A144338 -id:A144338 - cp's OEIS frontend

A280128 Expansion of Product_{k>=2} (1 + mu(k)^2*x^k), where mu(k) is the Moebius function (A008683).

1, 0, 1, 1, 0, 2, 1, 2, 2, 2, 3, 3, 3, 5, 4, 6, 7, 7, 9, 9, 11, 13, 14, 16, 19, 18, 24, 23, 28, 31, 33, 39, 42, 46, 52, 57, 63, 71, 76, 87, 92, 103, 113, 123, 135, 149, 161, 178, 193, 210, 231, 249, 274, 298, 323, 352, 382, 414, 451, 486, 528, 572, 617, 669

Offset: 0

Ilya Gutkovskiy, Dec 26 2016

nonn

Number of partitions of n into distinct squarefree parts > 1 (A144338).

			G.f. = 1 + x^2 + x^3 + 2*x^5 + x^6 + 2*x^7 + 2*x^8 + 2*x^9 + 3*x^10 + 3*x^11 + ...
a(10) = 3 because we have [10], [7, 3] and [5, 3, 2].

G. C. Greubel, Table of n, a(n) for n = 0..5000
Joerg Arndt, Matters Computational (The Fxtbook), section 16.4.3 "Partitions into square-free parts", pp.351-352
Eric Weisstein's World of Mathematics, Squarefree
Index entries for related partition-counting sequences

Cf. A005117, A008683, A087188, A144338, A280127.

with(numtheory): seq(coeff(series(mul(1+mobius(k)^2*x^k,k=2..n), x,n+1),x,n),n=0..70); # Muniru A Asiru, Jul 30 2018

nmax = 75; CoefficientList[Series[Product[1 + MoebiusMu[k]^2 x^k, {k, 2, nmax}], {x, 0, nmax}], x]

{a(n) = if(n < 0, 0, polcoeff( prod(k=2, n, 1 + issquarefree(k)*x^k + x*O(x^n)), n))}; /* Michael Somos, Dec 26 2016 */

G.f.: Product_{k>=2} (1 + mu(k)^2*x^k).

A282290 Expansion of (Sum_{p prime, i>=2} x^(p^i))(Sum_{j>=2} mu(j)^2x^j), where mu() is the Moebius function (A008683).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 2, 3, 1, 1, 3, 3, 1, 1, 3, 4, 1, 3, 3, 4, 1, 2, 3, 4, 2, 3, 6, 4, 3, 3, 4, 5, 1, 5, 7, 6, 3, 3, 7, 4, 3, 4, 7, 6, 3, 4, 5, 7, 2, 3, 5, 7, 4, 3, 4, 5, 4, 4, 7, 6, 4, 4, 8, 6, 4, 6, 7, 7, 2, 5, 7, 7, 2, 4, 9, 5, 4, 4, 7, 8, 4, 5, 9, 9, 4, 4, 7, 7, 5, 6, 8, 8, 5, 5, 8, 6, 4, 6, 8, 7, 5, 6, 6, 6, 2, 5, 10

Offset: 0

Ilya Gutkovskiy, Feb 11 2017

nonn

Number of ways of writing n as a sum of a proper prime power (A246547) and a squarefree number > 1 (A144338).

Conjecture: a(n) > 0 for all n > 8.

			a(19) = 4 because we have [16, 3], [15, 4], [11, 8] and [10, 9].

Ilya Gutkovskiy, Extended graphical example
Eric Weisstein's World of Mathematics, Prime Power
Eric Weisstein's World of Mathematics, Squarefree

Cf. A005117, A008683, A098983, A144338, A246547.

nmax = 110; CoefficientList[Series[Sum[Sign[PrimeOmega[i] - 1] Floor[1/PrimeNu[i]] x^i, {i, 2, nmax}] Sum[MoebiusMu[j]^2 x^j, {j, 2, nmax}], {x, 0, nmax}], x]

G.f.: (Sum_{p prime, i>=2} x^(p^i))*(Sum_{j>=2} mu(j)^2*x^j).

A363919 a(n) = n^excess(n), where excess(n) = A046660(n).

Original entry on oeis.org

1, 1, 1, 4, 1, 1, 1, 64, 9, 1, 1, 12, 1, 1, 1, 4096, 1, 18, 1, 20, 1, 1, 1, 576, 25, 1, 729, 28, 1, 1, 1, 1048576, 1, 1, 1, 1296, 1, 1, 1, 1600, 1, 1, 1, 44, 45, 1, 1, 110592, 49, 50, 1, 52, 1, 2916, 1, 3136, 1, 1, 1, 60, 1, 1, 63, 1073741824, 1, 1, 1, 68

Offset: 1

Peter Luschny and Michael De Vlieger, Jul 14 2023

nonn

			108 = 2^2 * 3^3 => excess(108) = 5 - 2 => a(108) = 108^3 = 1259712.

