A087436 -id:A087436 - cp's OEIS frontend

A336466 Fully multiplicative with a(p) = A000265(p-1) for any prime p, where A000265(k) gives the odd part of k.

1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 5, 1, 3, 3, 1, 1, 1, 1, 9, 1, 3, 5, 11, 1, 1, 3, 1, 3, 7, 1, 15, 1, 5, 1, 3, 1, 9, 9, 3, 1, 5, 3, 21, 5, 1, 11, 23, 1, 9, 1, 1, 3, 13, 1, 5, 3, 9, 7, 29, 1, 15, 15, 3, 1, 3, 5, 33, 1, 11, 3, 35, 1, 9, 9, 1, 9, 15, 3, 39, 1, 1, 5, 41, 3, 1, 21, 7, 5, 11, 1, 9, 11, 15, 23, 9, 1, 3, 9, 5, 1, 25, 1, 51, 3, 3

Offset: 1

Antti Karttunen, Jul 22 2020

For the comment here, we extend the definition of the second kind of Cunningham chain (see Wikipedia-article) so that also isolated primes for which neither (p+1)/2 nor 2p-1 is a prime are considered to be in singular chains, that is, in chains of the length one. If we replace one or more instances of any particular odd prime factor p in n with any odd prime q in such a chain, so that m = (q^k)*n / p^(e-k), where e is the exponent of p of n, and k <= e is the number of instances of p replaced with q, then it holds that a(m) = a(n), and by induction, the value stays invariant for any number of such replacements. Note also that A001222, but not necessarily A001221 will stay invariant in such changes.
For example, if some of the odd prime divisors p of n are in A005382, then replacing it with 2p-1 (i.e., the corresponding terms of A005383), gives a new number m, for which a(m) = a(n). And vice versa, the same is true for any of the prime divisors > 3 of n that are in A005383, then replacing any one of them with (p+1)/2 will not affect the result. For example, a(37*37*37) = a(19*37*73) = 729 as 37 is both in A005382 and in A005383.
a(n) = A053575(n) for squarefree n (A005117). - Antti Karttunen, Mar 16 2021

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Wikipedia, Cunningham chain

Cf. A000265, A003958, A005117, A005382, A005383, A053575, A329697, A333787, A335915, A336396, A336467, A336468, A339870, A339876 [= a(A122111(n))], A339903 [= a(A003961(n))], A340084 [= gcd(n-1, a(n))], A340085, A340086.

Cf. also A336460, A336470, A339974.

Array[Times @@ Map[If[# <= 2, 1, (# - 1)/2^IntegerExponent[# - 1, 2]] &, Flatten[ConstantArray[#1, #2] & @@@ FactorInteger[#]]] &, 105] (* Michael De Vlieger, Jul 24 2020 *)

A000265(n) = (n>>valuation(n,2));
A336466(n) = { my(f=factor(n)); prod(k=1,#f~,A000265(f[k,1]-1)^f[k,2]); };

a(n) = A000265(A003958(n)) = A000265(A333787(n)).
a(A000010(n)) = A336468(n) = a(A053575(n)).
A329697(a(n)) = A336396(n) = A329697(n) - A087436(n).
a(n) = A335915(n) / A336467(n). - Antti Karttunen, Mar 16 2021

A369002 Numbers k for which k' / gcd(k,k') is even, where k' stands for the arithmetic derivative of k, A003415.

Original entry on oeis.org

1, 9, 12, 15, 16, 20, 21, 25, 28, 33, 35, 39, 44, 49, 51, 52, 55, 57, 65, 68, 69, 76, 77, 81, 85, 87, 91, 92, 93, 95, 108, 111, 115, 116, 119, 121, 123, 124, 129, 133, 135, 141, 143, 144, 145, 148, 155, 159, 161, 164, 169, 172, 177, 180, 183, 185, 187, 188, 189, 192, 201, 203, 205, 209, 212, 213, 215, 217, 219, 221

Offset: 1

Antti Karttunen, Jan 14 2024

nonn

From Antti Karttunen, Feb 09 2024: (Start)
Numbers k for which A276085(k) is a multiple of four.
Even terms in this sequence are all multiples of four.
A multiplicative semigroup; if m and n are in the sequence then so is m*n.
(End)
Appears to be products of an even number of terms from {4} U A065091 (counting repetitions). - Peter Munn, Jul 15 2024

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A003415, A083345, A087436, A235127, A276085, A369001 (characteristic function), A369003 (complement).
Positions of even terms in A083345.
Subsequence of A368998, which is a subsequence of A235992.
Setwise difference A003159 \ A373142.
Disjoint union of A369005 and A373265.
Disjoint union of A373138 and A373267.
Disjoint union of A369976 and A369977.
Other subsequences: A046337 (odd terms in this sequence), A373259.

