A006881 -id:A006881 - cp's OEIS frontend

A046386 Products of exactly four distinct primes.

210, 330, 390, 462, 510, 546, 570, 690, 714, 770, 798, 858, 870, 910, 930, 966, 1110, 1122, 1155, 1190, 1218, 1230, 1254, 1290, 1302, 1326, 1330, 1365, 1410, 1430, 1482, 1518, 1554, 1590, 1610, 1722, 1770, 1785, 1794, 1806, 1830, 1870, 1914, 1938, 1974

Offset: 1

Patrick De Geest, Jun 15 1998

A squarefree subsequence of A033993. Numbers like 420 = 2^2*3*5*7 with at least one prime exponent greater than 1 in the prime signature are excluded here. - R. J. Mathar, Apr 03 2011

Numbers such that omega(n) = bigomega(n) = 4. - Michel Marcus, Dec 15 2015

			210 = 2*3*5*7;
330 = 2*3*5*11;
390 = 2*3*5*13;
462 = 2*3*7*11.

T. D. Noe, Table of n, a(n) for n = 1..10000

Products of exactly k distinct primes, for k = 1 to 6: A000040, A006881. A007304, A046386, A046387, A067885.
Cf. A001221 (omega), A001222 (bigomega), A014613 (bigomega(N) = 4) and A033993 (omega(N) = 4).
Cf. A046402 (4 palindromic prime factors).

fQ[n_] := Last /@ FactorInteger[n] == {1, 1, 1, 1}; Select[ Range[2000], fQ[ # ] &] (* Robert G. Wilson v, Aug 04 2005 *)
Select[Range[2000],PrimeNu[#]==PrimeOmega[#]==4&] (* Harvey P. Dale, Jan 05 2025 *)

is(n)=factor(n)[,2]==[1,1,1,1]~ \\ Charles R Greathouse IV, Sep 17 2015

is(n) = omega(n)==4 && bigomega(n)==4 \\ Hugo Pfoertner, Dec 18 2018

list(lim)=my(v=List()); forprime(p=2,sqrtnint(lim\=1,4), forprime(q=p+1,sqrtnint(lim\p,3), forprime(r=q+2,sqrtint(lim\p\q), my(t=p*q*r); forprime(s=r+2,lim\t, listput(v,t*s))))); Set(v) \\ Charles R Greathouse IV, Dec 05 2024

from math import isqrt
from sympy import primepi, primerange, integer_nthroot
def A046386(n):
    def f(x): return int(n+x-sum(primepi(x//(k*m*r))-c for a,k in enumerate(primerange(integer_nthroot(x,4)[0]+1),1) for b,m in enumerate(primerange(k+1,integer_nthroot(x//k,3)[0]+1),a+1) for c,r in enumerate(primerange(m+1,isqrt(x//(k*m))+1),b+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
    return bisection(f) # Chai Wah Wu, Aug 29 2024

Intersection of A014613 (product of 4 primes) and A033993 (divisible by 4 distinct primes). - M. F. Hasler, Mar 24 2022

A339846 Number of even-length factorizations of n into factors > 1.

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 2, 0, 1, 1, 3, 0, 2, 0, 2, 1, 1, 0, 4, 1, 1, 1, 2, 0, 3, 0, 3, 1, 1, 1, 5, 0, 1, 1, 4, 0, 3, 0, 2, 2, 1, 0, 6, 1, 2, 1, 2, 0, 4, 1, 4, 1, 1, 0, 6, 0, 1, 2, 6, 1, 3, 0, 2, 1, 3, 0, 8, 0, 1, 2, 2, 1, 3, 0, 6, 3, 1, 0, 6, 1, 1, 1, 4, 0, 6, 1, 2, 1, 1, 1, 10, 0, 2, 2, 5, 0, 3, 0, 4, 3

