A074794 -id:A074794 - cp's OEIS frontend

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

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

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

A074795 Number of numbers k <= n such that tau(k) == 0 (mod 3) where tau(k) = A000005(k) is the number of divisors of k.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 1, 1, 2, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 5, 5, 5, 5, 5, 6, 6, 6, 7, 7, 7, 7, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 10, 11, 11, 11, 11, 12, 13, 13, 14, 14, 14, 14, 14, 14, 14, 14, 15, 15, 15, 16, 16, 16, 16, 16, 17, 17, 17, 17, 18, 18, 18, 19, 20, 20, 20, 20, 20, 20, 20, 20, 21

Offset: 1

Benoit Cloitre, Sep 07 2002

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000
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, A059269, A074794, A074796.

Accumulate[Boole[Divisible[DivisorSigma[0,Range[90]],3]]] (* Harvey P. Dale, Jan 11 2015 *)

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

a(n) is asymptotic to c*n with c = 0.26....

The constant is c = 1 - zeta(3)/zeta(2) = 1 - 6*zeta(3)/Pi^2 = 0.2692370305 ... (Sathe, 1945). - Amiram Eldar, Aug 29 2020

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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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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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A074795 Number of numbers k <= n such that tau(k) == 0 (mod 3) where tau(k) = A000005(k) is the number of divisors of k.

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