A048691 -id:A048691 - cp's OEIS frontend

A061680 a(n) = gcd(d(n^2), d(n)).

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 3, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 3, 1, 1, 1, 1, 3, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 3, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 3, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 3, 1, 3, 1, 1, 1, 3, 1, 3, 3, 1, 1, 1, 1, 1, 1

Offset: 1

Labos Elemer, Jun 18 2001

nonn

			This GCD can only be odd since d(n^2) is odd.
For n = 4608: a(4608) = gcd(d(21233664), d(4608)) = gcd(95, 30) = 5.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..2000 from Harry J. Smith)

Cf. A000005, A000290, A048691.

Table[GCD[DivisorSigma[0,n],DivisorSigma[0,n^2]],{n,110}] (* Harvey P. Dale, Sep 03 2023 *)

a(n) = gcd(numdiv(n^2), numdiv(n)); \\ Harry J. Smith, Jul 26 2009

a(n) = {my(e = factor(n)[,2]); gcd(vecprod(apply(x -> 2*x+1, e)), vecprod(apply(x -> x+1, e)));} \\ Amiram Eldar, Dec 02 2023

a(n) = gcd(A000005(A000290(n)), A000005(n)) = gcd(A048691(n), A000005(n)).

Offset changed from 0 to 1 by Harry J. Smith, Jul 26 2009

A063774 Numbers k such that the number of divisors of k^2 is a square.

Original entry on oeis.org

1, 6, 10, 14, 15, 16, 21, 22, 26, 33, 34, 35, 36, 38, 39, 46, 51, 55, 57, 58, 62, 65, 69, 74, 77, 81, 82, 85, 86, 87, 91, 93, 94, 95, 100, 106, 111, 115, 118, 119, 122, 123, 129, 133, 134, 141, 142, 143, 145, 146, 155, 158, 159, 161, 166, 177, 178, 183, 185, 187, 194

Offset: 1

Jason Earls, Aug 15 2001

The numbers of terms not exceeding 10^k, for k = 1, 2, ..., are 3, 35, 326, 3275, 33090, 332435, 3327555, 33283964, 332868092, 3328794682, ... . Apparently, the asymptotic density of this sequence exists and equals 0.3328... . - Amiram Eldar, Nov 28 2023

			n=2: a(2) = 6 because the number of divisors of 6^2 is 9, a square.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harry J. Smith)

Cf. A000290, A048691.

Subsequences: A030229, A238748.

Select[Range[200],IntegerQ[Sqrt[DivisorSigma[0,#^2]]]&] (* Harvey P. Dale, Jun 06 2012 *)

j=[]; for(n=1,500,a=numdiv(n^2); if(issquare(a),j=concat(j,n))); j

n=0; for (m=1, 10^9, if(issquare(numdiv(m^2)), write("b063774.txt", n++, " ", m); if (n==1000, break))) \\ Harry J. Smith, Aug 30 2009

is(n)=my(f=factor(n)[,2]); issquare(prod(i=1,#f,2*f[i]+1)) \\ Charles R Greathouse IV, Sep 18 2015

{n: A048691(n) in A000290}. - R. J. Mathar, Aug 09 2012

A086165 a(n) = |{ (x,y,z) | x < y < z and lcm(x,y,z) = n}|.

Original entry on oeis.org

0, 0, 0, 1, 0, 4, 0, 3, 1, 4, 0, 15, 0, 4, 4, 6, 0, 15, 0, 15, 4, 4, 0, 33, 1, 4, 3, 15, 0, 44, 0, 10, 4, 4, 4, 48, 0, 4, 4, 33, 0, 44, 0, 15, 15, 4, 0, 58, 1, 15, 4, 15, 0, 33, 4, 33, 4, 4, 0, 133, 0, 4, 15, 15, 4, 44, 0, 15, 4, 44, 0, 100, 0, 4, 15, 15, 4, 44, 0, 58, 6, 4, 0, 133, 4, 4, 4, 33, 0

Offset: 1

Yuval Dekel (dekelyuval(AT)hotmail.com), Sep 13 2003

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

Cf. A048691, A070919, A086222.

