cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A088725 Numbers having no divisors d>1 such that also d+1 is a divisor.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 37, 38, 39, 41, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 55, 57, 58, 59, 61, 62, 63, 64, 65, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 79, 81, 82, 83, 85, 86, 87, 88, 89, 91
Offset: 1

Views

Author

Reinhard Zumkeller, Oct 12 2003

Keywords

Comments

Complement of A088723.
Union of A132895 and A005408, the odd numbers. - Ray Chandler, May 29 2008
The numbers of terms not exceeding 10^k, for k = 1, 2, ..., are 9, 79, 778, 7782, 77813, 778055, 7780548, 77805234, 778052138, 7780519314, ... . Apparently, the asymptotic density of this sequence exists and equals 0.77805... . - Amiram Eldar, Jun 14 2022

Examples

			From _Gus Wiseman_, Oct 16 2019: (Start)
The sequence of terms together with their divisors > 1 begins:
   1: {}
   2: {2}
   3: {3}
   4: {2,4}
   5: {5}
   7: {7}
   8: {2,4,8}
   9: {3,9}
  10: {2,5,10}
  11: {11}
  13: {13}
  14: {2,7,14}
  15: {3,5,15}
  16: {2,4,8,16}
  17: {17}
  19: {19}
  21: {3,7,21}
  22: {2,11,22}
  23: {23}
  25: {5,25}
(End)

Crossrefs

Positions of 0's and 1's in A129308.
Positions of 0's and 1's in A328457 (also).
Numbers whose divisors (including 1) have no non-singleton runs are A005408.
The number of runs of divisors of n is A137921(n).
The longest run of divisors of n has length A055874(n).
Cf. A000005, A027750, A060680, A088722, A088723 (complement), A088724, A088726, A328166, A328450.

Programs

  • Mathematica
    Select[Range[100],FreeQ[Differences[Rest[Divisors[#]]],1]&] (* Harvey P. Dale, Sep 16 2017 *)
  • PARI
    isok(n) = {my(d=setminus(divisors(n), [1])); #setintersect(d, apply(x->x+1, d)) == 0;} \\ Michel Marcus, Oct 28 2019

Formula

A088722(a(n)) = 0.

Extensions

Extended by Ray Chandler, May 29 2008

A088722 Number of divisors d>1 of n such that d+1 also divides n.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 1, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 4, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 3, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 2, 0, 0, 0, 1, 0, 1, 0, 0, 0
Offset: 1

Views

Author

Reinhard Zumkeller, Oct 12 2003

Keywords

Comments

Also, number of partitions of n into two distinct parts (s,t), sWesley Ivan Hurt, Jan 16 2022

Examples

			n=144: divisors(144) = {1,2,3,4,6,8,9,12,16,18,24,36,48,72,144}, there are a(144) = 3 divisors d>1 such that also d+1 divides 144: (2,3), (3,4) and (8,9).

Links

Crossrefs

Cf. A059841, A088723, A088724, A088725, A088726, A129308.

Programs

  • Mathematica
    Table[DivisorSum[n, 1 &, And[# > 1, Divisible[n, # + 1]] &], {n, 105}] (* Michael De Vlieger, Jul 12 2017 *)
  • PARI
    A088722(n) = sumdiv(n,d,(d>1)&&!(n%(d+1))); \\ Antti Karttunen, Jul 12 2017
  • PARI
    first(n) = my(v = vector(n),k); for(i=2,sqrtint(n),k=i*(i+1); for(j=1, n\k, v[j*k]++)); v \\ David A. Corneth, Jul 12 2017

Formula

a(A088723(n)) > 0, a(A088724(n)) = 1, a(A088725(n)) = 0.
a(A088726(n)) = n, a(k) <> n, for n < A088726(n).
a(2n+1) = 0. - Ray Chandler, May 29 2008
a(n) = Sum_{d|n, (d+1)|n, d>1} 1. - Wesley Ivan Hurt, Jan 16 2022
From Amiram Eldar, Dec 31 2023: (Start)
a(n) = A129308(n) - A059841(n).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1/2. (End)

Extensions

Extended by Ray Chandler, May 29 2008

A088723 Numbers k with at least one divisor d>1 such that d+1 also divides k.

Original entry on oeis.org

6, 12, 18, 20, 24, 30, 36, 40, 42, 48, 54, 56, 60, 66, 72, 78, 80, 84, 90, 96, 100, 102, 108, 110, 112, 114, 120, 126, 132, 138, 140, 144, 150, 156, 160, 162, 168, 174, 180, 182, 186, 192, 198, 200, 204, 210, 216, 220, 222, 224, 228, 234, 240, 246, 252, 258, 260
Offset: 1

Views

Author

Reinhard Zumkeller, Oct 12 2003

Keywords

Comments

Complement of A088725.
Complement of A132895 relative to A005843, the even numbers. - Ray Chandler, May 29 2008
The numbers of terms that do not exceed 10^k, for k = 1, 2, ..., are 1, 21, 222, 2218, 22187, 221945, 2219452, 22194766, 221947862, 2219480686, ... . Apparently, the asymptotic density of this sequence exists and equals 0.22194... . - Amiram Eldar, Apr 20 2025

Links

Crossrefs

Cf. A088722, A088724, A088725, A088726.
Cf. A027750, A008588 (subsequence).

Programs

  • Haskell
    a088723 n = a088723_list !! (n-1)
a088723_list = filter f [2..] where
   f x = 1 `elem` (zipWith (-) (tail divs) divs)
         where divs = tail $ a027750_row x
-- Reinhard Zumkeller, Dec 16 2013
  • Mathematica
    Select[Range[300],MemberQ[Differences[Select[Divisors[#], #>1&]], 1]&]  (* Harvey P. Dale, Apr 03 2011 *)
  • PARI
    isok(k) = if(k%2, 0, if(!(k%3), 1, fordiv(k, d, if(d > 1 && !(k % (d+1)), return(1))); 0)); \\ Amiram Eldar, Apr 20 2025

Formula

A088722(a(n)) > 0.

Extensions

Extended by Ray Chandler, May 29 2008

A088726 Smallest numbers having exactly n divisors d>1 such that also d+1 is a divisor.

Original entry on oeis.org

1, 6, 12, 72, 60, 180, 360, 420, 840, 1260, 3780, 2520, 5040, 13860, 36960, 41580, 27720, 55440, 83160, 166320, 277200, 491400, 471240, 360360, 1113840, 720720, 1081080, 3341520, 2162160, 2827440, 5405400, 4324320, 12972960, 6126120
Offset: 0

Views

Author

Reinhard Zumkeller, Oct 12 2003

Keywords

Comments

A088722(a(n))=n, A088722(k)<>n, for n

Links

Crossrefs

Cf. A088722, A088723, A088724, A088725.

Formula

a(n) = 2*A130317(n+1) for n>0. - Chandler

Extensions

Extended by Ray Chandler, May 29 2008
Showing 1-4 of 4 results.

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