A057521 -id:A057521 - cp's OEIS frontend

A056674 Number of squarefree divisors which are not unitary. Also number of unitary divisors which are not squarefree.

0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 2, 0, 0, 0, 1, 0, 2, 0, 2, 0, 0, 0, 2, 1, 0, 1, 2, 0, 0, 0, 1, 0, 0, 0, 3, 0, 0, 0, 2, 0, 0, 0, 2, 2, 0, 0, 2, 1, 2, 0, 2, 0, 2, 0, 2, 0, 0, 0, 4, 0, 0, 2, 1, 0, 0, 0, 2, 0, 0, 0, 3, 0, 0, 2, 2, 0, 0, 0, 2, 1, 0, 0, 4, 0, 0, 0, 2, 0, 4, 0, 2, 0, 0, 0, 2, 0, 2, 2, 3, 0, 0, 0, 2, 0

Offset: 1

Labos Elemer, Aug 10 2000

Numbers of unitary and of squarefree divisors are identical, although the 2 sets are usually different, so sizes of parts outside overlap are also equal to each other.

			For n = 252, it has 18 divisors, 8 are unitary, 8 are squarefree, 2 belong to both classes, so 6 are squarefree but not unitary, thus a(252) = 6. The set {2,3,14,21,42} forms squarefree but non-unitary while the set {4,9,36,28,63,252} of same size gives the set of not squarefree but unitary divisors.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences computed from exponents in factorization of n.

Cf. A000005, A007913, A034444, A055229, A055231, A056671.

Table[DivisorSum[n, 1 &, And[SquareFreeQ@ #, ! CoprimeQ[#, n/#]] &], {n, 105}] (* Michael De Vlieger, Jul 19 2017 *)
f[p_, e_] := If[e == 1, 2, 1]; a[1] = 0; a[n_] := 2^Length[fct = FactorInteger[n]] - Times @@ (f @@@ fct); Array[a, 100] (* Amiram Eldar, Jul 24 2024 *)

A034444(n) = (2^omega(n));
A057521(n) = { my(f=factor(n)); prod(i=1, #f~, if(f[i, 2]>1, f[i, 1]^f[i, 2], 1)); } \\ Charles R Greathouse IV, Aug 13 2013
A055231(n) = n/A057521(n);
A056674(n) = (A034444(n) - numdiv(A055231(n)));
\\ Or:
A055229(n) = { my(c=core(n)); gcd(c, n/c); }; \\ Charles R Greathouse IV, Nov 20 2012
A056674(n) = ((2^omega(n)) - numdiv(core(n)/A055229(n)));
\\ Antti Karttunen, Jul 19 2017

a(n) = {my(f = factor(n), e = f[,2]); 2^omega(f) - prod(i = 1, #e, if(e[i] == 1, 2, 1));} \\ Amiram Eldar, Jul 24 2024

from sympy import gcd, primefactors, divisor_count
from sympy.ntheory.factor_ import core
def a055229(n):
    c=core(n)
    return gcd(c, n//c)
def a056674(n): return 2**len(primefactors(n)) - divisor_count(core(n)//a055229(n))
print([a056674(n) for n in range(1, 101)]) # Indranil Ghosh, Jul 19 2017

a(n) = A034444(n) - A056671(n) = A034444(n) - A000005(A055231(n)) = A034444(n) - A000005(A007913(n)/A055229(n)).

A254503 Möbius transform of A034448.

Original entry on oeis.org

1, 2, 3, 2, 5, 6, 7, 4, 6, 10, 11, 6, 13, 14, 15, 8, 17, 12, 19, 10, 21, 22, 23, 12, 20, 26, 18, 14, 29, 30, 31, 16, 33, 34, 35, 12, 37, 38, 39, 20, 41, 42, 43, 22, 30, 46, 47, 24, 42, 40, 51, 26, 53, 36, 55, 28, 57, 58, 59, 30, 61, 62, 42, 32, 65, 66, 67, 34, 69, 70

Offset: 1

Álvar Ibeas, Jan 31 2015

Álvar Ibeas, Table of n, a(n) for n = 1..10000

Cf. A000010 (totient), A001694 (powerful), A005117 (squarefree), A034448 (usigma), A057521 (powerful part), A055231 (unitary squarefree kernel).