for n from 1 to 100 do a[n] := 0:for x from 1 to n do for y from x+1 to n do for z from y+1 to n do if(lcm(x,y,z)=n) then a[n] := a[n]+1:fi:od:od:od:od:seq(a[j],j=1..200); # Sascha Kurz, Sep 22 2003

f1[p_, e_] := (e+1)^3 - e^3; f2[p_, e_] := 2*e + 1; a[1] = 0; a[n_] := (Times @@ f1 @@@ (f = FactorInteger[n]) - 3 * Times @@ f2 @@@f + 2) / 6; Array[a, 100] (* Amiram Eldar, Sep 03 2023 *)

A048691(n) = numdiv(n^2);
A070919(n) = sumdiv(n, d, (numdiv(d)^3)*moebius(n/d));
A086165(n) = ((A070919(n)-3*A048691(n)+2)/6); \\ Antti Karttunen, May 19 2017, after Jovovic's formula

a(n) = {my(e = factor(n)[, 2]); (vecprod(apply(x->(x+1)^3-x^3, e)) - 3*vecprod(apply(x->2*x+1, e)) + 2) / 6;} \\ Amiram Eldar, Sep 03 2023

a(n) = (A070919(n) - 3*A048691(n) + 2)/6. - Vladeta Jovovic, Dec 01 2004

a(n) = A086222(n) - A048691(n). - Ridouane Oudra, Aug 14 2025

More terms from Sascha Kurz, Sep 22 2003

A093616 a(n) is the smallest k such that k*n has exactly as many divisors as n^2.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 8, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 8, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72

Offset: 1

Reinhard Zumkeller, Apr 06 2004

nonn

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

Cf. A000005, A000290, A048691, A093617, A093618.

a[n_] := For[k = 1, True, k++, If[DivisorSigma[0, k*n] == DivisorSigma[0, n^2], Return[k]]]; Array[a, 72] (* Jean-François Alcover, Aug 14 2014 *)

a(n) = {my(k = 1, d = numdiv(n^2)); while(numdiv(k*n) != d, k++); k;} \\ Amiram Eldar, Apr 15 2024

A000005(a(n)*n) = A000005(n^2) and A000005(m*n) <> A000005(n^2) for m < a(n).

a(A093617(n)) < n, a(A093618(n)) = n.

A093617 Numbers m such that there exists a number k less than m with k*m and m^2 having an equal number of divisors.

Original entry on oeis.org

18, 50, 75, 90, 98, 108, 126, 144, 147, 150, 198, 234, 242, 245, 294, 300, 306, 324, 338, 342, 350, 363, 384, 400, 414, 450, 490, 500, 507, 522, 525, 540, 550, 558, 578, 588, 600, 605, 630, 640, 648, 650, 666, 720, 722, 726, 735, 738, 756, 774, 784, 825

Offset: 1

Reinhard Zumkeller, Apr 06 2004

nonn

From Amiram Eldar, Apr 15 2024: (Start)
All the terms are nonsquarefree numbers (A013929).
The number k is of the form j^2*A007913(m).
The numbers of terms that do not exceed 10^k, for k = 1, 2, ..., are 0, 5, 64, 678, 6954, 69867, 699511, 6996322, 69962916, 699616048, ... . Apparently, the asymptotic density of this sequence exists and equals 0.06996... . (End)

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..300 from Vincenzo Librandi)

Complement of A093618.

Cf. A000005, A000290, A007913, A013929, A048691, A093616.

A093616[n_] := For[k = 1, True, k++, If[DivisorSigma[0, k*n] == DivisorSigma[0, n^2], Return[k]]]; Select[Range[1000], A093616[#] < # &] (* Jean-François Alcover, Aug 14 2014 *)
f[p_, e_] := p^(e + Mod[e, 2]); q[n_] := Module[{fct = FactorInteger[n], d, m, k = 1}, d = Times @@ ((2*# + 1) & /@ fct[[;; , 2]]); s = Times @@ f @@@ fct; m = Sqrt[n^2/s]; While[k < m && DivisorSigma[0, k^2*s] != d, k++]; k < m]; Select[Range[1000], q] (* Amiram Eldar, Apr 15 2024 *)

is(n) = {my(f = factor(n), d = prod(i = 1, #f~, 2*f[i, 2] + 1), s = prod(i = 1, #f~, f[i, 1]^(f[i, 2] + f[i, 2]%2)), m = sqrtint(n^2/s), k = 1); while(k < m && numdiv(k^2 * s) != d, k++); k < m;} \\ Amiram Eldar, Apr 15 2024

A093616(a(n)) < n.

A093618 Numbers m such that for all k less than m the number of divisors of k*m and m^2 are different.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74

Offset: 1

Reinhard Zumkeller, Apr 06 2004

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1800 from Vincenzo Librandi)

Complement of A093617.

