A030230 -id:A030230 - cp's OEIS frontend

A308063 Number of ordered factorizations of n into numbers with an odd number of distinct prime divisors.

1, 1, 1, 2, 1, 2, 1, 4, 2, 2, 1, 5, 1, 2, 2, 8, 1, 5, 1, 5, 2, 2, 1, 12, 2, 2, 4, 5, 1, 7, 1, 16, 2, 2, 2, 14, 1, 2, 2, 12, 1, 7, 1, 5, 5, 2, 1, 28, 2, 5, 2, 5, 1, 12, 2, 12, 2, 2, 1, 21, 1, 2, 5, 32, 2, 7, 1, 5, 2, 7, 1, 37, 1, 2, 5, 5, 2, 7, 1, 28, 8, 2, 1, 21, 2, 2, 2, 12, 1, 21

Offset: 1

Ilya Gutkovskiy, May 10 2019

nonn

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

Cf. A008578, A030230, A050334, A074206, A092248.

terms = 90; A[] = 0; Do[A[x] = x + Sum[Boole[OddQ[PrimeNu[k]]] A[x^k], {k, 1, terms}] + O[x]^(terms + 1) // Normal, terms + 1]; Rest[CoefficientList[A[x], x]]
f[n_] := f[n] = Boole[OddQ[PrimeNu[n]]]; a[n_] := If[n == 1, n, Sum[If[d < n, f[n/d] a[d], 0], {d, Divisors[n]}]]; Table[a[n], {n, 1, 90}]

a(n) = if(n == 1, 1, sumdiv(n, d, if(dAmiram Eldar, Jul 03 2025

G.f. A(x) satisfies: A(x) = x + Sum_{k>=1} A(x^A030230(k)).
a(1) = 1; a(n) = Sum_{d|n, dA092248(n/d)*a(d).
a(n) = 1 if n is in A008578.

A214195 Numbers with the number of distinct prime factors a multiple of 3.

Original entry on oeis.org

1, 30, 42, 60, 66, 70, 78, 84, 90, 102, 105, 110, 114, 120, 126, 130, 132, 138, 140, 150, 154, 156, 165, 168, 170, 174, 180, 182, 186, 190, 195, 198, 204, 220, 222, 228, 230, 231, 234, 238, 240, 246, 252, 255, 258, 260, 264, 266, 270, 273, 276, 280, 282, 285

Offset: 1

Enrique Pérez Herrero, Jul 07 2012

nonn

If GCD(a(n),a(m))=1, then a(n)*a(m) is also in this sequence. - Enrique Pérez Herrero, Nov 23 2013

Enrique Pérez Herrero, Table of n, a(n) for n = 1..5000

Subsequences include A033992, A067885, A007304 and A147573.

Cf. A145784, A030230, A030231.

Select[Range[1000],Mod[PrimeNu[#],3]==0&]

is(n)=omega(n)%3==0 \\ Charles R Greathouse IV, Sep 14 2015

A010872(A001221(a(n))) = 0.

A327669 Sum of divisors of n that have an odd number of distinct prime factors.

Original entry on oeis.org

0, 2, 3, 6, 5, 5, 7, 14, 12, 7, 11, 9, 13, 9, 8, 30, 17, 14, 19, 11, 10, 13, 23, 17, 30, 15, 39, 13, 29, 40, 31, 62, 14, 19, 12, 18, 37, 21, 16, 19, 41, 54, 43, 17, 17, 25, 47, 33, 56, 32, 20, 19, 53, 41, 16, 21, 22, 31, 59, 104, 61, 33, 19, 126, 18, 82, 67, 23, 26, 84

Offset: 1

Ilya Gutkovskiy, Sep 21 2019

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..20000

Cf. A000203, A030230, A049060, A092248, A285799, A318677, A327670.

with(numtheory):
a:= n-> add(`if`(nops(factorset(d))::odd, d, 0), d=divisors(n)):
seq(a(n), n=1..80);  # Alois P. Heinz, Sep 21 2019

a[n_] := DivisorSum[n, # &, OddQ[PrimeNu[#]] &]; Table[a[n], {n, 1, 70}]

G.f.: Sum_{k>=1} A030230(k) * x^A030230(k) / (1 - x^A030230(k)).
L.g.f.: log(B(x)) = Sum_{n>=1} a(n) * x^n / n, where B(x) = g.f. of A285799.
a(n) = Sum_{d|n} d * A092248(d).
a(n) = A000203(n) - A327670(n).
a(p) = p, where p is prime.

A328856 Number of factorizations of n into distinct numbers with an odd number of distinct prime factors.