Table[DivisorSum[n, MoebiusMu[#]^2*EulerPhi[n/#] &, CoprimeQ[n/#, #] &], {n, 70}] (* Michael De Vlieger, Jun 27 2018 *)
f[p_, e_] := (p - 1)*p^(e - 1); f[p_, 1] := p; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Aug 27 2023 *)

a(n) = {my(f = factor(n)); for (i=1, #f~, if ((e=f[i, 2]) > 1, f[i, 1] = eulerphi(f[i, 1]^e); f[i, 2] = 1);); factorback(f);} \\ Michel Marcus, Feb 06 2015

a(n) = sumdiv(n, d, if(gcd(n/d, d) == 1, moebius(d)^2 * eulerphi(n/d))); \\ Daniel Suteu, Jun 27 2018

a(n) = phi(A057521(n)) * A055231(n).
If n is squarefree, a(n) = n; if n is powerful, a(n) = phi(n).
Multiplicative with a(p) = p; a(p^e) = phi(p^e), for e > 1.
Dirichlet g.f.: zeta(s-1) / zeta(2s-1).
a(n) = Sum_{d|n, gcd(n/d, d) = 1} mu(d)^2 * phi(n/d). - Daniel Suteu, Jun 27 2018
Sum_{k=1..n} a(k) ~ n^2 / (2*zeta(3)). - Vaclav Kotesovec, Jan 11 2019

A338540 Numbers having exactly three non-unitary prime factors.

Original entry on oeis.org

900, 1764, 1800, 2700, 3528, 3600, 4356, 4500, 4900, 5292, 5400, 6084, 6300, 7056, 7200, 8100, 8712, 8820, 9000, 9800, 9900, 10404, 10584, 10800, 11025, 11700, 12100, 12168, 12348, 12600, 12996, 13068, 13500, 14112, 14400, 14700, 15300, 15876, 16200, 16900, 17100

Offset: 1

Amiram Eldar, Nov 01 2020

nonn

Numbers k such that A056170(k) = A001221(A057521(k)) = 3.
Numbers divisible by the squares of exactly three distinct primes.
Subsequence of A318720 and first differs from it at n = 123.
The asymptotic density of this sequence is (eta_1^3 - 3*eta_1*eta_2 + 2*eta_3)/Pi^2 = 0.0032920755..., where eta_j = Sum_{p prime} 1/(p^2-1)^j (Pomerance and Schinzel, 2011).

			900 = 2^2 * 3^2 * 5^2 is a term since it has exactly 3 prime factors, 2, 3 and 5, that are non-unitary.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Carl Pomerance and Andrzej Schinzel, Multiplicative Properties of Sets of Residues, Moscow Journal of Combinatorics and Number Theory, Vol. 1, No. 1 (2011), pp. 52-66. See pp. 61-62.

Subsequence of A013929, A318720 and A327877.
Cf. A001221, A056170, A057521, A190641, A338539, A338541, A338542.
Cf. A154945 (eta_1), A324833 (eta_2), A324834 (eta_3).

Select[Range[17000], Count[FactorInteger[#][[;;,2]], _?(#1 > 1 &)] == 3 &]

A338541 Numbers having exactly four non-unitary prime factors.

Original entry on oeis.org

44100, 88200, 108900, 132300, 152100, 176400, 213444, 217800, 220500, 260100, 264600, 298116, 304200, 308700, 324900, 326700, 352800, 396900, 426888, 435600, 441000, 456300, 476100, 485100, 509796, 520200, 529200, 544500, 573300, 592900, 596232, 608400, 617400

Offset: 1

Amiram Eldar, Nov 01 2020

nonn

Numbers k such that A056170(k) = A001221(A057521(k)) = 4.
Numbers divisible by the squares of exactly four distinct primes.
The asymptotic density of this sequence is (eta_1^4 - 6*eta_1^2*eta_2 + 3*eta_2^2 + 8*eta_1*eta_3 - 6*eta_4)/(4*Pi^2) = 0.0000970457..., where eta_j = Sum_{p prime} 1/(p^2-1)^j (Pomerance and Schinzel, 2011).

			44100 = 2^2 * 3^2 * 5^2 * 7^2 is a term since it has exactly 4 prime factors, 2, 3, 5 and 7, that are non-unitary.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Carl Pomerance and Andrzej Schinzel, Multiplicative Properties of Sets of Residues, Moscow Journal of Combinatorics and Number Theory, Vol. 1, No. 1 (2011), pp. 52-66. See pp. 61-62.

