A056595 -id:A056595 - cp's OEIS frontend

A055205 Number of nonsquare divisors of n^2.

0, 1, 1, 2, 1, 5, 1, 3, 2, 5, 1, 9, 1, 5, 5, 4, 1, 9, 1, 9, 5, 5, 1, 13, 2, 5, 3, 9, 1, 19, 1, 5, 5, 5, 5, 16, 1, 5, 5, 13, 1, 19, 1, 9, 9, 5, 1, 17, 2, 9, 5, 9, 1, 13, 5, 13, 5, 5, 1, 33, 1, 5, 9, 6, 5, 19, 1, 9, 5, 19, 1, 23, 1, 5, 9, 9, 5, 19, 1, 17, 4, 5, 1, 33, 5, 5, 5, 13, 1, 33, 5, 9, 5, 5, 5

Offset: 1

Labos Elemer, Jun 19 2000

Seems to be equal to the number of unordered pairs of coprime divisors of n. (Checked up to 2*10^14.) - Charles R Greathouse IV, May 03 2013
Outline of a proof for this observation, R. J. Mathar, May 05 2013: (Start)
i) To construct the divisors of n, write n=product_i p_i^e_i as the standard prime power decomposition, take any subset of the primes p_i (including the empty set representing the 1) and run with the associated list exponents from 0 up to their individual e_i.
To construct the *nonsquare* divisors of n, ensure that one or more of the associated exponents is/are odd. (The empty set is interpreted as 1^0 with even exponent.) To construct the nonsquare divisors of n^2, the principle remains the same, although the exponents may individually range from 0 up to 2*e_i.
The nonsquare divisor is therefore a nonempty product of prime powers (at least one) with odd exponents times  a (potentially empty) product of prime powers (of different primes) with even exponents.
The nonsquare divisors of n^2 have exponents from 0 up to 2*e_i, but the subset of exponents in the "even/square" factor has e_i candidates (range 2, 4, .., 2*e_i) and in the "odd/nonsquare" factor also only e_i candidates (range 1,3,5,2*e_i-1).
ii) To construct the pairs of coprime divisors of n, take any two non-intersecting subsets of  the set of p_i (possibly the empty subset which represents the factor 1), and let the exponents run from 1 up to their individual e_i in each of the two products.
iii) The bijection between the sets constructed in i) and ii) is given by mapping the two non-intersection prime sets onto each other, and observing that the numbers of compositions of exponents have the same orders in both cases.
(End)

			n = 8, d(64) = 7 and from the 7 divisors {1,4,16,64} are square and the remaining 3 = a(8).
n = 12, d(144) = 15, from which 6 divisors are squares {1,4,9,16,36,144} so a(12) = d(144)-d(12) = 9
a(60) = (number of terms of finite A171425) = 33. [_Reinhard Zumkeller_, Dec 08 2009]

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)

Cf. A000005, A000290, A048691, A056595.

Cf. A001620, A013661, A082633, A201994, A306016.

a055205 n = length [d | d <- [1..n^2], n^2 `mod` d == 0, a010052 d == 0]
-- Reinhard Zumkeller, Aug 15 2011

Table[Count[Divisors[n^2], d_ /;  ! IntegerQ[Sqrt[d]]], {n, 1, 95}] (* Jean-François Alcover, Mar 22 2011 *)
Table[DivisorSigma[0,n^2]-DivisorSigma[0,n],{n,100}] (* Harvey P. Dale, Sep 02 2017 *)

a(n)=my(f=factor(n)[,2]);prod(i=1,#f,2*f[i]+1)-prod(i=1,#f,f[i]+1) \\ Charles R Greathouse IV, May 02 2013

a(n) = A000005(n^2)-A000005(n) because the number of square divisors of n^2 equals the number of divisors of n.
a(n) = A056595(A000290(n)).
a(n) = A048691(n) - A000005(n). - Reinhard Zumkeller, Dec 08 2009
Sum_{k=1..n} a(k) ~ (n/zeta(2)) * (log(n)^2/2 + c_1 * log(n) + c_2), where c_1 =  3*gamma - 2*zeta'(2)/zeta(2) - zeta(2) - 1 = 0.226634..., c_2 = 3*gamma^2 - (2*gamma - 1)*zeta(2) - 3*gamma_1 + (1 - 3*gamma)*(2*zeta'(2)/zeta(2) + 1) + (2*zeta'(2)/zeta(2))^2 - 2*zeta''(2)/zeta(2) = -0.0529271..., gamma is Euler's constant (A001620), and gamma_1 is the first Stieltjes constant (A082633). - Amiram Eldar, Dec 01 2023

A293514 a(n) = Product_{d|n, d>1} prime(A286561(n,d)), where A286561(n,d) gives the highest exponent of d dividing n.

