A119416 -id:A119416 - cp's OEIS frontend

A062011 a(n) = 2tau(n) = 2A000005(n).

2, 4, 4, 6, 4, 8, 4, 8, 6, 8, 4, 12, 4, 8, 8, 10, 4, 12, 4, 12, 8, 8, 4, 16, 6, 8, 8, 12, 4, 16, 4, 12, 8, 8, 8, 18, 4, 8, 8, 16, 4, 16, 4, 12, 12, 8, 4, 20, 6, 12, 8, 12, 4, 16, 8, 16, 8, 8, 4, 24, 4, 8, 12, 14, 8, 16, 4, 12, 8, 16, 4, 24, 4, 8, 12, 12, 8, 16, 4, 20, 10, 8, 4, 24, 8, 8, 8, 16

Offset: 1

Ahmed Fares (ahmedfares(AT)my-deja.com), Jul 12 2001

Old definition was "Number of cyclic subgroups of the group C_n X C_2 (where C_n is the cyclic group with n elements)."
More generally, the number of cyclic subgroups of the group C_n X C_m is Sum_{i|n, j|m} phi(i)*phi(j)/phi(lcm(i,j)), where phi=Euler totient function, cf. A000010. - Vladeta Jovovic, Jul 15 2001
Number of divisors of p*n, where p is any prime not dividing n. - Reinhard Zumkeller, May 17 2006
From Enrique Pérez Herrero, Jul 21 2011: (Start)
If p(x) is a polynomial with integer coefficients, and if r is an integer zero of p(x), then r is a divisor of the constant term c_0 of p(x). Under this theorem, p(x) can have a(c_0) possible integer roots.
a(n) is the number of integer divisors of n, while A000005(n) is the number of positive divisors. (End)
Number of solutions to the Diophantine equation i*j = n*i + j. - Robert G. Wilson v, Apr 10 2019
a(n) is also the number of times n appears in the triangle A333119, or equivalently, the number of positive integer solutions of the equation A333119(x, y) = n for y < x. - Stefano Spezia, Oct 05 2022

Harry J. Smith, Table of n, a(n) for n = 1..1000
Omar E. Pol, Illustration of initial terms
Index entries for sequences related to groups

Cf. A000005, A060648, A060710.
Cf. A087560, A119416.
Cf. A333119.

A062011[n_] := 2*DivisorSigma[0,n]; Array[A062011,50] (* Enrique Pérez Herrero, Jul 15 2011 *)

for (n=1, 1000, write("b062011.txt", n, " ", 2*numdiv(n))) \\ Harry J. Smith, Jul 29 2009

a(n) = A000005(A087560(n)) = A000005(A119416(n)). - Reinhard Zumkeller, May 17 2006

L.g.f.: -log(Product_{k>=1} (1 - x^k)^(2/k)) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, Mar 18 2018

More terms from Vladeta Jovovic, Jul 14 2001

Edited by N. J. A. Sloane, Sep 20 2018, replacing old definition (which was of course correct) with a simple formula.

A087560 Smallest m > n such that gcd(m, n^2) = n.

Original entry on oeis.org

2, 6, 6, 12, 10, 30, 14, 24, 18, 30, 22, 60, 26, 42, 30, 48, 34, 90, 38, 60, 42, 66, 46, 120, 50, 78, 54, 84, 58, 210, 62, 96, 66, 102, 70, 180, 74, 114, 78, 120, 82, 210, 86, 132, 90, 138, 94, 240, 98, 150, 102, 156, 106, 270, 110, 168, 114, 174, 118, 420, 122

Offset: 1

Reinhard Zumkeller, Oct 24 2003

nonn

Equals n multiplied by the least nontrivial number coprime to n. - Amarnath Murthy, Nov 20 2005

Ivan Neretin, Table of n, a(n) for n = 1..10000

Cf. A000005, A053669, A062011, A119416, A249270.

Table[n*Select[Prime[Range[Log2[n] + 1]], ! Divisible[n, #] &][[1]], {n, 61}] (* Ivan Neretin, May 21 2015 *)

a(n) = forprime(p = 2, , if(n%p, return(n*p))); \\ Amiram Eldar, Feb 01 2025

a(n) = n*A053669(n).
A000005(a(n)) = 2*A000005(n) = A062011(n). - Reinhard Zumkeller, May 17 2006
Sum_{k=1..n} ~ c * n^2 / 2, where c = A249270. - Amiram Eldar, Feb 01 2025

A381500 a(n) = A019565(A187769(n)).

