A211338 -id:A211338 - cp's OEIS frontend

A059269 Numbers m for which the number of divisors, tau(m), is divisible by 3.

4, 9, 12, 18, 20, 25, 28, 32, 36, 44, 45, 49, 50, 52, 60, 63, 68, 72, 75, 76, 84, 90, 92, 96, 98, 99, 100, 108, 116, 117, 121, 124, 126, 132, 140, 144, 147, 148, 150, 153, 156, 160, 164, 169, 171, 172, 175, 180, 188, 196, 198, 200, 204, 207, 212, 220, 224, 225, 228

Offset: 1

Avi Peretz (njk(AT)netvision.net.il), Jan 24 2001

tau(n) is divisible by 3 iff at least one prime in the prime factorization of n has exponent of the form 3*m + 2. This sequence is an extension of the sequence A038109 in which the numbers has at least one prime with exponent 2 (the case of m = 0 here ) in their prime factorization.
The union of A211337 and A211338 is the complementary sequence to this one. - Douglas Latimer, Apr 12 2012
Numbers whose cubefree part (A050985) is not squarefree (A005117). - Amiram Eldar, Mar 09 2021

			a(7) = 28 is a term because the number of divisors of 28, d(28) = 6, is divisible by 3.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Eckford Cohen, Arithmetical Notes, XIII. A Sequal to Note IV, Elemente der Mathematik, Vol. 18 (1963), pp. 8-11.
S. S. Pillai, On a congruence property of the divisor function, J. Indian Math. Soc. (N. S.), Vol. 6, (1942), pp. 118-119.
L. G. Sathe, On a congruence property of the divisor function, American Journal of Mathematics, Vol. 67, No. 3 (1945), pp. 397-406.

Cf. A000005, A005117, A038109, A050985, A211337, A211338, A253905.

Characteristic function: A353470.

with(numtheory): for n from 1 to 1000 do if tau(n) mod 3 = 0 then printf(`%d,`,n) fi: od:

Select[Range[230], Divisible[DivisorSigma[0, #], 3] &] (* Amiram Eldar, Jul 26 2020 *)

is(n)=vecmax(factor(n)[,2]%3)==2 \\ Charles R Greathouse IV, Apr 10 2012

is(n)=numdiv(n)%3==0 \\ Charles R Greathouse IV, Sep 18 2015

Conjecture: a(n) ~ k*n where k = 1/(1 - Product(1 - (p-1)/(p^(3*i)))) = 3.743455... where p ranges over the primes and i ranges over the positive integers. - Charles R Greathouse IV, Apr 13 2012
The asymptotic density of this sequence is 1 - zeta(3)/zeta(2) = 1 - 6*zeta(3)/Pi^2 = 0.2692370305... (Sathe, 1945). Therefore, the above conjecture, a(n) ~ k*n, is true, but k = 1/(1-6*zeta(3)/Pi^2) = 3.7141993349... - Amiram Eldar, Jul 26 2020
A001248 UNION A030515 UNION A030627 UNION A030630 UNION A030633 UNION A030636 UNION ... - R. J. Mathar, May 05 2023

More terms from James Sellers, Jan 24 2001

A211337 Numbers k for which the number of divisors, tau(k), is congruent to 1 modulo 3.

Original entry on oeis.org

1, 6, 8, 10, 14, 15, 21, 22, 26, 27, 33, 34, 35, 38, 39, 46, 48, 51, 55, 57, 58, 62, 64, 65, 69, 74, 77, 80, 82, 85, 86, 87, 91, 93, 94, 95, 106, 111, 112, 115, 118, 119, 120, 122, 123, 125, 129, 133, 134, 141, 142, 143, 145, 146, 155, 158, 159, 161, 162

Offset: 1

Douglas Latimer, Apr 07 2012

nonn

Any term a(n) can be expressed as 1 term from A211484 times 1 nonzero term from A000578. - Douglas Latimer, Apr 20 2012