Offset: 1

Gus Wiseman, Dec 28 2020

nonn

			The a(n) factorizations for n = 12, 16, 24, 36, 48, 72, 96, 120:
  2*6  2*8      3*8      4*9      6*8      8*9      2*48         2*60
  3*4  4*4      4*6      6*6      2*24     2*36     3*32         3*40
       2*2*2*2  2*12     2*18     3*16     3*24     4*24         4*30
                2*2*2*3  3*12     4*12     4*18     6*16         5*24
                         2*2*3*3  2*2*2*6  6*12     8*12         6*20
                                  2*2*3*4  2*2*2*9  2*2*3*8      8*15
                                           2*2*3*6  2*2*4*6      10*12
                                           2*3*3*4  2*3*4*4      2*2*5*6
                                                    2*2*2*12     2*3*4*5
                                                    2*2*2*2*2*3  2*2*2*15
                                                                 2*2*3*10

The case of set partitions (or n squarefree) is A024430.
The case of partitions (or prime powers) is A027187.
The ordered version is A174725, odd: A174726.
The odd-length factorizations are counted by A339890.
A001055 counts factorizations, with strict case A045778.
A001358 lists semiprimes, with squarefree case A006881.
A027187 counts partitions of even length, ranked by A028260.
A058696 counts partitions of even numbers, ranked by A300061.
A316439 counts factorizations by product and length.
A340102 counts odd-length factorizations into odd factors.
Cf. A002033, A007716, A027193, A050320, A058695, A074206, A236913, A320655, A320656, A320732.

g:= proc(n, k, t) option remember; `if`(n>k, 0, t)+
      `if`(isprime(n), 0, add(`if`(d>k, 0, g(n/d, d, 1-t)),
          d=numtheory[divisors](n) minus {1, n}))
    end:
a:= n-> `if`(n=1, 1, g(n$2, 0)):
seq(a(n), n=1..100);  # Alois P. Heinz, Dec 30 2020

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[Select[facs[n],EvenQ@Length[#]&]],{n,100}]

A339846(n, m=n, e=1) = if(1==n, e, sumdiv(n, d, if((d>1)&&(d<=m), A339846(n/d, d, 1-e)))); \\ Antti Karttunen, Oct 22 2023

a(n) + A339890(n) = A001055(n).

Data section extended up to a(105) by Antti Karttunen, Oct 22 2023

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

Original entry on oeis.org

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

A007774 Numbers that are divisible by exactly 2 different primes; numbers n with omega(n) = A001221(n) = 2.

Original entry on oeis.org

6, 10, 12, 14, 15, 18, 20, 21, 22, 24, 26, 28, 33, 34, 35, 36, 38, 39, 40, 44, 45, 46, 48, 50, 51, 52, 54, 55, 56, 57, 58, 62, 63, 65, 68, 69, 72, 74, 75, 76, 77, 80, 82, 85, 86, 87, 88, 91, 92, 93, 94, 95, 96, 98, 99, 100, 104, 106, 108, 111, 112, 115, 116, 117, 118

Offset: 1

Luke Pebody (ltp1000(AT)hermes.cam.ac.uk)

nonn

Every group of order p^a * q^b is solvable (Burnside, 1904). - Franz Vrabec, Sep 14 2008

Characteristic function for a(n): floor(omega(n)/2) * floor(2/omega(n)) where omega(n) is the number of distinct prime factors of n. - Wesley Ivan Hurt, Jan 10 2013

			20 is a term because 20 = 2^2*5 with two distinct prime divisors 2, 5.

T. D. Noe, Table of n, a(n) for n = 1..1000
W. Burnside, On groups of order p^alpha q^beta, Proc. London Math. Soc. (2) 1 (1904), 388-392.
Hans Montanus and Ron Westdijk, Cellular Automation and Binomials, Green Blue Mathematics (2022), p. 90.