Cf. A000005, A000290, A048691, A093616.

A093616[n_] := For[k = 1, True, k++, If[DivisorSigma[0, k*n] == DivisorSigma[0, n^2], Return[k]]]; Select[Range[100], A093616[#] == # &] (* Jean-François Alcover, Aug 14 2014 *)

is(n) = {my(k = 1, d = numdiv(n^2)); while(k < n && numdiv(k*n) != d, k++); k == n;} \\ Amiram Eldar, Apr 15 2024

A093616(a(n)) = n.

Corrected by Franklin T. Adams-Watters, Dec 19 2006

A166722 a(n) is the number of divisors of A166721(n).

Original entry on oeis.org

1, 3, 5, 9, 7, 15, 21, 27, 11, 25, 45, 13, 35, 33, 63, 75, 39, 81, 49, 17, 55, 105, 135, 99, 19, 65, 51, 189, 77, 125, 117, 147, 225, 165, 57, 243, 91, 175, 23, 85, 315, 195, 297, 153, 231, 95, 405, 245, 69, 375, 351, 119, 275, 441, 171, 121, 273, 567, 495, 255, 525

Offset: 1

Alexander Isaev (i2357(AT)mail.ru), Oct 20 2009

This is a permutation of the odd numbers A005408. - Alois P. Heinz, Mar 04 2018

			a(8) = A000005(A166721(8)) = A000005(900) = A000005(2^2 * 3^2 * 5^2) = (2+1)*(2+1)*(2+1) = 27.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..300 from Alois P. Heinz)

Cf. A000005, A005408, A048691, A136404, A152674, A166721.

a(n) = A000005(A166721(n)).

Proper definition (and removal of obscure Comments entries) by Jon E. Schoenfield, Mar 03 2018

A182139 Inverse Moebius transform of A061142.

Original entry on oeis.org

1, 3, 3, 7, 3, 9, 3, 15, 7, 9, 3, 21, 3, 9, 9, 31, 3, 21, 3, 21, 9, 9, 3, 45, 7, 9, 15, 21, 3, 27, 3, 63, 9, 9, 9, 49, 3, 9, 9, 45, 3, 27, 3, 21, 21, 9, 3, 93, 7, 21, 9, 21, 3, 45, 9, 45, 9, 9, 3, 63, 3, 9, 21, 127, 9, 27, 3, 21, 9, 27, 3, 105, 3, 9, 21, 21, 9

Offset: 1

Enrique Pérez Herrero, Apr 14 2012

a(n) is multiplicative with a(p^e) = -1 + 2^(e+1).
If s is squarefree then a(s) = A048691(s).
More generally: Let a_q(n) be multiplicative with a_q(p^e) = (q^(e+1)-1)/ (q-1) for prime p, e >= 0 and some fixed integer q. Then a_q(n) is the inverse Moebius transform of the completely multiplicative sequence b_q(n) = q^bigomega(n) with b_q(p) = q and b_q(1) = 1. For q = 1 see a_q(n) = A000005(n) and b_q(n) = A000012(n), for q = 0 see a_q(n) = A000012(n) and b_q(n) = A000007(n) with offset 1, and for q = -1 see a_q(n) = A010052(n) with offset 1 and b_q(n) = A008836(n). - Werner Schulte, Feb 20 2019

			a(12) = a(2^2 * 3^1) = (-1 + 2^(2+1)) * (-1 + 2^(1+1)) = 7 * 3 = 21; or, using the divisors set {1,2,3,4,6,12}: 2^0 + 2^1 + 2^1 + 2^2 + 2^2 + 2^3 = 21.

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

Cf. A001222, A061142, A048691.

t[n_] := DivisorSum[n, 2^PrimeOmega[#]&]; Table[t[n], {n,100}]

for(n=1, 100, print1(direuler(p=2, n, 1/(1 - X)/(1 - 2*X))[n], ", ")) \\ Vaclav Kotesovec, Mar 14 2023

a(n) = Sum_{d|n} 2^Omega(d) = Sum_{d|n} 2^A001222(d) = Sum_{d|n} A061142(d).

Dirichlet g.f.: zeta(s)^3 * Product_{p prime} 1/(1 - 1/(p^s - 1)^2).

A276553 Numbers n such that n^2 and (n + 1)^2 have the same number of divisors.