PARI
```
\\ See A369001
```

For all n >= 1, A235127(a(n)) == A087436(a(n)) (mod 2). - Antti Karttunen, Feb 09 2024

A072978 Numbers of the form m*2^Omega(m), where m>1 is odd and Omega(m)=A001222(m), the number of prime factors of m.

Original entry on oeis.org

1, 6, 10, 14, 22, 26, 34, 36, 38, 46, 58, 60, 62, 74, 82, 84, 86, 94, 100, 106, 118, 122, 132, 134, 140, 142, 146, 156, 158, 166, 178, 194, 196, 202, 204, 206, 214, 216, 218, 220, 226, 228, 254, 260, 262, 274, 276, 278, 298, 302, 308, 314, 326, 334, 340, 346

Offset: 1

Reinhard Zumkeller, Aug 20 2002

(number of odd prime factors) = (number of even prime factors).
A000400, A011557, A001023, A001024, A009965, A009966 and A009975 are subsequences. - Reinhard Zumkeller, Jan 06 2008
Subsequence of A028260. - Reinhard Zumkeller, Sep 20 2008

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

Cf. A001222, A007814, A087436.

Cf. A000400, A011557, A001023, A001024, A009965, A009966, A009975, A028260.

Join[{1}, Select[Range[2, 500, 2], First[#] == Total[Rest[#]] & [FactorInteger[#][[All, 2]]] &]] (* Paolo Xausa, Feb 19 2025 *)

isok(k) = {my(v = valuation(k, 2)); bigomega(k >> v) == v;} \\ Amiram Eldar, May 15 2025

from math import prod, isqrt
from sympy import primerange, integer_nthroot, primepi
def A072978(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 g(x,a,b,c,m): yield from (((d,) for d in enumerate(primerange(b,isqrt(x//c)+1),a)) if m==2 else (((a2,b2),)+d for a2,b2 in enumerate(primerange(b,integer_nthroot(x//c,m)[0]+1),a) for d in g(x,a2,b2,c*b2,m-1)))
    def h(x,n): return sum(primepi(x//prod(c[1] for c in a))-a[-1][0] for a in g(x,1,3,1,n))
    def f(x): return int(n+x-primepi(x>>1)-sum(h(x>>m,m) for m in range(2,x.bit_length()+1))) if x>1 else 1
    return bisection(f,n,n) # Chai Wah Wu, Apr 10 2025

A007814(a(n)) = A087436(a(n)). - Reinhard Zumkeller, Jan 06 2008

A336158 The least number with the prime signature of the odd part of n: a(n) = A046523(A000265(n)).

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 2, 1, 4, 2, 2, 2, 2, 2, 6, 1, 2, 4, 2, 2, 6, 2, 2, 2, 4, 2, 8, 2, 2, 6, 2, 1, 6, 2, 6, 4, 2, 2, 6, 2, 2, 6, 2, 2, 12, 2, 2, 2, 4, 4, 6, 2, 2, 8, 6, 2, 6, 2, 2, 6, 2, 2, 12, 1, 6, 6, 2, 2, 6, 6, 2, 4, 2, 2, 12, 2, 6, 6, 2, 2, 16, 2, 2, 6, 6, 2, 6, 2, 2, 12, 6, 2, 6, 2, 6, 2, 2, 4, 12, 4, 2, 6, 2, 2, 30

Offset: 1

Antti Karttunen, Jul 11 2020

nonn

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

Cf. A000265, A001227, A005087, A046523, A064989, A087436, A336156, A336157, A336159, A336160.

A000265(n) = (n>>valuation(n,2));
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
A336158(n) = A046523(A000265(n));

from math import prod
from sympy import factorint, prime
def A336158(n): return prod(prime(i+1)**e for i,e in enumerate(sorted(factorint(n>>(~n&n-1).bit_length()).values(),reverse=True))) # Chai Wah Wu, Sep 16 2022

a(n) = A046523(A000265(n)) = A046523(A064989(n)).
A000005(a(n)) = A001227(n).
A001221(a(n)) = A005087(n).
A001222(a(n)) = A087436(n).