Subsequence of A085736; A256617 is a subsequence.
Row 2 of A125666.
Cf. A001358 (products of two primes), A014612 (products of three primes), A014613 (products of four primes), A014614 (products of five primes), where the primes are not necessarily distinct.
Cf. A006881, A046386, A046387, A067885 (product of exactly 2, 4, 5, 6 distinct primes respectively).
Cf. A000040, A112801.

a007774 n = a007774_list !! (n-1)
a007774_list = filter ((== 2) . a001221) [1..]
-- Reinhard Zumkeller, Aug 02 2012

with(numtheory,factorset):f := proc(n) if nops(factorset(n))=2 then RETURN(n) fi; end;

Select[Range[0,6! ],Length[FactorInteger[ # ]]==2&] (* Vladimir Joseph Stephan Orlovsky, Apr 22 2010 *)
Select[Range[120],PrimeNu[#]==2&] (* Harvey P. Dale, Jun 03 2020 *)

is(n)=omega(n)==2 \\ Charles R Greathouse IV, Apr 01 2013

from sympy import primefactors
A007774_list = [n for n in range(1,10**5) if len(primefactors(n)) == 2] # Chai Wah Wu, Aug 23 2021

Expanded definition. - N. J. A. Sloane, Aug 22 2021

A320656 Number of factorizations of n into squarefree semiprimes. Number of multiset partitions of the multiset of prime factors of n, into strict pairs.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0

Offset: 1

Gus Wiseman, Oct 18 2018

nonn

			The a(4620) = 6 factorizations into squarefree semiprimes:
  4620 = (6*10*77)
  4620 = (6*14*55)
  4620 = (6*22*35)
  4620 = (10*14*33)
  4620 = (10*21*22)
  4620 = (14*15*22)
The a(4620) = 6 multiset partitions into strict pairs:
  {{1,2},{1,3},{4,5}}
  {{1,2},{1,4},{3,5}}
  {{1,2},{1,5},{3,4}}
  {{1,3},{1,4},{2,5}}
  {{1,3},{2,4},{1,5}}
  {{1,4},{2,3},{1,5}}
The a(69300) = 10 factorizations into squarefree semiprimes:
  69300 = (6*6*35*55)
  69300 = (6*10*15*77)
  69300 = (6*10*21*55)
  69300 = (6*10*33*35)
  69300 = (6*14*15*55)
  69300 = (6*15*22*35)
  69300 = (10*10*21*33)
  69300 = (10*14*15*33)
  69300 = (10*15*21*22)
  69300 = (14*15*15*22)
The a(69300) = 10 multiset partitions into strict pairs:
  {{1,2},{1,2},{3,4},{3,5}}
  {{1,2},{1,3},{2,3},{4,5}}
  {{1,2},{1,3},{2,4},{3,5}}
  {{1,2},{1,3},{2,5},{3,4}}
  {{1,2},{1,4},{2,3},{3,5}}
  {{1,2},{2,3},{1,5},{3,4}}
  {{1,3},{1,3},{2,4},{2,5}}
  {{1,3},{1,4},{2,3},{2,5}}
  {{1,3},{2,3},{2,4},{1,5}}
  {{1,4},{2,3},{2,3},{1,5}}.
The a(210) = 3 factorizations into squarefree semiprimes: 210 = (6*35) = (10*21) = (14*15). - _Antti Karttunen_, Nov 02 2022

Antti Karttunen, Table of n, a(n) for n = 1..100000
Index entries for sequences computed from exponents in factorization of n

Cf. A001055, A002110, A006881, A079275, A007716, A007717, A045778, A123023, A318871, A318953, A320462, A320655, A320658, A320659.

bepfacs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[bepfacs[n/d],Min@@#>=d&]],{d,Select[Rest[Divisors[n]],SquareFreeQ[#]&&PrimeOmega[#]==2&]}]];
Table[Length[bepfacs[n]],{n,100}]

A320656(n, m=n) = if(1==n, 1, my(s=0); fordiv(n, d, if((d>1)&&(d<=m)&&issquarefree(d)&&2==bigomega(d), s += A320656(n/d, d))); (s)); \\ Antti Karttunen, Nov 02 2022

a(A002110(n)) = A123023(n). - Antti Karttunen, Nov 02 2022

Data section extended up to a(120) and the secondary offset added by Antti Karttunen, Nov 02 2022

A048146 Sum of non-unitary divisors of n.