Subsequence of A013929 and A318720.
Cf. A001221, A056170, A057521, A190641, A338539, A338540, A338542.
Cf. A154945 (eta_1), A324833 (eta_2), A324834 (eta_3), A324835 (eta_4).

Select[Range[620000], Count[FactorInteger[#][[;;,2]], _?(#1 > 1 &)] == 4 &]

A338542 Numbers having exactly five non-unitary prime factors.

Original entry on oeis.org

5336100, 7452900, 10672200, 12744900, 14905800, 15920100, 16008300, 18404100, 21344400, 22358700, 23328900, 25489800, 26680500, 29811600, 31472100, 31840200, 32016600, 36072036, 36808200, 37088100, 37264500, 37352700, 38234700, 39312900, 42380100, 42688800, 43956900

Offset: 1

Amiram Eldar, Nov 01 2020

nonn

Numbers k such that A056170(k) = A001221(A057521(k)) = 5.
Numbers divisible by the squares of exactly five distinct primes.
The asymptotic density of this sequence is (eta_1^5 - 10*eta_1^3*eta_2 + 15*eta_1*eta_2^2 + 20*eta_1^2*eta_3 - 20*eta_2*eta_3 - 30*eta_1*eta_4 + 24*eta_5)/(20*Pi^2) = 0.0000015673..., where eta_j = Sum_{p prime} 1/(p^2-1)^j (Pomerance and Schinzel, 2011).

			5336100 = 2^2 * 3^2 * 5^2 * 7^2 * 11^2 is a term since it has exactly 5 prime factors, 2, 3, 5, 7 and 11, that are non-unitary.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Carl Pomerance and Andrzej Schinzel, Multiplicative Properties of Sets of Residues, Moscow Journal of Combinatorics and Number Theory, Vol. 1, No. 1 (2011), pp. 52-66. See pp. 61-62.

Subsequence of A013929, A318720 and A327877.
Cf. A001221, A056170, A057521, A190641, A338539, A338540, A338541.
Cf. A154945 (eta_1), A324833 (eta_2), A324834 (eta_3), A324835 (eta_4), A324836 (eta_5).

Select[Range[2*10^7], Count[FactorInteger[#][[;;,2]], _?(#1 > 1 &)] == 5 &]

A349442 Dirichlet convolution of A000027 (the identity function) with A349350 (Dirichlet inverse of the powerful part of n).

Original entry on oeis.org

1, 1, 2, -1, 4, 2, 6, -3, -2, 4, 10, -2, 12, 6, 8, -1, 16, -2, 18, -4, 12, 10, 22, -6, -4, 12, -16, -6, 28, 8, 30, 5, 20, 16, 24, 2, 36, 18, 24, -12, 40, 12, 42, -10, -8, 22, 46, -2, -6, -4, 32, -12, 52, -16, 40, -18, 36, 28, 58, -8, 60, 30, -12, 7, 48, 20, 66, -16, 44, 24, 70, 6, 72, 36, -8, -18, 60, 24, 78, -4

Offset: 1

Antti Karttunen, Nov 18 2021

Multiplicative because both A000027 and A349350 are.

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A000027, A057521, A349350, A349441 (Dirichlet inverse), A349443 (sum with it).

A057521(n) = { my(f=factor(n)); prod(i=1, #f~, if(f[i, 2]>1, f[i, 1]^f[i, 2], 1)); }; \\ From A057521
memoA349350 = Map();
A349350(n) = if(1==n,1,my(v); if(mapisdefined(memoA349350,n,&v), v, v = -sumdiv(n,d,if(dA057521(n/d)*A349350(d),0)); mapput(memoA349350,n,v); (v)));
A349442(n) = sumdiv(n,d,d*A349350(n/d));

a(n) = Sum_{d|n} d * A349350(n/d).

A365552 The number of exponentially odd divisors of the powerful part of n.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 3, 1, 2, 1, 2, 1, 1, 1, 3, 2, 1, 3, 2, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 3, 2, 2, 1, 2, 1, 3, 1, 3, 1, 1, 1, 2, 1, 1, 2, 4, 1, 1, 1, 2, 1, 1, 1, 6, 1, 1, 2, 2, 1, 1, 1, 3, 3, 1, 1, 2, 1, 1, 1

Offset: 1

Amiram Eldar, Sep 08 2023

First differs from A095691 at n = 512.