The numbers of terms not exceeding 10^k, for k = 1, 2, ..., are 4, 36, 366, 3635, 36499, 365456, 3654240, 36538501, 365382167, 3653804173, ... . Conjecture: the asymptotic density of this sequence exists and equals 3*zeta(3)/Pi^2 = 0.3653814847007... (A346602), so, a(n) ~ k*n with k = Pi^2/(3*zeta(3)) = 2.73686555524... . This conjecture is true if this sequence and A211338 have the same density (see A059269). - Amiram Eldar, Jan 06 2024

			The divisors of 10  are: 1, 2, 5, 10 (4 divisors). 4 is congruent to 1 modulo 3. Thus 10 is a member of this sequence.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Douglas Latimer)

This is an extension of A030513 (numbers with 4 divisors).
The union of A059269 and A211338 is the complementary sequence to this one.
The definition of this sequence uses A000005 (the number of divisors of n).
Cf. A000578, A074794, A211484, A346602.

Select[Range[162], Mod[DivisorSigma[0, #], 3] == 1 &] (* T. D. Noe, Apr 21 2012 *)

{plnt=1 ; mxind=100 ;for(k=1, 10^6,
if(numdiv(k) % 3 == 1, print(k); plnt++; if(mxind+1 ==  plnt, break() )))}

Conjecture: a(n) ~ k*n where k = 2/prod(1 - (p-1)/(p^(3*k))) = 2.7290077... where p ranges over the primes and k ranges over the positive integers. - Charles R Greathouse IV, Apr 13 2012

A336590 Numbers k such that k/A008834(k) is squarefree, where A008834(k) is the largest cube dividing k.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 8, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, 24, 26, 27, 29, 30, 31, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 46, 47, 48, 51, 53, 54, 55, 56, 57, 58, 59, 61, 62, 64, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 80, 81, 82, 83, 85, 86, 87, 88

Offset: 1

Amiram Eldar, Jul 26 2020

nonn

Numbers such that none of the exponents in their prime factorization is of the form 3*m + 2.
Cohen (1962) proved that for a given number k >= 2 the asymptotic density of numbers whose exponents in their prime factorization are of the forms k*m or k*m + 1 only is zeta(k)/zeta(2). In this sequence k = 3, and therefore its asymptotic density is zeta(3)/zeta(2) = 6*zeta(3)/Pi^2 = 0.7307629694... (A253905).
Also, numbers k whose number of divisors, A000005(k), is not divisible by 3, i.e., complement of A059269.

			6 is a term since 6 = 2^1 * 3^1 and 1 is not of the form 3*m + 2.
9 is not a term since 9 = 3^2 and 2 is of the form 3*m + 2.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eckford Cohen, Arithmetical notes. III. Certain equally distributed sets of integers, Pacific Journal of Mathematics, No. 12, Vol. 1 (1962), pp. 77-84.
Eckford Cohen, Arithmetical Notes, XIII. A Sequal to Note IV, Elemente der Mathematik, Vol. 18 (1963), pp. 8-11.
L. G. Sathe, On a congruence property of the divisor function, American Journal of Mathematics, Vol. 67, No. 3 (1945), pp. 397-406.

Cf. A000005, A002117, A005117, A008834, A013661, A059269, A253905.

Union of A211337 and A211338.

Select[Range[100], Max[Mod[FactorInteger[#][[;;,2]], 3]] < 2 &]

A074796 Number of numbers k <= n such that tau(k) == 2 (mod 3) where tau(k) = A000005(k) is the number of divisors of k.

Original entry on oeis.org

0, 1, 2, 2, 3, 3, 4, 4, 4, 4, 5, 5, 6, 6, 6, 7, 8, 8, 9, 9, 9, 9, 10, 11, 11, 11, 11, 11, 12, 13, 14, 14, 14, 14, 14, 14, 15, 15, 15, 16, 17, 18, 19, 19, 19, 19, 20, 20, 20, 20, 20, 20, 21, 22, 22, 23, 23, 23, 24, 24, 25, 25, 25, 25, 25, 26, 27, 27, 27, 28, 29, 29, 30, 30, 30, 30

Offset: 1

Benoit Cloitre, Sep 07 2002

nonn

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

Cf. A000005, A074794, A074795, A211338, A346602.