A379301 Positive integers whose prime indices include a unique composite number.

Original entry on oeis.org

7, 13, 14, 19, 21, 23, 26, 28, 29, 35, 37, 38, 39, 42, 43, 46, 47, 52, 53, 56, 57, 58, 61, 63, 65, 69, 70, 71, 73, 74, 76, 77, 78, 79, 84, 86, 87, 89, 92, 94, 95, 97, 101, 103, 104, 105, 106, 107, 111, 112, 113, 114, 115, 116, 117, 119, 122, 126, 129, 130, 131

Offset: 1

Gus Wiseman, Dec 25 2024

nonn

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.

			The prime indices of 70 are {1,3,4}, so 70 is in the sequence.
The prime indices of 98 are {1,4,4}, so 98 is not in the sequence.

For no composite parts we have A302540, counted by A034891 (strict A036497).
For all composite parts we have A320629, counted by A023895 (strict A204389).
For a unique prime part we have A331915, counted by A379304 (strict A379305).
Positions of one in A379300.
Partitions of this type are counted by A379302 (strict A379303).
A000040 lists the prime numbers, differences A001223.
A002808 lists the composite numbers, nonprimes A018252, differences A073783 or A065310.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A066247 is the characteristic function for the composite numbers.
A377033 gives k-th differences of composite numbers, see A073445, A377034-A377037.
Other counts of prime indices:
- A087436 postpositive, see A038550.
- A257991 odd, see A000041, A000070, A066207, A349158.
- A257992 even, see A000009, A038348, A066208, A379317.
- A257994 prime, see A000586, A000607, A076610, A331386.
- A330944 nonprime, see A002095, A096258, A320628, A330945.
- A379306 squarefree, see A302478, A379308, A379309, A379316.
- A379310 nonsquarefree, see A114374, A256012, A379307.
- A379311 old prime, see A379312-A379315.
Cf. A113646, A376680.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Length[Select[prix[#],CompositeQ]]==1&]

A105441 Numbers with at least two odd prime factors (not necessarily distinct).

Original entry on oeis.org

9, 15, 18, 21, 25, 27, 30, 33, 35, 36, 39, 42, 45, 49, 50, 51, 54, 55, 57, 60, 63, 65, 66, 69, 70, 72, 75, 77, 78, 81, 84, 85, 87, 90, 91, 93, 95, 98, 99, 100, 102, 105, 108, 110, 111, 114, 115, 117, 119, 120, 121, 123, 125, 126, 129, 130, 132, 133, 135, 138, 140, 141

Offset: 1

Reinhard Zumkeller, Apr 09 2005

Also polite numbers (A138591) that can be expressed as the sum of two or more consecutive integers in more than one ways. For example 9=4+5 and 9=2+3+4. Also 15=7+8, 15=4+5+6 and 15=1+2+3+4+5. - Jayanta Basu, Apr 30 2013

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Complement of A093641; A093642 is a subsequence.

Cf. A001227, A087436, A138591.

a105441 n = a105441_list !! (n-1)
a105441_list = filter ((> 2) . a001227) [1..]
-- Reinhard Zumkeller, May 01 2012

opf3Q[n_]:=Count[Flatten[Table[First[#],{Last[#]}]&/@FactorInteger[n]], ?OddQ]>1 (* _Harvey P. Dale, Jun 13 2011 *)

upTo(lim)=my(v=List(),p=7,m);forprime(q=8,lim,forstep(n=p+2,q-2,2,m=n;while(m<=lim,listput(v,m);m<<=1));p=q);forstep(n=p+2,lim,2,listput(v,n));vecsort(Vec(v)) \\ Charles R Greathouse IV, Aug 08 2011

is(n)=n>>=valuation(n,2); !isprime(n) && n>1 \\ Charles R Greathouse IV, Apr 30 2013

from sympy import primepi
def A105441(n):
    def f(x): return int(n+1+sum(primepi(x>>i) for i in range(x.bit_length())))
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m # Chai Wah Wu, Feb 02 2025

A087436(a(n)) > 1.

A001227(a(n)) > 2. [Reinhard Zumkeller, May 01 2012]

A379312 Positive integers whose prime indices include a unique 1 or prime number.