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

Cf. A013661, A095691, A057521, A322483.

f[p_, e_] := If[e == 1, 1, Floor[(e + 3)/2]]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = vecprod(apply(x -> if(x == 1, 1, (x+3)\2), factor(n)[, 2]));

a(n) = A322483(A057521(n)).
Multiplicative with a(p) = 1 and a(p^e) = floor((e+3)/2) for e >= 2.
Dirichlet g.f.: zeta(s) * zeta(2*s) * Product_{p prime} (1 + 1/p^(3*s) - 1/p^(4*s)).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = zeta(2) * Product_{p prime} (1 + 1/p^3 - 1/p^4) = 1.80989829762278336163... .

A375341 The maximum exponent in the prime factorization of the numbers that have exactly one non-unitary prime factor.

Original entry on oeis.org

2, 3, 2, 2, 4, 2, 2, 3, 2, 3, 2, 5, 3, 2, 2, 4, 2, 2, 2, 3, 3, 2, 2, 6, 2, 2, 2, 4, 4, 2, 3, 2, 2, 5, 2, 2, 3, 4, 2, 2, 3, 2, 2, 3, 2, 7, 2, 3, 3, 2, 2, 2, 2, 3, 2, 2, 5, 4, 2, 3, 2, 2, 2, 2, 4, 3, 2, 3, 6, 2, 2, 2, 4, 2, 2, 5, 2, 3, 2, 2, 4, 2, 5, 2, 2, 3, 3, 8, 2, 2, 3, 2, 3, 4, 2, 2, 2, 3, 2, 2, 2, 2, 3, 3

Offset: 1

Amiram Eldar, Aug 12 2024

The positive terms in A375339.

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

Cf. A005361, A048109, A051903, A060687, A082293, A154945, A190641, A375339, A375340.

s[n_] := Module[{e = Select[FactorInteger[n][[;; , 2]], # > 1 &]}, If[Length[e] == 1, e[[1]], Nothing]]; Array[s, 300]

lista(kmax) = {my(e); for(k = 1, kmax, e = select(x -> x > 1, factor(k)[,2]); if(#e == 1, print1(e[1], ", ")));}

a(n) = A051903(A190641(n)).
a(n) = A005361(A190641(n)).
a(n) = A375339(A190641(n)).
a(n) = A132349(A057521(A190641(n))).
a(n) = 2 if and only if A190641(n) is in A060687.
a(n) = 3 if and only if A190641(n) is in A048109.
a(n) <= 3 if and only if A190641(n) is in A082293.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{p prime} (2*p-1)/((p-1)*(p^2-1)) / Sum_{p prime} 1/(p^2-1) = A375340 / A154945 = 2.74622231282166656595... .
Asymptotic second raw moment:  = Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)^2 = Sum_{p prime} (4*p^2-3*p+1)/((p-1)^3*(p+1)) / Sum_{p prime} 1/(p^2-1) = 9.064902009520365378603... .
Asymptotic second central moment, or variance, is  - ^2 = 1.52316501808078192104... and the asymptotic standard deviation is sqrt( - ^2) = 1.23416571743051667098... .

A382061 Numbers whose number of divisors is divisible by their number of unitary divisors.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 8, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 24, 26, 27, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 46, 47, 51, 53, 54, 55, 56, 57, 58, 59, 61, 62, 65, 66, 67, 69, 70, 71, 72, 73, 74, 77, 78, 79, 82, 83, 85, 86, 87, 88, 89, 91, 93, 94, 95, 96, 97