Original entry on oeis.org

1, 2, 2, 6, 2, 8, 2, 20, 6, 8, 2, 48, 2, 8, 8, 84, 2, 48, 2, 48, 8, 8, 2, 320, 6, 8, 20, 48, 2, 128, 2, 264, 8, 8, 8, 864, 2, 8, 8, 320, 2, 128, 2, 48, 48, 8, 2, 2688, 6, 48, 8, 48, 2, 320, 8, 320, 8, 8, 2, 3072, 2, 8, 48, 1560, 8, 128, 2, 48, 8, 128, 2, 11520, 2, 8, 48, 48, 8, 128, 2, 2688, 84, 8, 2, 3072, 8, 8, 8, 320

Offset: 1

Antti Karttunen, Nov 11 2017

nonn

			For n = 24, its divisors larger than one are: 2, 3, 4, 6, 8, 12, 24. Only 2 has valuation > 1, namely A286561(24,2) = 3 (as 2^3 divides 24), while the other six have valuation 1. Thus a(24) = prime(1)^6 * prime(3) = 64*5 = 320.
For n = 64, its divisors larger than one are: 2, 4, 8, 16, 32, 64. We see that 2^6 = 4^3 = 8^2 = 64, while valuation of the last three 16, 32 and 64 is 1. Thus a(64) = prime(1)^3 * prime(2) * prime(3) * prime(6) = 2^3 * 3 * 5 * 13 = 1560.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences computed from exponents in factorization of n

Cf. A000040, A286561.

Cf. also A293515, A293524, A294876, A293442.

A293514(n) = { my(m=1); fordiv(n,d,if(d>1, m *= prime(valuation(n,d)))); m; };

a(n) = Product_{d|n, d>1} A000040(A286561(n,d)).
Other identities. For all n >= 1:
A001222(a(n)) = A032741(n).
A007814(a(n)) = A056595(n) [See A046951.]
1+A056239(a(n)) = A169594(n).
A064989(a(n)) = A293515(n).

A294873 a(n) = Product_{d|n, d>1, d = x^(2k-1) for some maximal k >= 1} prime(k).

Original entry on oeis.org

1, 2, 2, 2, 2, 8, 2, 6, 2, 8, 2, 16, 2, 8, 8, 6, 2, 16, 2, 16, 8, 8, 2, 96, 2, 8, 6, 16, 2, 128, 2, 30, 8, 8, 8, 32, 2, 8, 8, 96, 2, 128, 2, 16, 16, 8, 2, 192, 2, 16, 8, 16, 2, 96, 8, 96, 8, 8, 2, 1024, 2, 8, 16, 30, 8, 128, 2, 16, 8, 128, 2, 384, 2, 8, 16, 16, 8, 128, 2, 192, 6, 8, 2, 1024, 8, 8, 8, 96, 2, 1024, 8, 16, 8, 8, 8, 1920, 2, 16, 16, 32, 2, 128, 2

Offset: 1

Antti Karttunen, Nov 11 2017

nonn

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

Cf. A000040, A056595, A052409, A183096.

Cf. A293524, A294874, A294875.

A294873(n) = { my(m=1,e); fordiv(n,d, if(d>1, e = ispower(d); if(!e, m += m, if((e>1)&&(e%2), m *= prime((e+1)/2))))); m; };

a(n) = Product_{d|n, d>1, r = A052409(d) is odd} A000040((r+1)/2).
Other identities. For all n >= 1:
A001222(a(n)) = A056595(n).
A007814(a(n)) = A183096(n).

A285309 Sum of nonsquare divisors of n.

Original entry on oeis.org

0, 2, 3, 2, 5, 11, 7, 10, 3, 17, 11, 23, 13, 23, 23, 10, 17, 29, 19, 37, 31, 35, 23, 55, 5, 41, 30, 51, 29, 71, 31, 42, 47, 53, 47, 41, 37, 59, 55, 85, 41, 95, 43, 79, 68, 71, 47, 103, 7, 67, 71, 93, 53, 110, 71, 115, 79, 89, 59, 163, 61, 95, 94, 42, 83, 143, 67, 121, 95, 143, 71, 145, 73, 113, 98

Offset: 1

Ilya Gutkovskiy, Apr 16 2017

nonn

			a(6) = 11 because 6 has 4 divisors {1, 2, 3, 6} among which 3 are nonsquares {2, 3, 6} therefore 2 + 3 + 6 = 11.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences related to sums of divisors

Cf. A000037, A000203, A005117, A035316, A056595, A087153.