Original entry on oeis.org

2, 3, 5, 11, 14, 17, 21, 26, 31, 35, 38, 39, 41, 46, 57, 58, 59, 65, 67, 69, 74, 77, 83, 86, 87, 94, 95, 98, 106, 109, 111, 115, 119, 122, 127, 129, 141, 142, 143, 145, 146, 147, 157, 158, 159, 178, 179, 182, 183, 185, 191, 194, 202, 206, 209, 211, 213, 214

Offset: 1

Gus Wiseman, Dec 28 2024

nonn

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.

			The terms together with their prime indices begin:
    2: {1}
    3: {2}
    5: {3}
   11: {5}
   14: {1,4}
   17: {7}
   21: {2,4}
   26: {1,6}
   31: {11}
   35: {3,4}
   38: {1,8}
   39: {2,6}
   41: {13}
   46: {1,9}
   57: {2,8}
   58: {1,10}
   59: {17}
   65: {3,6}
   67: {19}
   69: {2,9}
   74: {1,12}
   77: {4,5}

These "old" primes are listed by A008578.
For no composite parts we have A302540, counted by A034891 (strict A036497).
For all composite parts we have A320629, counted by A023895 (strict A204389).
For a unique prime part we have A331915, counted by A379304 (strict A379305).
Positions of ones in A379311, see A379313.
Partitions of this type are counted by A379314, strict A379315.
A000040 lists the prime numbers, differences A001223.
A002808 lists the composite numbers, nonprimes A018252, differences A073783 or A065310.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A080339 is the characteristic function for the old prime numbers.
A376682 gives k-th differences of old prime numbers, see A030016, A075526.
Other counts of prime indices:
- A257991 odd, see A000041, A000070, A066207, A349158.
- A257992 even, see A000009, A038348, A066208, A379317.
- A257994 prime, see A000586, A000607, A076610, A331386.
- A330944 nonprime, see A002095, A096258, A320628, A330945.
- A379306 squarefree, see A302478, A379308, A379309, A379316.
- A379310 nonsquarefree, see A114374, A256012, A379307.
Cf. A038550, A073445, A087436, A173390, A379300, A379301, A379302.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[1000],Length[Select[prix[#],#==1||PrimeQ[#]&]]==1&]

A379316 Positive integers whose prime indices include a unique squarefree number.

Original entry on oeis.org

2, 3, 5, 11, 13, 14, 17, 21, 29, 31, 35, 38, 41, 43, 46, 47, 57, 59, 67, 69, 73, 74, 77, 79, 83, 91, 95, 98, 101, 106, 109, 111, 113, 115, 119, 122, 127, 137, 139, 142, 147, 149, 157, 159, 163, 167, 178, 179, 181, 183, 185, 191, 194, 199, 203, 206, 209, 211

Offset: 1

Gus Wiseman, Dec 29 2024

nonn

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.

			The terms together with their prime indices begin:
    2: {1}
    3: {2}
    5: {3}
   11: {5}
   13: {6}
   14: {1,4}
   17: {7}
   21: {2,4}
   29: {10}
   31: {11}
   35: {3,4}
   38: {1,8}
   41: {13}
   43: {14}
   46: {1,9}

For all squarefree parts we have A302478, zeros of A379310.
Positions of 1 in A379306.
For no squarefree parts we have A379307, counted by A114374, strict A256012.
Partitions of this type are counted by A379308, strict A379309.
A000040 lists the primes, differences A001223.
A005117 lists the squarefree numbers, differences A076259.
A008966 is the characteristic function for the squarefree numbers.
A013929 lists the nonsquarefree numbers, differences A078147.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
Other counts of prime indices:
- A257991 odd, see A000041, A000070, A066207, A349158.
- A257992 even, see A000009, A038348, A066208, A379317.
- A257994 prime, see A002095, A096258, A320628, A331386, A331915, A379304, A379305.
- A330944 nonprime, see A000586, A000607, A076610, A330945.
- A379300 composite, see A023895, A034891, A036497, A302540, A379301.
- A379311 prime or 1, see A204389, A320629, A379312-A379315.
Cf. A000720, A013928, A038550, A057627, A068361, A070321, A072284, A087436, A112929, A377430.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[1000],Length[Select[prix[#],SquareFreeQ]]==1&]

A366842 Number of integer partitions of n whose odd parts have a common divisor > 1.