Original entry on oeis.org

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

Offset: 1

Ilya Gutkovskiy, Oct 28 2019

nonn

			a(32) = 3 because 32 = 4 * 8 = 2 * 16.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences computed from exponents in factorization of n

Cf. A030230, A045778, A050332, A328855.

seq(n)={my(v=vector(n, k, k==1)); for(k=2, n, if(omega(k)%2, my(m=logint(n, k), p=(1 + x + O(x*x^m)), w=vector(n)); for(i=0, m, w[k^i]=polcoef(p, i)); v=dirmul(v, w))); v} \\ Andrew Howroyd, Oct 29 2019, In older versions of PARI, use polcoeff instead of polcoef. - Antti Karttunen, Oct 29 2019

A328856(n, k=n) = (((n<=k)&&((1==n)||(omega(n)%2))) + sumdiv(n, d, if(d > 1 && d <= k && d < n && (omega(d)%2), A328856(n/d, d-1)))); \\ Antti Karttunen, Oct 29 2019

Dirichlet g.f.: Product_{k>=1} (1 + A030230(k)^(-s)).

a(n) <= A045778(n). - Antti Karttunen, Oct 29 2019

More terms from Antti Karttunen, Oct 29 2019

A081400 a(n) = d(n) - bigomega(n) - A005361(n).

Original entry on oeis.org

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

Offset: 1

Labos Elemer, Mar 28 2003

sign

			Negative for true prime powers; zero for 1 and primes; see also A030231, A007304, A034683, A075819 etc. to judge about positivity or magnitude.

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

Cf. A000005, A001222, A005361, A030231, A030230, A007304, A034683, A075819.

a(n) = my(f=factor(n)); numdiv(n) - bigomega(n) - prod(k=1, #f~, f[k,2]); \\ Michel Marcus, May 25 2017

from sympy import primefactors, factorint, divisor_count
from operator import mul
def bigomega(n): return 0 if n==1 else bigomega(n/primefactors(n)[0]) + 1
def a005361(n):
    f=factorint(n)
    return 1 if n==1 else reduce(mul, [f[i] for i in f])
def a(n): return divisor_count(n) - bigomega(n) - a005361(n) # Indranil Ghosh, May 25 2017

a(n) = A000005(n) - A001222(n) - A005361(n).

A175264 a(n) = the middle prime among the distinct primes dividing {the n-th positive integer divisible by exactly an odd number of distinct primes}.

Original entry on oeis.org

2, 3, 2, 5, 7, 2, 3, 11, 13, 2, 17, 19, 23, 5, 3, 29, 3, 31, 2, 37, 41, 3, 43, 47, 7, 53, 59, 3, 61, 2, 3, 67, 5, 71, 73, 3, 79, 3, 83, 3, 89, 3, 97, 101, 3, 103, 5, 107, 109, 5, 113, 3, 3, 11, 5, 3, 127, 2, 5, 131, 3, 137, 3, 139, 5, 149, 3, 151, 7, 3, 157, 163, 5, 167, 3, 13, 5, 173, 3

Offset: 1

Leroy Quet, Mar 16 2010

nonn

			The 28th positive integer divisible by exactly an odd number of distinct primes is 60. 60 is factored as 2^2*3*5. The distinct primes dividing 60 are therefore 2, 3, and 5. Since the middle of these primes is 3, then a(28) = 3.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A030230

mp[n_]:=Module[{prs=Transpose[FactorInteger[n]][[1]],pos},pos= Floor[ Length[ prs]/2]+1;prs[[pos]]]; mp/@Select[Range[300],OddQ[PrimeNu[#]]&] (* Harvey P. Dale, Jun 20 2012 *)

More terms from R. J. Mathar, Oct 09 2010

A321962 Where the zeros in A123066 occur.

Original entry on oeis.org

1, 51, 53, 7955, 7959, 7961, 7985, 7987, 8245, 8805, 8807, 8809, 8813, 8815, 8817, 8819, 8821, 8825, 8829, 8847, 8851, 8853, 8855, 8857, 8859, 8873, 8877, 8879, 8969, 8973, 8975, 9063, 9079, 9081, 9083, 9089, 9091, 9093, 9095, 9097, 9163, 9165, 9215, 9219

Offset: 1

Peter Luschny, Dec 21 2018

nonn

Let pp(n) be the prime parity of n, defined as 1 if the number of distinct primes dividing n is odd and -1 if it is even; by convention pp(1) = 0. The cumulative sum of pp is A123066. We call the initial segment of the integers [1..n] balanced with respect to prime parity if the cumulative sum of pp(n) is 0. [1..a(n)] give the balanced segments.