Table[DivisorSum[n, # &, Mod[DivisorSigma[0, #], 2] == 0 &], {n, 1, 75}]
nmax = 75; Rest[CoefficientList[Series[Sum[(k + Floor[1/2 + Sqrt[k]]) x^(k + Floor[1/2 + Sqrt[k]])/(1 - x^(k + Floor[1/2 + Sqrt[k]])), {k, 1, nmax}], {x, 0, nmax}], x]]
Array[DivisorSum[#, # &, ! IntegerQ@ Sqrt@ # &] &, 75] (* Michael De Vlieger, Nov 23 2017 *)

a(n) = sumdiv(n, d, if (!issquare(d), d)); \\ Michel Marcus, Apr 17 2017

import gmpy
from sympy import divisors
def a(n): return sum([d for d in divisors(n) if gmpy.is_square(d)==0]) # Indranil Ghosh, Apr 18 2017

G.f.: Sum_{k>=1} A000037(k)*x^A000037(k)/(1 - x^A000037(k)).
a(n) = A000203(n) - A035316(n).
a(A005117(n)) = A000203(A005117(n)) - 1.
a(p^(2*k-1)) = a(p^(2*k)) = p*(p^(2*k) - 1)/(p^2 - 1) for p is a prime and k >= 1.

A291208 Number of noncube divisors of n.

Original entry on oeis.org

0, 1, 1, 2, 1, 3, 1, 2, 2, 3, 1, 5, 1, 3, 3, 3, 1, 5, 1, 5, 3, 3, 1, 6, 2, 3, 2, 5, 1, 7, 1, 4, 3, 3, 3, 8, 1, 3, 3, 6, 1, 7, 1, 5, 5, 3, 1, 8, 2, 5, 3, 5, 1, 6, 3, 6, 3, 3, 1, 11, 1, 3, 5, 4, 3, 7, 1, 5, 3, 7, 1, 10, 1, 3, 5, 5, 3, 7, 1, 8, 3, 3, 1, 11, 3, 3, 3, 6, 1, 11, 3, 5, 3, 3, 3, 10, 1, 5, 5, 8, 1, 7, 1, 6, 7

Offset: 1

Ilya Gutkovskiy, Aug 21 2017

nonn

			a(8) = 2 because 8 has 4 divisors {1, 2, 4, 8} among which 2 are noncubes {2, 4}.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences computed from exponents in factorization of n.

Cf. A000005, A007412, A056595, A061704.

Cf. A001620, A002117.

nmax = 105; Rest[CoefficientList[Series[Sum[(x^k - x^k^3)/((1 - x^k) (1 - x^k^3)), {k, 1, nmax}], {x, 0, nmax}], x]]
f1[p_, e_] := e + 1; f2[p_, e_] := 1 + Floor[e/3]; a[1] = 0; a[n_] := Module[{fct = FactorInteger[n]}, Times @@ f1 @@@ fct - Times @@ f2 @@@ fct]; Array[a, 100] (* Amiram Eldar, Jan 30 2025 *)

a(n) = sumdiv(n, d, !ispower(d, 3)); \\ Michel Marcus, Aug 21 2017

from math import prod
from sympy import factorint
def A291208(n):
    f = factorint(n).values()
    return prod(e+1 for e in f)-prod(e//3+1 for e in f) # Chai Wah Wu, Jun 05 2025

G.f.: Sum_{k>=1} x^A007412(k)/(1 - x^A007412(k)).
G.f.: Sum_{k>=1} (x^k - x^(k^3))/((1 - x^k)*(1 - x^(k^3))).
a(n) = A000005(n) - A061704(n).
From Amiram Eldar, Jan 30 2025: (Start)
Dirichlet g.f.: zeta(s) * (zeta(s) - zeta(3*s)).
Sum_{k=1..n} a(k) ~ n*(log(n) + 2*gamma - zeta(3) - 1), where gamma is Euler's constant (A001620). (End)

A293575 Difference between the number of proper divisors of n and the number of squares dividing n.