Original entry on oeis.org

0, 0, 0, 1, 0, 2, 1, 4, 1, 8, 3, 13, 6, 21, 10, 36, 15, 53, 28, 80, 41, 122, 63, 174, 97, 250, 140, 359, 201, 496, 299, 685, 410, 949, 575, 1284, 804, 1726, 1093, 2327, 1482, 3076, 2023, 4060, 2684, 5358, 3572, 6970, 4745, 9050, 6221, 11734, 8115, 15060, 10609

Offset: 0

Gus Wiseman, Oct 28 2023

nonn

			The a(3) = 1 through a(11) = 13 partitions:
  (3)  .  (5)    (3,3)  (7)      (3,3,2)  (9)        (5,5)      (11)
          (3,2)         (4,3)             (5,4)      (4,3,3)    (6,5)
                        (5,2)             (6,3)      (3,3,2,2)  (7,4)
                        (3,2,2)           (7,2)                 (8,3)
                                          (3,3,3)               (9,2)
                                          (4,3,2)               (4,4,3)
                                          (5,2,2)               (5,4,2)
                                          (3,2,2,2)             (6,3,2)
                                                                (7,2,2)
                                                                (3,3,3,2)
                                                                (4,3,2,2)
                                                                (5,2,2,2)
                                                                (3,2,2,2,2)

This is the odd case of A018783, complement A000837.
The even version is A047967.
The complement is counted by A366850, ranks A366846.
A000041 counts integer partitions, strict A000009.
A000740 counts relatively prime compositions.
A113685 counts partitions by sum of odds, stat A366528, w/o zeros A365067.
A168532 counts partitions by gcd.
A239261 counts partitions with (sum of odd parts) = (sum of even parts).
A289508 gives gcd of prime indices, positions of ones A289509.
Cf. A007359, A051424, A055922, A066208, A078374, A087436, A116598, A337485, A366843, A366844, A366845.

Table[Length[Select[IntegerPartitions[n], GCD@@Select[#,OddQ]>1&]], {n,0,30}]

from math import gcd
from sympy.utilities.iterables import partitions
def A366842(n): return sum(1 for p in partitions(n) if gcd(*(q for q in p if q&1))>1) # Chai Wah Wu, Oct 28 2023

A379300 Number of prime indices of n that are composite.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Dec 25 2024

nonn

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.

			The prime indices of 39 are {2,6}, so a(39) = 1.
The prime indices of 70 are {1,3,4}, so a(70) = 1.
The prime indices of 98 are {1,4,4}, so a(98) = 2.
The prime indices of 294 are {1,2,4,4}, a(294) = 2.
The prime indices of 1911 are {2,4,4,6}, so a(1911) = 3.
The prime indices of 2548 are {1,1,4,4,6}, so a(2548) = 3.

Positions of first appearances are A000420.
Positions of zero are A302540, counted by A034891 (strict A036497).
Positions of one are A379301, counted by A379302 (strict A379303).
A000040 lists the prime numbers, differences A001223.
A002808 lists the composite numbers, nonprimes A018252, differences A073783 or A065310.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A066247 is the characteristic function for the composite numbers.
A377033 gives k-th differences of composite numbers, see A073445, A377034-A377037.
Other counts of prime indices:
- A087436 postpositive, see A038550.
- A257991 odd, see A000041, A000070, A066207, A349158.
- A257992 even, see A000009, A038348, A066208, A379317.
- A257994 prime, see A000586, A000607, A076610, A331386, A331915, A379304, A379305.
- A330944 nonprime, see A002095, A096258, A320628, A330945.
- A379306 squarefree, see A302478, A379308, A379309, A379316.
- A379310 nonsquarefree, see A114374, A256012, A379307.
- A379311 old prime, see A379312-A379315.
Cf. A023895, A113646, A175804, A204389, A320629, A376680, A378373, A378456.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[Select[prix[n],CompositeQ]],{n,100}]

Totally additive with a(prime(k)) = A066247(k).

cp's OEIS Frontend

A336466 Fully multiplicative with a(p) = A000265(p-1) for any prime p, where A000265(k) gives the odd part of k.

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A369002 Numbers k for which k' / gcd(k,k') is even, where k' stands for the arithmetic derivative of k, A003415.

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A072978 Numbers of the form m*2^Omega(m), where m>1 is odd and Omega(m)=A001222(m), the number of prime factors of m.

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A336158 The least number with the prime signature of the odd part of n: a(n) = A046523(A000265(n)).

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A379301 Positive integers whose prime indices include a unique composite number.

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A105441 Numbers with at least two odd prime factors (not necessarily distinct).

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A379312 Positive integers whose prime indices include a unique 1 or prime number.

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A379316 Positive integers whose prime indices include a unique squarefree number.

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