Amiram Eldar, Table of n, a(n) for n = 1..8908 (terms below 10^10)
Harald Helfgott and Adrián Ubis, Primos, paridad y análisis, Publicaciones Matemáticas del Uruguay, Vol. 18 (2023), pp. 205-283; arXiv preprint, arXiv:1812.08707 [math.NT], 2018-2019.

Cf. A001222, A123066, A030230, A030231.

a_list := proc(len) local omega, c, L, j; c := 0; L := 1;
omega := n -> nops(numtheory[factorset](n));
for j from 2 to len do
   c := c + (-1)^omega(j);
   if c = 0 then L := L,j fi
od; L end: a_list(10000);

A123066[n_] := Join[{0}, Accumulate[Table[-(-1)^PrimeNu[j], {j,2,n}]]];
A321962List[n_] := Position[A123066[n], 0] // Flatten;
A321962List[10000]

A328502 Dirichlet g.f.: zeta(s-1) / (zeta(s) * zeta(s-2)).

Original entry on oeis.org

1, -3, -7, -2, -21, 21, -43, -4, -12, 63, -111, 14, -157, 129, 147, -8, -273, 36, -343, 42, 301, 333, -507, 28, -80, 471, -36, 86, -813, -441, -931, -16, 777, 819, 903, 24, -1333, 1029, 1099, 84, -1641, -903, -1807, 222, 252, 1521, -2163, 56, -252, 240, 1911, 314, -2757, 108, 2331

Offset: 1

Ilya Gutkovskiy, Oct 22 2019

Dirichlet inverse of A057660.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Olivier Bordelles, A Multidimensional Cesaro Type Identity and Applications, J. Int. Seq. 18 (2015) # 15.3.7.

Cf. A000010, A008683, A030230 (positions of negative terms), A057660, A101035.

a[1] = 1; a[n_] := -Sum[DivisorSigma[2, (n/d)^2]/DivisorSigma[1, (n/d)^2] a[d], {d, Most @ Divisors[n]}]; Table[a[n], {n, 1, 55}]
Table[DivisorSum[n, EulerPhi[n/#] MoebiusMu[#] #^2 &], {n, 1, 55}]
f[p_, e_] := If[e == 1, p - 1 - p^2, -p^(e - 1)*(p - 1)^2]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Dec 03 2022 *)

a(n)={sumdiv(n, d, eulerphi(n/d)*moebius(d)*d^2)} \\ Andrew Howroyd, Oct 25 2019

a(1) = 1; a(n) = -Sum_{d|n, dA057660(n/d) * a(d).
a(n) = Sum_{d|n} phi(n/d) * mu(d) * d^2.
Multiplicative with a(p) = p - 1 - p^2, and a(p^e) = -p^(e-1) * (p-1)^2, for e > 1. - Amiram Eldar, Dec 03 2022
a(n) = Sum_{k = 1..n} gcd(k, n)^2 * mu(gcd(k, n)) (follows from an identity of Cesàro. See, for example, Bordelles, Lemma 1). - Peter Bala, Jan 16 2024

A328855 Number of factorizations of n into numbers with an odd number of distinct prime factors.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 5, 1, 2, 1, 2, 1, 1, 1, 3, 2, 1, 3, 2, 1, 2, 1, 7, 1, 1, 1, 4, 1, 1, 1, 3, 1, 2, 1, 2, 2, 1, 1, 5, 2, 2, 1, 2, 1, 3, 1, 3, 1, 1, 1, 4, 1, 1, 2, 11, 1, 2, 1, 2, 1, 2, 1, 6, 1, 1, 2, 2, 1, 2, 1, 5, 5, 1, 1, 4, 1, 1, 1, 3, 1, 4

Offset: 1

Ilya Gutkovskiy, Oct 28 2019

nonn

			a(8) = 3 because 8 = 2 * 4 = 2 * 2 * 2.

Cf. A001055, A030230, A050330, A328856.

Dirichlet g.f.: Product_{k>=1} 1 / (1 - A030230(k)^(-s)).

cp's OEIS Frontend

A308063 Number of ordered factorizations of n into numbers with an odd number of distinct prime divisors.

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A214195 Numbers with the number of distinct prime factors a multiple of 3.

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A327669 Sum of divisors of n that have an odd number of distinct prime factors.

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A328856 Number of factorizations of n into distinct numbers with an odd number of distinct prime factors.

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A081400 a(n) = d(n) - bigomega(n) - A005361(n).

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A175264 a(n) = the middle prime among the distinct primes dividing {the n-th positive integer divisible by exactly an odd number of distinct primes}.

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A321962 Where the zeros in A123066 occur.

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A328502 Dirichlet g.f.: zeta(s-1) / (zeta(s) * zeta(s-2)).

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