Accumulate[Table[If[Mod[DivisorSigma[0,n],3]==2,1,0],{n,80}]] (* Harvey P. Dale, Apr 22 2018 *)

PARI
```
a(n)=sum(k=1,n,if(numdiv(k)%3-2,0,1))
```

a(n) is asymptotic to c*n with c = 0.38....

The constant is conjecturally 3*zeta(3)/Pi^2 = 0.3653814847007... (A346602). See A211338 for more details. - Amiram Eldar, Feb 01 2025

A353446 Let g be the inverse Möbius transform of the Eisenstein integer-valued function f defined in A353445. a(n) is twice the real part of g(n).

Original entry on oeis.org

2, 1, 1, 0, 1, 2, 1, 2, 0, -1, 1, 0, 1, 2, 2, 1, 1, 0, 1, 0, -1, -1, 1, 1, 0, 2, 2, 0, 1, 1, 1, 0, 2, -1, 2, 0, 1, 2, -1, 1, 1, 1, 1, 0, 0, -1, 1, 2, 0, 0, 2, 0, 1, 1, -1, 1, -1, 2, 1, 0, 1, -1, 0, 2, 2, 1, 1, 0, 2, 1, 1, 0, 1, 2, 0, 0, 2, 1, 1, -1, 1, -1, 1, 0, -1, 2, -1, 1, 1, 0, -1, 0, 2, -1, 2, 0, 1, 0, 0, 0, 1

Offset: 1

Antti Karttunen and Peter Munn, Apr 19 2022

sign

The imaginary part of g(n) is A353354(n)*(sqrt(3)/2)*i.
f(n), g(n), and so also a(n), are determined by the cubefree part of n, A050985(n). If the cubefree part is not squarefree, g(n) is 0; otherwise g(n) = x^(A195017(A050985(n))), where x = (1 + sqrt(3)*i)/2, the primitive 6th root of unity with positive imaginary part.
The above formula arises from g being multiplicative (because f is multiplicative). g(prime(m)^k) is 1 for k == 0 (mod 3), 0 for k == 2 (mod 3), and for k == 1 (mod 3) the result depends on the parity of m. g(prime(m)^(3k+1)) is 1+w for odd m, -w for even m, where w is the cube root of unity with positive imaginary part. 1+w and -w are the primitive 6th roots of unity.
So the range of g is the 6 sixth roots of unity and 0 itself: these are the 7 Eisenstein integers closest to 0, and they are clearly closed under multiplication. The range of (a(n)) is [-2..2]. g(n) and a(n) are 0 if and only if the cubefree part of n is not squarefree. (Compare with the Moebius function being 0 when its argument is not squarefree.) Otherwise a(n) is even if and only if n is in A332820.

Peter Munn, Table of n, a(n) for n = 1..10000
Peter Munn, Figure showing relationship of the Eisenstein integer g(n) to the presence of n in other sequences.
Eric Weisstein's World of Mathematics, Eisenstein Integer

Sequences used in a definition of this sequence: A008966, A050985, A090882, A087204, A195017, A353445.
See A353354 for the imaginary part.
Positions of particular values depend on A059269, A211337, A211338, A332820 as shown in the formula section.
Positions of even numbers: A353355.

