A161842 -id:A161842 - cp's OEIS frontend

A161901 Array read by rows in which row n lists the divisors of n, but if n is a square then the square root of n appears twice.

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

Offset: 1

Omar E. Pol, Jun 23 2009

Row n has length A161841(n). Row sums give A060866. - Omar E. Pol, Jan 06 2014

			Array begins:
....... 1,1;
....... 1,2;
....... 1,3;
..... 1,2,2,4;
....... 1,5;
..... 1,2,3,6;
....... 1,7;
..... 1,2,4,8;
..... 1,3,3,9;
..... 1,2,5,10;
....... 1,11;
... 1,2,3,4,6,12;
....... 1,13;
..... 1,2,7,14;
..... 1,3,5,15;
... 1,2,4,4,8,16;

Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos
Omar E. Pol, Illustration: Divisors and pi(x)

Cf. A000005, A027750, A038548, A161841, A161842.

dsr[n_]:=If[IntegerQ[Sqrt[n]],Sort[Join[{Sqrt[n]},Divisors[n]]],Divisors[ n]]; Array[ dsr,30]//Flatten (* Harvey P. Dale, Sep 27 2020 *)

Keyword tabf added by R. J. Mathar, Jun 28 2009

Definition clarified by Harvey P. Dale, Sep 27 2020

A161841 Number of factors, with repetition, in all distinct pairs (a <= b) such that a*b = n.

Original entry on oeis.org

2, 2, 2, 4, 2, 4, 2, 4, 4, 4, 2, 6, 2, 4, 4, 6, 2, 6, 2, 6, 4, 4, 2, 8, 4, 4, 4, 6, 2, 8, 2, 6, 4, 4, 4, 10, 2, 4, 4, 8, 2, 8, 2, 6, 6, 4, 2, 10, 4, 6, 4, 6, 2, 8, 4, 8, 4, 4, 2, 12, 2, 4, 6, 8, 4, 8, 2, 6, 4, 8, 2, 12, 2, 4, 6, 6, 4, 8, 2, 10, 6, 4, 2, 12, 4, 4, 4, 8, 2, 12, 4, 6, 4, 4, 4, 12, 2, 6, 6, 10

Offset: 1

Omar E. Pol, Jun 23 2009

			a(16)=6 because there are three distinct pairs (a <= b) such that a*b = n: the pairs (1,16), (2,8) and (4,4). So the number of factors, with repetition, in all the pairs is equal to 6.

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

Cf. A000005, A010052, A038548, A161842.

seq(numtheory:-tau(n) + `if`(issqr(n),1,0), n = 1 .. 200); # Robert Israel, Dec 23 2015

2*Ceiling[DivisorSigma[0, Range[100]]/2] (* Paolo Xausa, Feb 05 2025 *)

a(n) = A000005(n) + A010052(n) = A038548(n)*2.

Sum_{k=1..n} a(k) ~ (log(n) + 2*gamma - 1)*n + sqrt(n), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Jul 01 2021

cp's OEIS Frontend

A161901 Array read by rows in which row n lists the divisors of n, but if n is a square then the square root of n appears twice.

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