Original entry on oeis.org

0, 0, 0, 2, 0, 0, 0, 6, 3, 0, 0, 8, 0, 0, 0, 14, 0, 9, 0, 12, 0, 0, 0, 24, 5, 0, 12, 16, 0, 0, 0, 30, 0, 0, 0, 41, 0, 0, 0, 36, 0, 0, 0, 24, 18, 0, 0, 56, 7, 15, 0, 28, 0, 36, 0, 48, 0, 0, 0, 48, 0, 0, 24, 62, 0, 0, 0, 36, 0, 0, 0, 105, 0, 0, 20, 40, 0, 0, 0, 84, 39, 0, 0, 64, 0, 0, 0, 72, 0, 54, 0

Offset: 1

Labos Elemer

nonn

			If n = 1000, the 12 non-unitary divisors are {2, 4, 5, 10, 20, 25, 40, 50, 100, 200, 250, 500} and their sum is a(n) = a(1000) = 1206. a(16) = a(2^4) = (2^4 - 2) / (2 - 1)= 14.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A034444, A000203, A006881, A048105, A048106, A048107, A048109, A048111, A005117, A120944.

Cf. A002117, A072691.

us[n_Integer] := (d = Divisors[n]; l = Length[d]; k = 1; s = n; While[k < l, If[ GCD[ d[[k]], n/d[[k]] ] == 1, s = s + d[[k]]]; k++ ]; s); Table[ DivisorSigma[1, n] - us[n], {n, 1, 100} ]
(* Second program: *)
Table[DivisorSum[n, # &, ! CoprimeQ[#, n/#] &], {n, 91}] (* Michael De Vlieger, Nov 20 2017 *)

a(n)=my(f=factor(n)); sigma(f)-prod(i=1, #f~, f[i, 1]^f[i, 2]+1) \\ Charles R Greathouse IV, Jun 17 2015

from sympy.ntheory.factor_ import divisor_sigma, udivisor_sigma
def A048146(n): return divisor_sigma(n)-udivisor_sigma(n) # Chai Wah Wu, Aug 22 2024

a(n) = A000203(n) - A034448(n) = sigma(n) - usigma(n). a(1) = 0, a(p) = 0, a(pq) = 0, a(pq...z) = 0, a(p^k) = (p^k - p) / (p - 1), for p = primes (A000040), pq = product of two distinct primes (A006881), pq...z = product of k (k >=2) distinct primes p, q, ..., z (A120944), p^k = prime powers (A000961(n) for n > 1) k = natural numbers (A000027).

Sum_{k=1..n} a(k) ~ c * n^2, where c = (Pi^2/12) * (1 - 1/zeta(3)) = 0.1382506... . - Amiram Eldar, Dec 09 2022

Edited by Jaroslav Krizek, Mar 01 2009

A085986 Squares of the squarefree semiprimes (p^2*q^2).

Original entry on oeis.org

36, 100, 196, 225, 441, 484, 676, 1089, 1156, 1225, 1444, 1521, 2116, 2601, 3025, 3249, 3364, 3844, 4225, 4761, 5476, 5929, 6724, 7225, 7396, 7569, 8281, 8649, 8836, 9025, 11236, 12321, 13225, 13924, 14161, 14884, 15129, 16641, 17689, 17956, 19881

Offset: 1

Alford Arnold, Jul 06 2003

This sequence is a member of a family of sequences directly related to A025487. First terms and known sequences are listed below: 1, A000007; 2, A000040; 4, A001248; 6, A006881; 8, A030078; 12, A054753; 16, A030514; 24, A065036; 30, A007304; 32, A050997; 36, this sequence; 48, ?; 60, ?; 64, ?; ....
Subsequence of A077448. The numbers in A077448 but not in here are 1, the squares of A046386, the squares of A067885, etc. - R. J. Mathar, Sep 12 2008
a(4)-a(3)=29 and a(3)+a(4)=421 are both prime.  There are no other cases where the sum and difference of two members of this sequence are both prime. - Robert Israel and J. M. Bergot, Oct 25 2019

			A006881 begins 6 10 14 15 ... so this sequence begins 36 100 196 225 ...