Plot2 A363923 vs A205959.
Michael De Vlieger, Plot p(k)^e(k) at (x,y) = (n,k) for n = 1..2^11, with a color function representing e(k) where black represents e(k) = 1, red e(k) = 2, through magenta. The bar at bottom represents a(n) = 1 in black, a(n) a composite prime power in gold, a(n) neither squarefree nor semiprime in blue, and a(n) such that all e(k) > 1 in purple.
Michael De Vlieger, Notes on certain sequences relating BigOmega(n), omega(n), and rad(n), 2023.

Cf. A001221, A001222, A005117, A046660, A053763, A056170, A060687, A013929, A205959, A337945, A363923.

using Nemo
exc(n::fmpz) = sum(e - 1 for (p, e) in factor(n))
A363919(n::fmpz) = n < 2 ? fmpz(1) : n^exc(n)
println([A363919(fmpz(n)) for n in 1:68])

with(NumberTheory):
A363919 := n -> n^(NumberOfPrimeFactors(n) - NumberOfPrimeFactors(n, 'distinct')):
# Alternative:
a := n -> local i: n^add(i[2] - 1, i in ifactors(n)[2]): seq(a(n), n = 1..68);

Array[#^(PrimeOmega[#] - PrimeNu[#]) &, 120]

a(n) = my(f=factor(n)[, 2]); n^(vecsum(f)-#f); \\ Michel Marcus, Jul 16 2023

from sympy import factorint
def A363919(n): return n**sum(map(lambda e:e-1,factorint(n).values())) # Chai Wah Wu, Jul 18 2023

def A363919(n):
    if n < 2: return 1
    return n^sum(p[1] - 1 for p in list(factor(n)))
print([A363919(n) for n in srange(1, 69)])

a(n) = n^(Sum_{p in Factors(n)} (mult(p) - 1)), where Factors(n) is the integer factorization of n and mult(p) the multiplicity of the prime factor p.
a(n) = A363923(n) / A205959(n).
a(n) = n^A046660(n) = n^(A001222(n) - A001221(n)).
a(n) = 1 or divisible by at least one squared prime.
a(n) = 1 <=> n is squarefree (A005117).
a(n) != 1 <=> A056170(n) != 0.
a(n) = n <=> n = A060687(n - 1) for n >= 2.
a(2^n) = 2^(n*(n - 1)) = A053763(n).
a(n) <= 2^(lb(n)*(lb(n)-1)), where lb(n) = floor(log_{2}(n)).
a(n) is even <=> n = 2*A337945(n).
a(n) > 1 is odd <=> n = A053850(n).
n is prime => a(n) = 1. ('prime' means term of A000040).
n is prime product => a(n) = 1. ('prime product' means term of A144338).
n is proper prime power => a(n) is proper prime power. ('proper prime power' means term of A246547).
Moebius(a(n)) = [a(n) = 1], where [ ] denotes the Iverson bracket.

A381588 If n = Product (p_j^k_j) then a(n) = Product (lcm(p_j, k_j)), with a(1) = 1.

Original entry on oeis.org

1, 2, 3, 2, 5, 6, 7, 6, 6, 10, 11, 6, 13, 14, 15, 4, 17, 12, 19, 10, 21, 22, 23, 18, 10, 26, 3, 14, 29, 30, 31, 10, 33, 34, 35, 12, 37, 38, 39, 30, 41, 42, 43, 22, 30, 46, 47, 12, 14, 20, 51, 26, 53, 6, 55, 42, 57, 58, 59, 30, 61, 62, 42, 6, 65, 66, 67, 34, 69, 70, 71

Offset: 1

Paolo Xausa, Feb 28 2025

			a(18) = 12 because 18 = 2^1*3^2, lcm(2,1) = 2, lcm(3,2) = 6 and 2*6 = 12.
a(300) = 30 because 300 = 2^2*3^1*5^2, lcm(2,2) = 2, lcm(3,1) = 3, lcm(5,2) = 10 and 2*3*10 = 60.

Paolo Xausa, Table of n, a(n) for n = 1..10000

Cf. A008473, A008477, A035306, A144338 (fixed points), A369008 (analogous for gcd).

A381588[n_] := Times @@ LCM @@@ FactorInteger[n];
Array[A381588, 100]

a(n) = my(f=factor(n)); prod(i=1, #f~, lcm(f[i,1], f[i,2])); \\ Michel Marcus, Mar 02 2025

a(p) = p, for p prime.

