A211485 -id:A211485 - cp's OEIS frontend

A211338 Numbers k for which the number of divisors, tau(k), is congruent to 2 modulo 3.

2, 3, 5, 7, 11, 13, 16, 17, 19, 23, 24, 29, 30, 31, 37, 40, 41, 42, 43, 47, 53, 54, 56, 59, 61, 66, 67, 70, 71, 73, 78, 79, 81, 83, 88, 89, 97, 101, 102, 103, 104, 105, 107, 109, 110, 113, 114, 127, 128, 130, 131, 135, 136, 137, 138, 139, 149, 151, 152, 154

Offset: 1

Douglas Latimer, Apr 07 2012

nonn

The product of any 2 terms a(i)*a(j) is not a member of the sequence.
Any term a(n) can be expressed as 1 term (required to be greater than 1) from A211485 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, 37, 368, 3681, 36596, 365336, 3653499, 36537962, 365381169, 3653826361, ... . 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 A211337 have the same density (see A059269). - Amiram Eldar, Jan 06 2024

			The divisors of 16 are: 1, 2, 4, 8, 16 (5 divisors). 5 is congruent to 2 modulo 3. Thus 16 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 A000040 (the prime numbers, which each have 2 divisors).
The union of A059269 and A211337 is the complementary sequence to this one.
The definition of this sequence uses A000005 (the number of divisors of n).
Cf. A000578, A074796, A211485, A346602.

Select[Range[154], Mod[DivisorSigma[0, #], 3] == 2 &] (* T. D. Noe, Apr 21 2012 *)

{plnt=1 ; mxind=100 ;for(k=1, 10^6,
if(numdiv(k) % 3 == 2, 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

A211484 Numbers for which the canonical prime factorization contains only an even number of exponents, all of which are congruent to 1 modulo 3.

Original entry on oeis.org

1, 6, 10, 14, 15, 21, 22, 26, 33, 34, 35, 38, 39, 46, 48, 51, 55, 57, 58, 62, 65, 69, 74, 77, 80, 82, 85, 86, 87, 91, 93, 94, 95, 106, 111, 112, 115, 118, 119, 122, 123, 129, 133, 134, 141, 142, 143, 145, 146, 155, 158, 159, 161, 162, 166, 176, 177, 178, 183

Offset: 1

Douglas Latimer, Apr 12 2012

By convention 1 is included as the first term, in order to simplify the statement of certain results involving this sequence.

			6 is included, as its canonical prime factorization (2^1)*(3^1) contains an even number of exponents, all of which are congruent to 1 modulo 3.

Douglas Latimer, Table of n, a(n) for n = 1..1000

This sequence includes A030229 and hence A006881. The definition is similar to that for A211485.

pfQ[n_]:=Module[{f=Transpose[FactorInteger[n]][[2]]},EvenQ[Length[f]] && Union[ Mod[f,3]]=={1}]; Join[{1},Select[Range[200],pfQ]] (* Harvey P. Dale, Mar 24 2016 *)

{plnt=1; k=1; print1(k, ", "); plnt++;
mxind=76 ; mxind++ ; for(k=2, 10^6,
M=factor(k);passes=1;
sz = matsize(M)[1];
for(k=1,sz, if(sz%2 != 0, passes=0;break()); if( M[k,2] % 3 != 1, passes=0));
if( passes == 1 , print1(k, ", "); plnt++) ; if(mxind ==  plnt, break() ))}

is(n,f=factor(n))=omega(f)%2==0 && factorback(f[,2]%3)==1 \\ Charles R Greathouse IV, Sep 07 2017

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A211338 Numbers k for which the number of divisors, tau(k), is congruent to 2 modulo 3.

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A211484 Numbers for which the canonical prime factorization contains only an even number of exponents, all of which are congruent to 1 modulo 3.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI