A030231 -id:A030231 - cp's OEIS frontend

A279458 Numbers k such that number of distinct primes dividing k is even and number of prime divisors (counted with multiplicity) of k is even.

1, 6, 10, 14, 15, 21, 22, 24, 26, 33, 34, 35, 36, 38, 39, 40, 46, 51, 54, 55, 56, 57, 58, 62, 65, 69, 74, 77, 82, 85, 86, 87, 88, 91, 93, 94, 95, 96, 100, 104, 106, 111, 115, 118, 119, 122, 123, 129, 133, 134, 135, 136, 141, 142, 143, 144, 145, 146, 152, 155, 158, 159, 160, 161, 166, 177, 178, 183, 184, 185, 187, 189

Offset: 1

Ilya Gutkovskiy, Dec 12 2016

Intersection of A028260 and A030231.
Numbers k such that A000035(A001221(k)) = 0 and A000035(A001222(k)) = 0.
Numbers k such that A076479(k) = 1 and A008836(k) = 1.

			24 is in the sequence because 24 = 2^3*3 therefore omega(24) = 2 {2,3} is even and bigomega(24) = 4 {2,2,2,3} is even.

G. C. Greubel, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Distinct Prime Factors.
Eric Weisstein's World of Mathematics, Prime Factor.

Cf. A000035, A001221, A001222, A008836, A028260, A030231, A076479, A279456, A279457.

Select[Range[220], Mod[PrimeNu[#1], 2] == Mod[PrimeOmega[#1], 2] == 0 & ]

is(k) = {my(f = factor(k)); !(omega(f) % 2) && !(bigomega(f) % 2);} \\ Amiram Eldar, Sep 17 2024

A286225 Number of compositions (ordered partitions) of n into parts with an even number of distinct prime divisors.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 7, 9, 13, 18, 25, 34, 46, 61, 83, 112, 153, 209, 286, 387, 526, 713, 969, 1317, 1794, 2437, 3312, 4497, 6110, 8302, 11290, 15347, 20865, 28354, 38533, 52361, 71167, 96721, 131464, 178672, 242834, 330020, 448532, 609590, 828511, 1126037, 1530418, 2079977, 2826896, 3841998

Offset: 0

Ilya Gutkovskiy, May 04 2017

nonn

			a(8) = 4 because we have [6, 1, 1], [1, 6, 1], [1, 1, 6] and [1, 1, 1, 1, 1, 1, 1, 1].

Amiram Eldar, Table of n, a(n) for n = 0..7000
Eric Weisstein's World of Mathematics, Distinct Prime Factors
Index entries for sequences related to compositions

Cf. A030231, A285798, A286221, A286224.

nmax = 53; CoefficientList[Series[1/(1 - Sum[Boole[EvenQ[PrimeNu[k]]] x^k, {k, 1, nmax}]), {x, 0, nmax}], x]

G.f.: 1/(1 - Sum_{k>=1} x^A030231(k)).

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.

A275665 Numbers n such that n and sopf(n) are relatively prime, where sopf(n) (A008472) is the sum of the distinct primes dividing n.

Original entry on oeis.org

1, 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, 119, 122, 123, 124, 129, 133, 134, 135, 136, 141, 142, 143, 144, 145, 146, 147, 148, 152, 153, 155, 158, 159, 160, 161, 162, 164, 165

Offset: 1

Charles R Greathouse IV, Aug 04 2016

nonn

Hall shows that the density of this sequence is 6/Pi^2, so a(n) ~ (Pi^2/6)n.

Differs from A267114, from A030231, and from A007774 (shifted by one index) first at n=93. - R. J. Mathar, Aug 22 2016

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
R. B. Hall, On the probability that n and f(n) are relatively prime, Acta Arithmetica 17 (1970), pp. 169-183.

Cf. A008472, A082300, A267114, A030231, A007774.

Select[Range@ 165, CoprimeQ[#, Total@ FactorInteger[#][[All, 1]]] &] (* Michael De Vlieger, Aug 06 2016 *)

sopf(n)=vecsum(factor(n)[,1])
is(n)=gcd(sopf(n),n)==1

A327670 Sum of divisors of n that have an even number of distinct prime factors.

Original entry on oeis.org

1, 1, 1, 1, 1, 7, 1, 1, 1, 11, 1, 19, 1, 15, 16, 1, 1, 25, 1, 31, 22, 23, 1, 43, 1, 27, 1, 43, 1, 32, 1, 1, 34, 35, 36, 73, 1, 39, 40, 71, 1, 42, 1, 67, 61, 47, 1, 91, 1, 61, 52, 79, 1, 79, 56, 99, 58, 59, 1, 64, 1, 63, 85, 1, 66, 62, 1, 103, 70, 60, 1, 169, 1, 75, 91

Offset: 1

Ilya Gutkovskiy, Sep 21 2019

nonn

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

Cf. A000961 (positions of 1's), A000203, A030231, A049060, A285798, A318676, A327669.

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

a[n_] := DivisorSum[n, # &, EvenQ[PrimeNu[#]] &]; Table[a[n], {n, 1, 75}]

G.f.: Sum_{k>=1} A030231(k) * x^A030231(k) / (1 - x^A030231(k)).
L.g.f.: log(B(x)) = Sum_{n>=1} a(n) * x^n / n, where B(x) = g.f. of A285798.
a(n) = A000203(n) - A327669(n).

A336222 Numbers k such that the square root of the largest square dividing k has an even number of prime divisors (counted with multiplicity).