A050985(n) = { my(f=factor(n)); f[, 2] = apply(x->(x % 3), f[, 2]); factorback(f); }; \\ From A050985
A087204(n) = ([2, 1, -1, -2, -1, 1][1+(n%6)]);
A195017(n) = { my(f); if(1==n, 0, f=factor(n); sum(i=1, #f~, f[i,2] * (-1)^(1+primepi(f[i,1])))); };
A353446(n) = { my(u=A050985(n)); issquarefree(u) * A087204(abs(A195017(u))); };

a(n) = A008966(m) * A087204(A090882(m)) = A008966(m) * A087204(|A195017(m)|), where m = A050985(n), the cubefree part of n, and A008966(.) is the characteristic function of squarefree numbers.
For n >= 1, -2 <= a(n) <= 2.
{n : a(n) = -2} = {A211338} INTERSECT {A332820}.
{n : a(n) = -1} = {A211337} \ {A332820}.
{n : a(n) = 0} = {A059269}.
{n : a(n) = 1} = {A211338} \ {A332820}.
{n : a(n) = 2} = {A211337} INTERSECT {A332820}.

A074943 a(n) = tau(n) mod 3.

Original entry on oeis.org

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

Offset: 1

Benoit Cloitre, Oct 04 2002

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

Cf. A000005 (tau), A002117, A010872, A059269, A211337, A211338.

Mod[DivisorSigma[0,Range[110]],3] (* Harvey P. Dale, Dec 22 2013 *)

PARI
```
a(n)=numdiv(n)%3
```

(define (A074943 n) (modulo (A000005 n) 3)) ;; Antti Karttunen, Jul 26 2017

From Amiram Eldar, Apr 16 2024: (Start)
a(n) = A010872(A000005(n)).
a(A059269(n)) = 0; a(A211337(n)) = 1; a(A211338(n)) = 2.
Conjecture: Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 9*zeta(3)/Pi^2 = 1.0961... . The conjecture is true if A211337 and A211338 have the same asymptotic density (see also A059269). (End)

A229972 Nonprime numbers whose product of proper divisors is a perfect cube.

Original entry on oeis.org

1, 8, 16, 24, 27, 30, 40, 42, 54, 56, 64, 66, 70, 78, 81, 88, 102, 104, 105, 110, 114, 125, 128, 130, 135, 136, 138, 152, 154, 165, 170, 174, 182, 184, 186, 189, 190, 192, 195, 216, 222, 230, 231, 232, 238, 240, 246, 248, 250, 255, 258, 266, 273, 282, 285

Offset: 1

Derek Orr, Oct 04 2013

A nonprime number m is a term if and only if m is a cube or the number of divisors of m is of the form 3k+2. - Chai Wah Wu, Mar 09 2016

			The set of proper divisors of 8 is {1,2,4} and 1*2*4 = 2^3 so 8 is in the sequence.

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A007956.

Union of A000578 and (intersection of A002808 and A211338).

Select[Range[343],!PrimeQ[#]&&IntegerQ[(Apply[Times,Divisors[#]]/#)^(1/3)]&] (* Farideh Firoozbakht Oct 10 2013 *)
Select[Range[300],!PrimeQ[#]&&IntegerQ[Surd[Times@@Most[Divisors[ #]],3]]&] (* Harvey P. Dale, Oct 24 2017 *)
m = 7; Union[Range[m]^3, Select[Range[m^3], !PrimeQ[#] && Mod[DivisorSigma[0, #], 3] == 2 &]] (* Amiram Eldar, Jul 07 2022 *)

for(n=1,10^3,d=divisors(n);p=prod(i=1,#d-1,d[i]);if(p!=1&&ispower(p,3),print1(n,", ")))

from gmpy2 import iroot
from sympy import divisor_count, isprime
A229972_list = [i for i in range(1,10**3) if not isprime(i) and (iroot(i,3)[1] or divisor_count(i) % 3 == 2)] # Chai Wah Wu, Mar 10 2016

