl-Heinz Hofmann, Mar 01 2023

nonn

			a(1)     =   4849845 = 3*5*7*11*13*17*19
a(9663)  = 253808555 = 5*7*11*13*17*19*157
a(9961)  = 258573315 = 3*5*7*11*13*17*1013
a(10000) = 259173915 = 3*5*7*11*13*41*421

Karl-Heinz Hofmann, Table of n, a(n) for n = 1..10000

Cf. A065091, A046388 (2 distinct odd primes).
Cf. A046389 (3 distinct odd primes), A046390 (4 distinct odd primes).
Cf. A046391 (5 distinct odd primes), A168352 (6 distinct odd primes).

import numpy
from sympy import nextprime, sieve, primepi
k_upto = 14 * 10**6
array = numpy.zeros(k_upto,dtype="i4")
sieve_max_number = primepi(nextprime(k_upto // 255255))
for s in range(2,sieve_max_number):
    array[sieve[s]:k_upto][::sieve[s]] += 1
for s in range(2,sieve_max_number):
    array[sieve[s]**2:k_upto][::sieve[s]**2] = 0
print([k for k in range(1,k_upto,2) if array[k] == 7])

from math import prod, isqrt
from sympy import primerange, integer_nthroot, primepi
def A361075(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,1,2,1,7)))
    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,n,n) # Chai Wah Wu, Sep 10 2024

cp's OEIS Frontend

A112643 Odd squarefree abundant numbers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A069158 a(n) = Product{d|n} mu(d), product over positive divisors, d, of n, where mu(d) = Moebius function (A008683).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Magma

Maple

Mathematica

PARI

Formula

A168352 Products of 6 distinct odd primes.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Python

Python

Formula

Extensions

A278569 Numbers of the form p^i*q^j*r^k where p,q,r are distinct odd primes and i,j,k >= 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A112644 Odd and squarefree abundant numbers not divisible by 5.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A361075 Products of exactly 7 distinct odd primes.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Python

Python

A278569 Numbers of the form p^iq^jr^k where p,q,r are distinct odd primes and i,j,k >= 1.