Original entry on oeis.org

1, 2, 3, 6, 5, 10, 15, 30, 7, 14, 21, 35, 42, 70, 105, 210, 11, 22, 33, 55, 77, 66, 110, 165, 154, 231, 385, 330, 462, 770, 1155, 2310, 13, 26, 39, 65, 91, 143, 78, 130, 195, 182, 273, 455, 286, 429, 715, 1001, 390, 546, 910, 1365, 858, 1430, 2145, 2002, 3003

Offset: 0

Michael De Vlieger, Peter Munn, and Antti Karttunen, Feb 27 2025

The squarefree numbers, ordered first by largest prime factor (dividing the sequence into rows), then by number of prime factors, then lexicographically by their prime factors (written in descending order).

We index (a(n)) from offset 0, matching the choice for A019565 and similar sequences.

			Table begins:
  Row 0:  1;
  Row 1:  2;
  Row 2:  3,  6;
  Row 3:  5, 10, 15, 30;
  Row 4:  7, 14, 21, 35, 42, 70, 105, 210;
  Row 5: 11, 22, 33, 55, 77, 66, 110, 165, 154, 231, 385, 330, 462, 770, 1155, 2310;
  ...
Table of a(n) for n = 0..31, demonstrating relationship of this sequence with s = A187769:
          <-factors                    <-factors
   n  a(n)  2 3 5 7  s(n)  |   n   a(n)  2 3 5 7 11 s(n)
  -------------------------|----------------------------
   0    1   .          0   |  16    11   . . . . x   16
   1    2   x          1   |  17    22   x . . . x   17
   2    3   . x        2   |  18    33   . x . . x   18
   3    6   x x        3   |  19    55   . . x . x   20
   4    5   . . x      4   |  20    77   . . . x x   24
   5   10   x . x      5   |  21    66   x x . . x   19
   6   15   . x x      6   |  22   110   x . x . x   21
   7   30   x x x      7   |  23   165   . x x . x   22
   8    7   . . . x    8   |  24   154   x . . x x   25
   9   14   x . . x    9   |  25   231   . x . x x   26
  10   21   . x . x   10   |  26   385   . . x x x   28
  11   35   . . x x   12   |  27   330   x x x . x   23
  12   42   x x . x   11   |  28   462   x x . x x   27
  13   70   x . x x   13   |  29   770   x . x x x   29
  14  105   . x x x   14   |  30  1155   . x x x x   30
  15  210   x x x x   15   |  31  2310   x x x x x   31
  -------------------------|----------------------------
            1 2 4 8  s(n)  |             1 2 4 8 16 s(n)
             bits->                         bits->

Michael De Vlieger, Table of n, a(n) for n = 0..16383
Michael De Vlieger, Log log scatterplot of a(n), n = 0..2^16-1.
Michael De Vlieger, Plot p | a(n) at (x,y) = (n, pi(p)), n = 0..2047, 12X vertical exaggeration.
Michael De Vlieger, Fan style binary tree showing a(n), n = 0..2048, with a color function showing smallest and greatest terms in each row in green and red, respectively.

Cf. A019565, A119416, A163866, A187769, A253550, A294648, A344085.

A002110, A098012 are subsequences.

a187769 = {{0}}~Join~Table[SortBy[Range[2^n, 2^(n + 1) - 1], DigitCount[#, 2, 1] &], {n, 0, 8}] // Flatten; a019565[x_] := Times @@ Prime@ Flatten@ Position[#, 1] &@ Reverse@ IntegerDigits[x, 2]; Map[a019565, a187769]

a(n) = A019565(A187769(n)).
As an irregular triangle T(n,k), where row 0 = {1}:
For n > 1, omega(T(n,1)) = 1, omega(T(n, 2^(n-1))) = n, thus row n is divided into n segments S such that with S, omega(T(n,k)) = m, where m = 1..n. (See A187769 for the lengths of segments associated with Pascal's triangle A007318.)
S(-1,-1) = (1).
For n >= 0:
S(n-1, n) = (); S(n, -1) = ();
for 0 <= m <= n, S(n,m) = ( A253550'(S(n-1, m)), A119416'(S(n-1, m-1)) ), where Axxx'((i_1, i_2, ..., i_j)) denotes Axxx(i_1), Axxx(i_2), ..., Axxx(i_j).
a(A163866(n)) = A098012(n).

cp's OEIS Frontend

A062011 a(n) = 2tau(n) = 2A000005(n).

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A087560 Smallest m > n such that gcd(m, n^2) = n.

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A381500 a(n) = A019565(A187769(n)).

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cp's OEIS Frontend

A062011 a(n) = 2*tau(n) = 2*A000005(n).

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A087560 Smallest m > n such that gcd(m, n^2) = n.

Views

Author

Keywords

Comments

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Crossrefs

Programs

Mathematica

PARI

Formula

A381500 a(n) = A019565(A187769(n)).

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Author

Keywords

Comments

Examples

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Crossrefs

Programs

Mathematica

Formula

A062011 a(n) = 2tau(n) = 2A000005(n).