A072490 Number of squarefree numbers (excluding 1) less than n.

Original entry on oeis.org

0, 0, 1, 2, 2, 3, 4, 5, 5, 5, 6, 7, 7, 8, 9, 10, 10, 11, 11, 12, 12, 13, 14, 15, 15, 15, 16, 16, 16, 17, 18, 19, 19, 20, 21, 22, 22, 23, 24, 25, 25, 26, 27, 28, 28, 28, 29, 30, 30, 30, 30, 31, 31, 32, 32, 33, 33, 34, 35, 36, 36, 37, 38, 38, 38, 39

Offset: 1

Amarnath Murthy, Jul 14 2002

nonn

			a(10) = 5 as the squarefree numbers less than 10 are 2,3,5,6 and 7.

Cf. A005117, A013928, A144338.

a(n) = sum(k=2, n-1, issquarefree(k)); \\ Michel Marcus, Sep 15 2019

from math import isqrt
from sympy import mobius
def A072490(n): return int(sum(mobius(k)*((n-1)//k**2) for k in range(1, isqrt(n-1)+1)))-1 if n>1 else 0 # Chai Wah Wu, Aug 19 2024

a(n) = A013928(n) - 1, n > 1.

G.f.: (x/(1 - x)) * Sum_{k>=2} mu(k)^2*x^k. - Ilya Gutkovskiy, Sep 14 2019

Name clarified and more terms from Ilya Gutkovskiy, Sep 14 2019

A280169 Expansion of Product_{k>=2} 1/(1 - mu(2k-1)^2x^(2*k-1)), where mu() is the Moebius function (A008683).

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 1, 1, 1, 1, 2, 2, 2, 3, 3, 4, 4, 5, 6, 6, 8, 9, 10, 11, 13, 14, 17, 18, 21, 24, 26, 30, 33, 38, 42, 47, 53, 58, 65, 73, 80, 90, 99, 110, 122, 134, 149, 164, 181, 199, 220, 242, 266, 292, 321, 352, 386, 424, 463, 507, 554, 606, 662, 722, 788, 860, 936, 1020, 1111, 1208, 1314, 1428, 1553, 1685, 1829, 1984, 2152

Offset: 0

Ilya Gutkovskiy, Dec 27 2016

nonn

Number of partitions of n into odd squarefree parts > 1.

			a(13) = 3 because we have [13], [7, 3, 3] and [5, 5, 3].

Joerg Arndt, Matters Computational (The Fxtbook), section 16.4.3 "Partitions into square-free parts", pp.351-352
Eric Weisstein's World of Mathematics, Squarefree
Index entries for related partition-counting sequences

Cf. A005117, A008683, A056911, A073576, A134345, A144338, A280127.

nmax = 76; CoefficientList[Series[Product[1/(1 - MoebiusMu[2 k - 1]^2 x^(2 k - 1)), {k, 2, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=2} 1/(1 - mu(2*k-1)^2*x^(2*k-1)).

A280197 Expansion of 1/(1 - Sum_{k>=2} mu(k)^2*x^k), where mu(k) is the Moebius function (A008683).

Original entry on oeis.org

1, 0, 1, 1, 1, 3, 3, 6, 8, 12, 20, 28, 45, 68, 102, 159, 238, 367, 557, 849, 1298, 1973, 3015, 4592, 7002, 10679, 16276, 24822, 37841, 57696, 87971, 134119, 204497, 311783, 475370, 724786, 1105053, 1684853, 2568837, 3916642, 5971587, 9104711, 13881698, 21165024, 32269721, 49200718, 75014949, 114373158, 174381511

Offset: 0

Ilya Gutkovskiy, Dec 28 2016

nonn

Number of compositions (ordered partitions) into squarefree parts > 1 (A144338).

			a(5) = 3 because we have [5], [3, 2] and [2, 3].

Robert Israel, Table of n, a(n) for n = 0..5456
Eric Weisstein's World of Mathematics, Squarefree
Index entries for sequences related to compositions

Cf. A005117, A008683, A073576, A144338, A280127, A280194.

N:= 100: # for a(0)..a(N)
g:= 1/(1-add(numtheory:-mobius(k)^2*x^k, k=2..N)):
S:= series(g,x,N+1):
seq(coeff(S,x,j),j=0..N); # Robert Israel, Dec 29 2016

nmax = 48; CoefficientList[Series[1/(1 - Sum[MoebiusMu[k]^2 x^k, {k, 2, nmax}]), {x, 0, nmax}], x]

G.f.: 1/(1 - Sum_{k>=2} mu(k)^2*x^k).

A033198 Discriminants of real quadratic number fields.