Original entry on oeis.org

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

Offset: 1

Juri-Stepan Gerasimov, Oct 14 2017

sign

The difference between the number of ways of writing n = m + k and the number of ways of writing n = r*s, where m|k and r|s.

First occurrence of k beginning with k=-1: 1, 2, 8, 6, 12, 36, 24, 30, 72, 96, 60, 2097152, 216, 576, 120, 210, 1152, 240, 864, etc. - Robert G. Wilson v, Nov 28 2017

			a(6) = 2 because 2 is difference of number of ways of writing n = 1 + 5 = 2 + 4 = 3 + 3 where 1|5, 2|4, 3|3 and number of ways of writing n = 1*6 where 1|6.

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

One less than A056595.
Cf. A000430, A032741, A046951, A166684, A080257.
Cf. A001620, A013661.

f[n_] := Block[{d = Divisors@ n}, Length@ d - Length[ Select[ d, IntegerQ@ Sqrt@# &]] - 1];; Array[f, 105] (* Robert G. Wilson v, Nov 28 2017 *)

a(n) = A032741(n) - A046951(n).
a(n) = A056595(n) - 1. - Antti Karttunen, Oct 30 2017
a(n) = 0 iff n is a prime or a square of a prime, A000430. - Robert G. Wilson v, Nov 28 2017
Sum_{k=1..n} a(k) ~ n*log(n) + (2*gamma - zeta(2) - 2)*n, where gamma is Euler's constant (A001620). - Amiram Eldar, Dec 01 2023

A345320 Sum of the divisors of n whose square does not divide n.

Original entry on oeis.org

0, 2, 3, 4, 5, 11, 7, 12, 9, 17, 11, 25, 13, 23, 23, 24, 17, 35, 19, 39, 31, 35, 23, 57, 25, 41, 36, 53, 29, 71, 31, 56, 47, 53, 47, 79, 37, 59, 55, 87, 41, 95, 43, 81, 74, 71, 47, 117, 49, 87, 71, 95, 53, 116, 71, 117, 79, 89, 59, 165, 61, 95, 100, 112, 83, 143, 67, 123, 95, 143, 71, 183, 73, 113, 118, 137, 95, 167, 79, 179, 108, 125, 83

Offset: 1

Wesley Ivan Hurt, Jun 13 2021

nonn

Inverse Möbius transform of n+n^(1/2)*((-1)^tau(n)-1)/2. - Wesley Ivan Hurt, Jul 07 2025

			a(16) = 24; The divisors of 16 whose square does not divide 16 are 8 and 16. The sum of the divisors is then 8 + 16 = 24.

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

Cf. A000005 (tau), A000203 (sigma), A056595, A069290.

Table[Sum[k (Ceiling[n/k^2] - Floor[n/k^2]) (1 - Ceiling[n/k] + Floor[n/k]), {k, n}], {n, 80}]
sdnd[n_]:=Total[Select[Divisors[n],Mod[n,#^2]!=0&]]; Array[sdnd,100] (* Harvey P. Dale, Jul 07 2025 *)

a(n) = sumdiv(n, d, if (n % d^2, d)); \\ Michel Marcus, Jun 13 2021

from math import prod
from sympy import factorint
def A345320(n):
    f = factorint(n).items()
    return (prod(p**(q+1)-1 for p, q in f) - prod(p**(q//2+1)-1 for p, q in f))//prod(p-1 for p, q in f) # Chai Wah Wu, Jun 14 2021

a(n) = Sum_{k=1..n} k * (ceiling(n/k^2) - floor(n/k^2)) * (1 - ceiling(n/k) + floor(n/k)).
a(n) = A000203(n) - A069290(n). - Rémy Sigrist, Jun 14 2021
a(n) = Sum_{d|n} (d+d^(1/2)*((-1)^tau(d)-1)/2). - Wesley Ivan Hurt, Jul 07 2025

A349330 a(n) = Sum_{d|n} d^c(d), where c is the characteristic function of squares (A010052).

Original entry on oeis.org

1, 2, 2, 6, 2, 4, 2, 7, 11, 4, 2, 9, 2, 4, 4, 23, 2, 14, 2, 9, 4, 4, 2, 11, 27, 4, 12, 9, 2, 8, 2, 24, 4, 4, 4, 55, 2, 4, 4, 11, 2, 8, 2, 9, 14, 4, 2, 28, 51, 30, 4, 9, 2, 16, 4, 11, 4, 4, 2, 15, 2, 4, 14, 88, 4, 8, 2, 9, 4, 8, 2, 58, 2, 4, 30, 9, 4, 8, 2, 28, 93, 4, 2, 15, 4, 4, 4

Offset: 1

Wesley Ivan Hurt, Nov 15 2021

nonn

For each divisor d of n, add d if d is a square, otherwise add 1 [see example].