Original entry on oeis.org

2, 14, 15, 21, 33, 34, 38, 44, 57, 75, 81, 85, 86, 93, 94, 98, 116, 118, 122, 133, 135, 141, 142, 145, 147, 158, 171, 177, 201, 202, 205, 213, 214, 217, 218, 230, 244, 253, 272, 285, 296, 298, 301, 302, 326, 332, 334, 375, 381, 387, 393, 394, 405, 429, 434, 445

Offset: 1

K. D. Bajpai, Apr 10 2017

nonn

Except for a(1), all the terms are composite.

			We see that 14^2 = 196, the divisors of which are 1, 2, 4, 7, 14, 28, 49, 98, 196, and there are nine of them. And we see that 15^2 = 225, the divisors of which are 1, 3, 5, 9, 15, 25, 45, 75, 225, and there are nine of them. Both 14^2 and 15^2 have the same number of divisors, hence 14 is in the sequence.
And we see that 16^2 = 256, the divisors of which are the powers of 2 from 2^0 to 2^8, that's nine divisors. Both 15^2 and 16^2 have the same number of divisors, hence 15 is also in the sequence.
But 16 is not in the sequence, since 17 is prime and 17^2 consequently only has three divisors.

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

Cf. A000290, A048691, A062832, A276542, A284378.
Cf. A052213 (a subsequence).
Positions of zeros in A284570.

N:= 1000: # to get all terms <= N
T:= map(t -> numtheory:-tau(t^2), [$1..N+1]):
select(t -> T[t]=T[t+1], [$1..N]); # Robert Israel, Apr 10 2017

Select[Range[1000], DivisorSigma[0, #^2] == DivisorSigma[0, (# + 1)^2] &]

k=[]; for(n=1, 1000, a=numdiv(n^2); b=numdiv((n+1)^2); if(a==b, k=concat(k, n))); k

from sympy.ntheory import divisor_count
print([n for n in range(1, 501) if divisor_count(n**2) == divisor_count((n + 1)**2)]) # Indranil Ghosh, Apr 10 2017
(Scheme, with Antti Karttunen's IntSeq-library) (define A276553 (ZERO-POS 1 1 A284570)) ;; Antti Karttunen, Apr 15 2017

A283262 Numbers m such that tau(m^2) is a prime.

Original entry on oeis.org

2, 3, 4, 5, 7, 8, 9, 11, 13, 17, 19, 23, 25, 27, 29, 31, 32, 37, 41, 43, 47, 49, 53, 59, 61, 64, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 121, 125, 127, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227

Offset: 1

Jaroslav Krizek, Mar 08 2017

nonn

tau(m) is the number of positive divisors of m (A000005).
Numbers m such that A000005(A000290(m)) = A048691(m) is a prime.
Union of A000040 (primes) and A051676.
Supersequence of A055638 (sigma(m^2) is prime).
Subsequence of A000961 (powers of primes).
Prime powers p^e with 2e+1 prime (e >= 1).
See A061285(m) = the smallest number k such that tau(k^2) = m-th prime.

			tau(4^2) = tau(16) = 5 (prime).

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000005, A000290, A000961, A048691, A051676, A055638, A061285.

[n: n in [2..100000] | IsPrime(NumberOfDivisors(n^2))];

N:= 1000: # to get all terms <= N
es:= select(t -> isprime(2*t+1), [$1..ilog2(N)]):
Ps:= select(isprime, [2,seq(i,i=3..N,2)]):
sort(select(`<=`, [seq(seq(p^e,e=es),p=Ps)],N)): # Robert Israel, Mar 16 2017

Select[Range@ 227, PrimeQ[DivisorSigma[0, #^2]] &] (* Michael De Vlieger, Mar 09 2017 *)

isok(n)=isprime(numdiv(n^2)) \\ Indranil Ghosh, Mar 09 2017

cp's OEIS Frontend

A061680 a(n) = gcd(d(n^2), d(n)).

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A063774 Numbers k such that the number of divisors of k^2 is a square.

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A086165 a(n) = |{ (x,y,z) | x < y < z and lcm(x,y,z) = n}|.

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A093616 a(n) is the smallest k such that k*n has exactly as many divisors as n^2.

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A093617 Numbers m such that there exists a number k less than m with k*m and m^2 having an equal number of divisors.

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A093618 Numbers m such that for all k less than m the number of divisors of k*m and m^2 are different.

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A166722 a(n) is the number of divisors of A166721(n).

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