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, 26, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 46, 47, 48, 51, 53, 55, 57, 58, 59, 61, 62, 65, 66, 67, 69, 70, 71, 72, 73, 74, 77, 78, 79, 80, 81, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95

Offset: 1

Amiram Eldar, Jul 12 2020

nonn

Numbers k such that A000188(k) is a term of A028260.
The squarefree numbers (A005117) are terms of this sequence since if k is squarefree, then A000188(k) = 1, 1 has 0 prime divisors, and 0 is even.
A number k is a term if and only if its powerful part, A057521(k), is a term.
The asymptotic density of this sequence is 7/10 (Cohen, 1964).
The corresponding sequence of numbers k such that the square root of the largest square dividing k has an even number of distinct prime divisors, i.e., numbers k such that A000188(k) is a term of A030231, is A333634.

			2 is a term since the largest square dividing 2 is 1, sqrt(1) = 1, 1 has 0 prime divisors, and 0 is even.
16 is a term since the largest square dividing 16 is 16, sqrt(16) = 4, 4 = 2 * 2 has 2 prime divisors, and 2 is even.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eckford Cohen, Some asymptotic formulas in the theory of numbers, Trans. Amer. Math. Soc., Vol. 112 (1964), pp. 214-227.

Cf. A000188, A001222, A005117, A028260, A030231, A057521, A333634.

f[p_, e_] := p^Floor[e/2]; Select[Range[100], EvenQ[PrimeOmega[Times @@ (f @@@ FactorInteger[#])]] &]

A336223 Numbers k such that the largest square dividing k is a unitary divisor of k and its square root has an even number of distinct prime divisors.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 46, 47, 51, 53, 55, 57, 58, 59, 61, 62, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 100, 101, 102, 103

Offset: 1

Amiram Eldar, Jul 12 2020

nonn

First differs from A333634 at n = 47.
Terms k of A335275 such that A000188(k) is a term of A030231.
Numbers whose powerful part (A057521) is a square term of A030231.
The squarefree numbers (A005117) are terms of this sequence since if k is squarefree, then the largest square dividing k is 1 which is a unitary divisor, sqrt(1) = 1 has 0 prime divisors, and 0 is even.
The asymptotic density of this sequence is (Product_{p prime} (1 - 1/(p^2*(p+1))) + Product_{p prime} (1 - (2*p+1)/(p^2*(p+1))))/2 = (0.881513... + 0.394391...)/2 = 0.637952807730728551636349961980617856650450613867264... (Cohen, 1964; the first product is A065465).

			36 is a term since the largest square dividing 36 is 36, which is a unitary divisor, sqrt(36) = 6, 6 = 2 * 3 has 2 distinct prime divisors, and 2 is even.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eckford Cohen, Some asymptotic formulas in the theory of numbers, Trans. Amer. Math. Soc., Vol. 112 (1964), pp. 214-227.

Intersection of A333634 and A335275.

Cf. A000188, A001221, A005117, A030231, A057521, A065465.

seqQ[n_] := EvenQ @ Length[(e = Select[FactorInteger[n][[;; , 2]], # > 1 &])] && AllTrue[e, EvenQ[#] &]; Select[Range[100], seqQ]

A357375 Number of ordered factorizations of n into numbers > 1 with an even number of distinct prime divisors.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 1, 1, 1, 2, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 2, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 3, 0, 1, 1, 1, 1, 0, 0, 1, 0, 1, 0, 2, 1, 1, 1, 1, 0, 2

Offset: 1

Ilya Gutkovskiy, Sep 25 2022

nonn

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

Cf. A030231, A050334, A074206, A308063, A357374.

f[n_] := Boole[EvenQ[PrimeNu[n]]]; a[1] = 1; a[n_] := a[n] = Sum[If[d < n, f[n/d] a[d], 0], {d, Divisors[n]}]; Table[a[n], {n, 1, 90}]

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).

A105642 Composite nonsquares and noncubes.

Original entry on oeis.org

6, 10, 12, 14, 15, 18, 20, 21, 22, 24, 26, 28, 30, 32, 33, 34, 35, 38, 39, 40, 42, 44, 45, 46, 48, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99, 102, 104, 105, 106, 108

Offset: 1

Zak Seidov, May 03 2005

nonn

Cf. A007774, A024619, A030231, A056760, A064040, A084227, A100658.

nn=200; Complement[Range[nn], Prime[Range[PrimePi[nn]]], Range[Sqrt[nn]]^2, Range[nn^(1/3)]^3]  (* Harvey P. Dale, Jan 26 2011 *)

is(n)=n>1 && !isprime(n) && !issquare(n) && !ispower(n,3) \\ Charles R Greathouse IV, Oct 19 2015

a(n) = n + O(n/log n). - Charles R Greathouse IV, Oct 19 2015

cp's OEIS Frontend

A279458 Numbers k such that number of distinct primes dividing k is even and number of prime divisors (counted with multiplicity) of k is even.

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A286225 Number of compositions (ordered partitions) of n into parts with an even 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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A275665 Numbers n such that n and sopf(n) are relatively prime, where sopf(n) (A008472) is the sum of the distinct primes dividing n.

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A327670 Sum of divisors of n that have an even number of distinct prime factors.

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A336222 Numbers k such that the square root of the largest square dividing k has an even number of prime divisors (counted with multiplicity).

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A336223 Numbers k such that the largest square dividing k is a unitary divisor of k and its square root has an even number of distinct prime divisors.

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A357375 Number of ordered factorizations of n into numbers > 1 with an even number of distinct prime divisors.

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