Subsequence of A036785 and of A077448.
Subsequence of A062503.
Cf. A025487.
Cf. A085548, A085964.

[k^2:k in [1..150]| IsSquarefree(k) and #PrimeDivisors(k) eq 2]; // Marius A. Burtea, Oct 24 2019

f[n_]:=Sort[Last/@FactorInteger[n]]=={2,2}; Select[Range[20000], f] (* Vladimir Joseph Stephan Orlovsky, Aug 14 2009 *)
Select[Range[200],PrimeOmega[#]==2&&SquareFreeQ[#]&]^2 (* Harvey P. Dale, Mar 07 2013 *)

list(lim)=my(v=List(), x=sqrtint(lim\=1), t); forprime(p=2, x\2, t=p; forprime(q=2, min(x\t,p-1), listput(v, (t*q)^2))); Set(v) \\ Charles R Greathouse IV, Sep 22 2015

is(n)=factor(n)[,2]==[2,2]~ \\ Charles R Greathouse IV, Oct 19 2015

from math import isqrt
from sympy import primepi, primerange
def A085986(n):
    def f(x): return int(n+x+(t:=primepi(s:=isqrt(x)))+(t*(t-1)>>1)-sum(primepi(x//k) for k in primerange(1, s+1)))
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    return m**2 # Chai Wah Wu, Aug 18 2024

a(n) = A006881(n)^2.

Sum_{n>=1} 1/a(n) = (P(2)^2 - P(4))/2 = (A085548^2 - A085964)/2 = 0.063767..., where P is the prime zeta function. - Amiram Eldar, Jul 06 2020

A320911 Numbers with an even number of prime factors (counted with multiplicity) that can be factored into squarefree semiprimes.

Original entry on oeis.org

1, 6, 10, 14, 15, 21, 22, 26, 33, 34, 35, 36, 38, 39, 46, 51, 55, 57, 58, 60, 62, 65, 69, 74, 77, 82, 84, 85, 86, 87, 90, 91, 93, 94, 95, 100, 106, 111, 115, 118, 119, 122, 123, 126, 129, 132, 133, 134, 140, 141, 142, 143, 145, 146, 150, 155, 156, 158, 159

Offset: 1

Gus Wiseman, Oct 23 2018

nonn

A squarefree semiprime (A006881) is a product of any two distinct primes.

Also numbers with an even number x of prime factors, whose prime multiplicities do not exceed x/2.

			360 is in the sequence because it can be factored into squarefree semiprimes as (6*6*10).
4620 is in the sequence, and can be factored into squarefree semiprimes in 6 ways: (6*10*77), (6*14*55), (6*22*35), (10*14*33), (10*21*22), (14*15*22).

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A001055, A001222, A001358, A005117, A006881, A007717, A028260, A320655, A320656, A320891, A320892, A320893, A320894, A320912.

sqfsemfacs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[sqfsemfacs[n/d],Min@@#>=d&]],{d,Select[Rest[Divisors[n]],And[SquareFreeQ[#],PrimeOmega[#]==2]&]}]];
Select[Range[100],And[EvenQ[PrimeOmega[#]],sqfsemfacs[#]!={}]&]

A065036 Product of the cube of a prime (A030078) and a different prime.

Original entry on oeis.org

24, 40, 54, 56, 88, 104, 135, 136, 152, 184, 189, 232, 248, 250, 296, 297, 328, 344, 351, 375, 376, 424, 459, 472, 488, 513, 536, 568, 584, 621, 632, 664, 686, 712, 776, 783, 808, 824, 837, 856, 872, 875, 904, 999, 1016, 1029, 1048, 1096, 1107, 1112

Offset: 1

Alford Arnold, Nov 04 2001

This sequence appears on row 8 of the list illustrated in A064839 and is similar to A054753 which appears on row 6. Previous rows are generated by A000007, A000040, A001248, A006881, A030078 respectively.