Original entry on oeis.org

8, 12, 5, 24, 28, 40, 44, 13, 56, 60, 17, 76, 21, 88, 92, 104, 29, 120, 124, 33, 136, 140, 37, 152, 156, 41, 168, 172, 184, 188, 204, 53, 220, 57, 232, 236, 61, 248, 65, 264, 268, 69, 280, 284, 73, 296, 77, 312, 316, 328, 332, 85, 344, 348, 89, 364, 93, 376, 380, 97

Offset: 1

N. J. A. Sloane

David A. Cox, "Primes of the Form x^2 + n y^2", Wiley, 1989, p. 103.

Cf. A144338.

with(numtheory): a:= proc(n) if issqrfree(n) then RETURN(piecewise(n mod 4=1,n,4*n)) else RETURN(NULL) fi: end: seq(a(n),n=2..150); # C. Ronaldo (aga_new_ac(AT)hotmail.com), Jan 03 2005

Reap[For[n = 2, n <= 100, n++, If[SquareFreeQ[n], Sow[If[Mod[n, 4] == 1, n, 4 n]]]]][[2, 1]] (* Jean-François Alcover, Mar 22 2023 *)

For squarefree n >= 2, list n if n=1 mod 4 else 4n.

More terms from C. Ronaldo (aga_new_ac(AT)hotmail.com), Jan 03 2005

A280716 Expansion of Product_{k>=2} (1 + mu(2k-1)^2x^(2*k-1)), where mu() is the Moebius function (A008683).

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 2, 2, 1, 3, 2, 3, 3, 3, 4, 4, 3, 5, 4, 5, 6, 4, 8, 6, 8, 8, 9, 11, 10, 11, 14, 13, 14, 15, 16, 19, 16, 20, 22, 22, 23, 26, 29, 30, 31, 35, 39, 38, 43, 44, 49, 50, 52, 58, 59, 64, 67, 71, 77, 82, 85, 93, 97, 107, 108, 117, 125, 131, 138, 143, 157, 162, 168, 179, 194, 199

Offset: 0

Ilya Gutkovskiy, Jan 07 2017

nonn

Number of partitions of n into distinct odd squarefree parts > 1.

			a(18) = 3 because we have [15, 3], [13, 5] and [11, 7].

Joerg Arndt, Matters Computational (The Fxtbook), section 16.4.3 "Partitions into square-free parts", pp.351-352
Eric Weisstein's World of Mathematics, Squarefree
Index entries for related partition-counting sequences

Cf. A005117, A008683, A056911, A073576, A134337, A144338, A280128, A280169.

nmax = 84; CoefficientList[Series[Product[1 + MoebiusMu[2 k - 1]^2 x^(2 k - 1), {k, 2, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=2} (1 + mu(2*k-1)^2*x^(2*k-1)).

A284892 Numbers n > 1 such that all Hopf algebras of dimension n over algebraically closed fields of characteristic 0 are semisimple.

Original entry on oeis.org

2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 31, 33, 34, 35, 37, 38, 39, 41

Offset: 1

Charles R Greathouse IV, Apr 05 2017

All terms in this sequence are squarefree, but not all squarefree numbers are present: 1 and 30 are missing.

Proper subsequence of A144338. A000040 and A128907 are subsequences.

cp's OEIS Frontend

A280128 Expansion of Product_{k>=2} (1 + mu(k)^2*x^k), where mu(k) is the Moebius function (A008683).

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A282290 Expansion of (Sum_{p prime, i>=2} x^(p^i))*(Sum_{j>=2} mu(j)^2*x^j), where mu() is the Moebius function (A008683).

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A363919 a(n) = n^excess(n), where excess(n) = A046660(n).

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A381588 If n = Product (p_j^k_j) then a(n) = Product (lcm(p_j, k_j)), with a(1) = 1.

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A072490 Number of squarefree numbers (excluding 1) less than n.

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A280169 Expansion of Product_{k>=2} 1/(1 - mu(2*k-1)^2*x^(2*k-1)), where mu() is the Moebius function (A008683).

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A280197 Expansion of 1/(1 - Sum_{k>=2} mu(k)^2*x^k), where mu(k) is the Moebius function (A008683).

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A282290 Expansion of (Sum_{p prime, i>=2} x^(p^i))(Sum_{j>=2} mu(j)^2x^j), where mu() is the Moebius function (A008683).

A280169 Expansion of Product_{k>=2} 1/(1 - mu(2k-1)^2x^(2*k-1)), where mu() is the Moebius function (A008683).

A280716 Expansion of Product_{k>=2} (1 + mu(2k-1)^2x^(2*k-1)), where mu() is the Moebius function (A008683).