Corrected and edited by Farideh Firoozbakht Oct 10 2013

A084302 Remainder of tau(n) modulo 6.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 4, 3, 4, 2, 0, 2, 4, 4, 5, 2, 0, 2, 0, 4, 4, 2, 2, 3, 4, 4, 0, 2, 2, 2, 0, 4, 4, 4, 3, 2, 4, 4, 2, 2, 2, 2, 0, 0, 4, 2, 4, 3, 0, 4, 0, 2, 2, 4, 2, 4, 4, 2, 0, 2, 4, 0, 1, 4, 2, 2, 0, 4, 2, 2, 0, 2, 4, 0, 0, 4, 2, 2, 4, 5, 4, 2, 0, 4, 4, 4, 2, 2, 0, 4, 0, 4, 4, 4, 0, 2, 0, 0, 3, 2, 2, 2, 2, 2

Offset: 1

Labos Elemer, Jun 02 2003

The sums of the first 10^k terms, for k = 1, 2, ..., are 27, 236, 2275, 22166, 220070, 2195376, 21933228, 219259514, 2192385128, 21923168052, ... . Conjecture: the asymptotic mean of this sequence is 3*zeta(3)/zeta(2) = 3 * A253905 = 2.192288... . The conjecture is true if A211337 and A211338 have an equal asymptotic density (see also A059269). - Amiram Eldar, Jul 11 2024

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

Cf. A000005, A054763, A084299, A084300, A084301.

Cf. A059269, A211337, A211338

Mod[DivisorSigma[0,Range[110]],6] (* Harvey P. Dale, Sep 04 2020 *)

A084302(n) = (numdiv(n)%6); \\ Antti Karttunen, Jul 07 2017

a(n) = A000005(n) modulo 6.

A369209 Numbers whose number of divisors has the largest prime factor 3.

Original entry on oeis.org

4, 9, 12, 18, 20, 25, 28, 32, 36, 44, 45, 49, 50, 52, 60, 63, 68, 72, 75, 76, 84, 90, 92, 96, 98, 99, 100, 108, 116, 117, 121, 124, 126, 132, 140, 147, 148, 150, 153, 156, 160, 164, 169, 171, 172, 175, 180, 188, 196, 198, 200, 204, 207, 212, 220, 224, 225, 228

Offset: 1

Amiram Eldar, Jan 16 2024

Subsequence of A059269 and first differs from it at n = 36: A059269(136) = 44 has 15 = 3 * 5 divisors and thus is not a term of this sequence.
Numbers k such that A000005(k) is in A065119.
Numbers k such that A071188(k) = 3.
Equals the complement of A354181, without the terms of A036537 (i.e., complement(A354181) \ A036537).
The asymptotic density of this sequence is Product_{p prime} (1-1/p) * (Sum_{k>=1} 1/p^(A003586(k)-1)) - A327839 = 0.26087647470200496716... .

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

Cf. A000005, A003586, A006530, A036537, A065119, A336595, A071188, A211337, A211338, A327839, A354181.
Subsequence of A013929 and A059269.
Subsequences: A001248, A030627, A050997, A054753, A062503, A067259, A079395, A085986, A085987, A086975, A095990, A096156, A138032, A162143, A179643, A179645.

gpf[n_] := FactorInteger[n][[-1, 1]]; Select[Range[300], gpf[DivisorSigma[0, #]] == 3 &]

gpf(n) = if(n == 1, 1, vecmax(factor(n)[, 1]));
is(n) = gpf(numdiv(n)) == 3;

cp's OEIS Frontend

A059269 Numbers m for which the number of divisors, tau(m), is divisible by 3.

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A211337 Numbers k for which the number of divisors, tau(k), is congruent to 1 modulo 3.

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A336590 Numbers k such that k/A008834(k) is squarefree, where A008834(k) is the largest cube dividing k.

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A074796 Number of numbers k <= n such that tau(k) == 2 (mod 3) where tau(k) = A000005(k) is the number of divisors of k.

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A353446 Let g be the inverse Möbius transform of the Eisenstein integer-valued function f defined in A353445. a(n) is twice the real part of g(n).

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A074943 a(n) = tau(n) mod 3.

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A229972 Nonprime numbers whose product of proper divisors is a perfect cube.

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