Or, the numbers n such that 20=number of perfect partitions of n. - Juri-Stepan Gerasimov, Sep 26 2009

			a(4)= 56 since 56 = 2*2*2*7.

Cf. A000007, A000040, A001248, A002033, A006881, A030078, A054753, A064839, A089233.

Select[ Range[1500], Sort[ Transpose[ FactorInteger[ # ]] [[2]]] == {1, 3} & ]
Module[{upto=1200},Select[(Union[Flatten[{#[[1]]^3 #[[2]],#[[1]]#[[2]]^3}&/@Subsets[Prime[Range[upto/8]],{2}]]]),#<=upto&]] (* Harvey P. Dale, May 23 2015 *)

list(lim)=my(v=List(),t);forprime(p=2,(lim\2)^(1/3),t=p^3; forprime(q=2,lim\t,if(p==q,next);listput(v,t*q)));vecsort(Vec(v)) \\ Charles R Greathouse IV, Jul 20 2011

is(n)=my(f=factor(n)[,2]); f==[3,1]~||f==[1,3]~ \\ Charles R Greathouse IV, Oct 15 2015

from sympy import primepi, primerange, integer_nthroot
def A065036(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return n+x-sum(primepi(x//p**3) for p in primerange(integer_nthroot(x,3)[0]+1))+primepi(integer_nthroot(x,4)[0])
    return bisection(f,n,n) # Chai Wah Wu, Feb 21 2025

A002033(a(n)) = 20. - Juri-Stepan Gerasimov, Sep 26 2009
A089233(a(n)) = 3. - Reinhard Zumkeller, Sep 04 2013
A000005(a(n)) = 8. - Altug Alkan, Nov 11 2015

A338913 Greater prime index of the n-th semiprime.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Nov 20 2020

nonn

A semiprime is a product of any two prime numbers. A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

After the first three terms, there appear to be no adjacent equal terms.

			The semiprimes are:
  2*2, 2*3, 3*3, 2*5, 2*7, 3*5, 3*7, 2*11, 5*5, 2*13, ...
so the greater prime factors are:
  2, 3, 3, 5, 7, 5, 7, 11, 5, 13, ...
with indices:
  1, 2, 2, 3, 4, 3, 4, 5, 3, 6, ...

Amiram Eldar, Table of n, a(n) for n = 1..10000

A115392 lists positions of first appearances of each positive integer.
A270652 is the squarefree case, with lesser part A270650.
A338898 has this as second column.
A338912 is the corresponding lesser prime index.
A001221 counts distinct prime indices.
A001222 counts prime indices.
A001358 lists semiprimes, with odd/even terms A046315/A100484.
A006881 lists squarefree semiprimes, with odd/even terms A046388/A100484.
A087794/A176504/A176506 are product/sum/difference of semiprime indices.
A338910/A338911 list products of pairs of odd/even-indexed primes.
Cf. A037143, A056239, A065516, A084126, A084127, A112798, A128301, A289182, A320732, A320892, A338900, A338904, A338909.

Table[Max[PrimePi/@First/@FactorInteger[n]],{n,Select[Range[100],PrimeOmega[#]==2&]}]

a(n) = A000720(A084127(n)).

cp's OEIS Frontend

A046386 Products of exactly four distinct primes.

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A339846 Number of even-length factorizations of n into factors > 1.

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

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A007774 Numbers that are divisible by exactly 2 different primes; numbers n with omega(n) = A001221(n) = 2.

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A320656 Number of factorizations of n into squarefree semiprimes. Number of multiset partitions of the multiset of prime factors of n, into strict pairs.

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A048146 Sum of non-unitary divisors of n.

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