Inverse Möbius transform of n^c(n), where c = A010052. - Wesley Ivan Hurt, Mar 31 2025

			The divisors of 12 are 1, 2, 3, 4, 6, and 12 with squares 1 and 4, so a(12) = 1 + 1 + 1 + 4 + 1 + 1 = 9 (respectively).

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

Cf. A010052, A035316.

a[n_] := DivisorSum[n, If[IntegerQ @ Sqrt[#], #, 1] &]; Array[a, 100] (* Amiram Eldar, Nov 15 2021 *)

a(n) = sumdiv(n, d, if (issquare(d), d, 1)); \\ Michel Marcus, Nov 15 2021

a(n) = {my(f = factor(n), cf = f, res); cf[,2]\=2; res = numdiv(f)-prod(i = 1, #f~, cf[i, 2]+1); res+=prod(i = 1, #f~, ((f[i,1]^(2*(cf[i,2]+1))-1)/(f[i,1]^2-1))); res } \\ David A. Corneth, Nov 16 2021

a(p) = 2 iff p is prime. - Wesley Ivan Hurt, Nov 28 2021

a(n) = A035316(n) + A056595(n). - R. J. Mathar, Aug 18 2024

A345446 Number of semiprime divisors of n whose square does not divide n.

Original entry on oeis.org

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

Offset: 1

Wesley Ivan Hurt, Jun 19 2021

nonn

a(p) = 0 for p prime.

Cf. A001358, A056595.

Table[Count[Select[Divisors[n],PrimeOmega[#]==2&],?(!IntegerQ[n/#^2]&)],{n,120}] (* _Harvey P. Dale, Aug 10 2023 *)

a(n) = Sum_{d|n} [Omega(d) = 2] * (ceiling(n/d^2) - floor(n/d^2)), where [ ] is the Iverson bracket.

A361634 Integers whose number of square divisors is coprime to the number of their nonsquare divisors.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, 25, 26, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 46, 47, 48, 49, 51, 53, 55, 57, 58, 59, 61, 62, 64, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 80, 81, 82, 83, 85, 86, 87, 89, 91, 93, 94

Offset: 1

Waldemar Puszkarz, Mar 19 2023

nonn

Appears to be a supersequence of A210490, and so also of positive squares and squarefree numbers (A005117). The first term that belongs in here but not in A210490 is 48. The nonsquarefree terms that are not squares are of the form p^(4k)*a, where a is a squarefree number, p is prime, and k > 0. About half of perfect numbers are of this form; one example is 496 = 2^4*31. The sequence has an asymptotic density of about 0.6420.

			48 has 3 square divisors (1, 4, and 16) and 7 nonsquare ones. Consequently, gcd(3,7)=1, thus 48 is a term.

Index entries for sequences related to divisors of numbers

Cf. A210490, A005117 (subsequences), A046951 (number of square divisors), A056595 (number of nonsquare divisors).

Select[Range[100],CoprimeQ[Total@(Boole/@IntegerQ/@Sqrt/@Divisors[#]),DivisorSigma[0,#]-Total@(Boole/@IntegerQ/@Sqrt/@Divisors[#])]&]

for(n=1, 100, a=divisors(n); c=0; for(i=1, #a, issquare(a[i])&&c++); gcd(#a-c, c)==1&&print1(n, ", "))

isok(n) = gcd(numdiv(n), numdiv(sqrtint(n/core(n))))==1 \\ Andrew Howroyd, Mar 19 2023

cp's OEIS Frontend

A055205 Number of nonsquare divisors of n^2.

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A293514 a(n) = Product_{d|n, d>1} prime(A286561(n,d)), where A286561(n,d) gives the highest exponent of d dividing n.

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A294873 a(n) = Product_{d|n, d>1, d = x^(2k-1) for some maximal k >= 1} prime(k).

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A285309 Sum of nonsquare divisors of n.

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A291208 Number of noncube divisors of n.

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A293575 Difference between the number of proper divisors of n and the number of squares dividing n.

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A345320 Sum of the divisors of n whose square does not divide n.

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