Offset: 1

Amiram Eldar, Mar 14 2025

Numbers k such that A034444(k) | A000005(k).
The criterion according to which a number belongs to this sequence depends only on the prime signature of this number: if {e_1, e_2, ... } are the exponents in the prime factorization of k then k is a term if and only if A000005(k)/A344444(k) = Product_{i} (e_i + 1)/2 is an integer.
The exponentially odd numbers (A268335) are all terms, since their prime factorization has only odd exponents e_i, so (e_i + 1)/2 is an integer. This sequence first differs from A268335 at n = 53: a(53) = 72 = 2^3 * 3^2 is not a term of A268335. The next terms that are not in A268335 are 108, 200, 360, 392, 432, 500, ... .
All the squarefree numbers (A005117, which is a subsequence of A268335) are terms. These are the numbers k such that A034444(k) = A000005(k).
A number k is a term if and only if the powerful part of k, A057521(k), is a term. Therefore, the primitive terms of this sequence are the powerful terms, A382062.
The asymptotic density of this sequence is Sum_{n>=1} f(A382062(n)) = 0.72201619..., where f(n) = (n/zeta(2)) * Product_{prime p|n} (p/(p+1)).
The asymptotic density of a few subsequences can be evaluated more easily. For example:
1) Powerful numbers that are exponentially odd (A335988): When summing only over these numbers, the formula for the asymptotic density gives the density of the exponentially odd numbers: Product_{p prime} (1 - 1/(p*(p+1))) = 0.704442... (A065463).
2) Numbers of the form p^(2*k) * q^(2*m+1), where k and m >= 1, and p != q are primes: When summing only over these numbers, the density of the numbers whose powerful part is of this form is ((Sum_{p prime} p/((p^2-1)*(p+1))) * (Sum_{p prime} p^2/((p^4-1)*(p+1))) - Sum_{p prime} p^3/((p^2-1)^2*(p^2+1)*(p+1)^2)) / zeta(2) = 0.017174455422470834821... .

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

Cf. A000005, A013661, A034444, A057521, A065463, A382063.

Subsequences: A005117, A268335, A382062.

q[k_] := Divisible[DivisorSigma[0, k], 2^PrimeNu[k]]; Select[Range[100], q]

isok(k) = {my(f = factor(k)); !(numdiv(f) % (1<

2 is a term since A000005(2) = A034444(2) = 2, so 2 | 2.

24 is a term since A000005(24) = 8, A034444(24) = 4, and 4 | 8.

A382889 The largest square dividing the n-th cubefree number.

Original entry on oeis.org

1, 1, 1, 4, 1, 1, 1, 9, 1, 1, 4, 1, 1, 1, 1, 9, 1, 4, 1, 1, 1, 25, 1, 4, 1, 1, 1, 1, 1, 1, 36, 1, 1, 1, 1, 1, 1, 4, 9, 1, 1, 49, 25, 1, 4, 1, 1, 1, 1, 1, 4, 1, 1, 9, 1, 1, 1, 4, 1, 1, 1, 1, 1, 25, 4, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 9, 1, 4, 1, 1, 1, 1, 49, 9, 100

Offset: 1

Amiram Eldar, Apr 07 2025

Also, the powerful part of the n-th cubefree number.

All the terms are squares of squarefree numbers (A062503).

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

Cf. A002117, A004709, A008833, A057521, A062503, A371188 (positions of 1's).

Similar sequences: A382888, A382890, A382891.

f[p_, e_] := p^If[e == 1, 0, 2]; s[n_] := Module[{fct = FactorInteger[n]}, If[AllTrue[fct[[;; , 2]], # < 3 &], Times @@ f @@@ fct, Nothing]]; Array[s, 100]

list(lim) = {my(f); print1(1, ", "); for(k = 2, lim, f = factor(k); if(vecmax(f[, 2]) < 3, print1(prod(i = 1, #f~, f[i, 1]^if(f[i, 2] == 1, 0, 2)), ", ")));}

a(n) = A008833(A004709(n)).
a(n) = A057521(A004709(n)).
a(n) = A382890(n)^2.
a(n) = A004709(n)/A382891(n).
a(n) = (A004709(n)/A382888(n))^2.
a(A371188(n)) = 1.
Sum_{k=1..n} a(k) ~ c * n^(3/2) / 3, where c = zeta(3)^(3/2) * Product_{p prime} (1 + 1/p^(3/2) - 1/p^2 - 1/p^(5/2)) = 1.48513488319516447978... .

cp's OEIS Frontend

A056674 Number of squarefree divisors which are not unitary. Also number of unitary divisors which are not squarefree.

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A254503 Möbius transform of A034448.

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A338540 Numbers having exactly three non-unitary prime factors.

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A338541 Numbers having exactly four non-unitary prime factors.

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A338542 Numbers having exactly five non-unitary prime factors.

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A349442 Dirichlet convolution of A000027 (the identity function) with A349350 (Dirichlet inverse of the powerful part of n).

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A365552 The number of exponentially odd divisors of the powerful part of n.

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A375341 The maximum exponent in the prime factorization of the numbers that have exactly one non-unitary prime factor.

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