A065475 -id:A065475 - cp's OEIS frontend

A171526 Denominator of (n-th noncomposite/n).

1, 1, 1, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50

Offset: 1

Jaroslav Krizek, Dec 11 2009

Denominator of (A008578(n)/n).
From Enrique Navarrete, Mar 15 2025: (Start)
a(n+1) is the number of binary strings of length n with at most one 0 and either zero or at least two 1s.  For example, for n=2, a(3)=1  since the string is 11; for n=3, a(4)=4 since the strings are 111, 011, 101, 110.
a(n+1) is also the number of ordered set partitions of an n-set into 2 sets such that the first set is empty or has one element and the second set is empty or has at least two elements. (End)

Cf. A008578, A065475.

Module[{nn=300,c,len},c=Select[Range[nn],!CompositeQ[#]&];len=Length[ c]; #[[1]]/#[[2]]&/@Thread[{c,Range[len]}]]//Denominator (* or *) Join[{1,1,1}, Range[ 4,60]] (* Harvey P. Dale, Feb 08 2020 *)

a(1) = a(2) = a(3) = 1, a(n) = n for n >= 4.
From Enrique Navarrete, Mar 15 2025: (Start)
G.f.: x*(1-x+3*x^3-2*x^4)/(1-x)^2.
E.g.f.: x*exp(x)-x^2/2-x^3/3. (End)

Corrected by Jaroslav Krizek, Dec 16 2009

A171950 a(1)=1. a(n) = the absolute difference between (the sum of previous terms) and A000217(n-2), n>1.

Original entry on oeis.org

1, 1, 1, 0, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68

Offset: 1

Giovanni Teofilatto, Oct 20 2010

Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A000217, A065475.

[1,1,1,0] cat [n-2: n in [5..70]]; // G. C. Greubel, Apr 03 2019

LinearRecurrence[{2,-1},{1,1,1,0,3,4},70] (* Harvey P. Dale, Dec 10 2015;; a(1)=1 amended by Georg Fischer, Apr 03 2019 *)

a(n)=if(n>4,n-2,n<4) \\ Charles R Greathouse IV, Oct 27 2011

[1,1,1,0]+[n-2 for n in (5..70)] # G. C. Greubel, Apr 03 2019

a(n) = |Sum_{i=1..n-1} a(i) - A000217(n-2)|, n>1.

a(n) = n-2, n>=5. - R. J. Mathar, Oct 26 2010

Zero inserted, precise indices added in definition, keyword:less and two formulas added - R. J. Mathar, Oct 26 2010

A171950 and A181440 are two different edited versions of a sequence submitted by Giovanni Teofilatto. - N. J. A. Sloane, Oct 29 2010

A272025 Irregular triangle read by rows, n >= 1, 1 <= k <= A038548(n), in which T(n,k) is the sum of the k-th pair of conjugate divisors of n, or T(n,k) is the central divisor of n if such a pair does not exist.

Original entry on oeis.org

1, 3, 4, 5, 2, 6, 7, 5, 8, 9, 6, 10, 3, 11, 7, 12, 13, 8, 7, 14, 15, 9, 16, 8, 17, 10, 4, 18, 19, 11, 9, 20, 21, 12, 9, 22, 10, 23, 13, 24, 25, 14, 11, 10, 26, 5, 27, 15, 28, 12, 29, 16, 11, 30, 31, 17, 13, 11, 32, 33, 18, 12, 34, 14, 35, 19, 36, 12, 37, 20, 15, 13, 6, 38, 39, 21, 40, 16, 41, 22, 14, 13, 42, 43, 23, 17, 13

Offset: 1

Omar E. Pol, Apr 21 2016

			Triangle begins:
  1;
  3;
  4;
  5, 2;
  6;
  7, 5;
  8;
  9, 6;
  10, 3;
  11, 7;
  12;
  13, 8, 7;
  ...
For n = 9 the divisors of 9 are [1, 3, 9]. There is only one pair of conjugate divisors: [1, 9], and the central divisor is 3, so the 9th row of the triangle is [10, 3].
For n = 12 the divisors of 12 are [1, 2, 3, 4, 6, 12]. There are three pairs of conjugate divisors, they are [1, 12], [2, 6], [3, 4], so the 12th row of the triangle is [13, 8, 7].

Row sums give A000203.
Row lengths give A038548.
Right border gives A207376.
Column 1 is A065475.
Cf. A027750, A210959, A228814.

cp's OEIS Frontend

A171526 Denominator of (n-th noncomposite/n).

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A171950 a(1)=1. a(n) = the absolute difference between (the sum of previous terms) and A000217(n-2), n>1.

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A272025 Irregular triangle read by rows, n >= 1, 1 <= k <= A038548(n), in which T(n,k) is the sum of the k-th pair of conjugate divisors of n, or T(n,k) is the central divisor of n if such a pair does not exist.

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