A088726 -id:A088726 - cp's OEIS frontend

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

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

Reinhard Zumkeller, Oct 12 2003

nonn

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

			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)

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.

Select[Range[100],FreeQ[Differences[Rest[Divisors[#]]],1]&] (* Harvey P. Dale, Sep 16 2017 *)

isok(n) = {my(d=setminus(divisors(n), [1])); #setintersect(d, apply(x->x+1, d)) == 0;} \\ Michel Marcus, Oct 28 2019

A088722(a(n)) = 0.

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

Reinhard Zumkeller, Oct 12 2003

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

			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).

Antti Karttunen, Table of n, a(n) for n = 1..10000

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

Table[DivisorSum[n, 1 &, And[# > 1, Divisible[n, # + 1]] &], {n, 105}] (* Michael De Vlieger, Jul 12 2017 *)

A088722(n) = sumdiv(n,d,(d>1)&&!(n%(d+1))); \\ Antti Karttunen, Jul 12 2017

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

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)

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

Reinhard Zumkeller, Oct 12 2003

nonn

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

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A088722, A088724, A088725, A088726.

Cf. A027750, A008588 (subsequence).

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

Select[Range[300],MemberQ[Differences[Select[Divisors[#], #>1&]], 1]&]  (* Harvey P. Dale, Apr 03 2011 *)

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

A088722(a(n)) > 0.

Extended by Ray Chandler, May 29 2008

A130317 Smallest number having exactly n triangular divisors.

Original entry on oeis.org

1, 3, 6, 36, 30, 90, 180, 210, 420, 630, 1890, 1260, 2520, 6930, 18480, 20790, 13860, 27720, 41580, 83160, 138600, 245700, 235620, 180180, 556920, 360360, 540540, 1670760, 1081080, 1413720, 2702700, 2162160, 6486480, 3063060, 8288280

Offset: 1

Reinhard Zumkeller, May 23 2007

nonn

2*a(n) is smallest number having exactly n oblong divisors.

A007862(a(n)) = n and A007862(m) <> n for m < a(n).

			a(3)=6: A007862(6)=#{1,2*(2+1)/2,3*(3+1)/2}=3;
a(4)=36: A007862(36)=#{1,2*(2+1)/2,3*(3+1)/2,8*(8+1)/2}=4;

Ray Chandler and Donovan Johnson, Table of n, a(n) for n = 1..100 (first 58 terms from Ray Chandler)

Cf. A007862, A088726, A130279.

a(n) = my(k=1); while (sumdiv(k, d, ispolygonal(d,3)) != n, k++); k; \\ Michel Marcus, Jan 14 2022

a(n) = A088726(n-1)/2 for n>1. - Ray Chandler, Jun 24 2008

Extended by Ray Chandler, Jun 24 2008

A088724 Numbers having exactly one divisor d>1 such that also d+1 is a divisor.

Original entry on oeis.org

6, 18, 20, 40, 54, 56, 66, 78, 80, 100, 102, 110, 112, 114, 138, 140, 160, 162, 174, 182, 186, 198, 200, 222, 224, 234, 246, 258, 260, 272, 282, 318, 320, 340, 354, 364, 366, 392, 400, 402, 414, 426, 438, 448, 460, 474, 486, 498, 500, 506, 520, 522, 534, 544

Offset: 1

Reinhard Zumkeller, Oct 12 2003

nonn

Subsequence of A088723.

The numbers of terms not exceeding 10^k, for k = 1, 2, ..., are 1, 10, 100, 976, 9712, 97140, 971139, 9711054, 97109111, 971091745, ... . Apparently, the asymptotic density of this sequence exists and equals 0.097109... . - Amiram Eldar, Jul 09 2022

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A088722, A088723, A088725, A088726.

Select[Range[600],Count[Differences[Rest[Divisors[#]]],1]==1&] (* Harvey P. Dale, Sep 05 2015 *)

A088722(a(n)) = 1.

Extended by Ray Chandler, May 29 2008

A132895 Even numbers for which all divisors, with the exception of 1 and 2, are isolated. A positive divisor, k, of n is isolated if neither (k-1) nor (k+1) divides n.

Original entry on oeis.org

2, 4, 8, 10, 14, 16, 22, 26, 28, 32, 34, 38, 44, 46, 50, 52, 58, 62, 64, 68, 70, 74, 76, 82, 86, 88, 92, 94, 98, 104, 106, 116, 118, 122, 124, 128, 130, 134, 136, 142, 146, 148, 152, 154, 158, 164, 166, 170, 172, 176, 178, 184, 188, 190, 194, 196, 202, 206, 208, 212

Offset: 1

Emeric Deutsch, Oct 16 2007, Oct 19 2007

nonn

Obviously, all divisors of an odd number are isolated.

The numbers of terms that do not exceed 10^k, for k = 1, 2, ..., are 4, 29, 278, 2782, 27813, 278055, 2780548, 27805234, 278052138, 2780519314, ... . Apparently, the asymptotic density of this sequence exists and equals 0.27805... . - Amiram Eldar, Apr 20 2025

			28 is a term of the sequence because its divisors are 1, 2, 4, 7, 14, 28 and only 1 and 2 are non-isolated.
30 does not belong to the sequence because its divisors are 1, 2, 3, 4, 6, 8, 12, 24 and 1, 2, 3, 4 are non-isolated.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A112886, A133950, A133951, A088722-A088726.

with(numtheory): b:=proc(n) local div,ISO,i: div:=divisors(n): ISO:={}: for i to tau(n) do if member(div[i]-1,div)=false and member(div[i]+1,div)=false then ISO:=`union`(ISO,{div[i]}) end if end do end proc: a:=proc(n) if nops(b(n))= tau(n)-2 then n else end if end proc: seq(a(n), n=4..200);

Select[2*Range[120],Min[Differences[Rest[Divisors[#]]]]>1&] (* Harvey P. Dale, Jul 13 2022 *)

isok(k) = if(k%2, 0, if(!(k%3), 0, fordiv(k, d, if(d > 1 && !(k % (d+1)), return(0))); 1)); \\ Amiram Eldar, Apr 20 2025

from itertools import count, islice
from sympy import divisors
from sympy.ntheory.primetest import is_square
def A132895_gen(startvalue=2): # generator of terms >= startvalue
    return filter(lambda n:all(d==1 or not is_square((d<<3)+1) for d in divisors(n>>1,generator=True)), count(max(startvalue+(startvalue&1),2),2))
A132895_list = list(islice(A132895_gen(),40)) # Chai Wah Wu, Jun 07 2025

a(n) = 2*A112886(n). - Ray Chandler, May 29 2008

Corrected and extended by Ray Chandler, May 29 2008

A287142 Least k such that the number of pairs of consecutive divisors of k equals n.

Original entry on oeis.org

1, 2, 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, 16576560, 28274400

Offset: 0

Michel Lagneau, May 20 2017

nonn

a(n) is even for n > 0.
We observe numbers of the decimal form (abcabc) = 360360, 720720 and numbers of the decimal form (abcabc0) = 1081080, 2162160, 5405400, 4324320, 6126120.
Observation and questions: many terms are products of powers of a contiguous set of the smallest primes. Many early terms of a(n) are in A002182; e.g., a(35) - A002182(44). The smallest exception outside of the empty product a(0) = 1 is a(22) = 491400 = 2^3 * 3^3 * 5^2 * 7 * 13. In other words, many terms have A006530(a(n)) < A053669(a(n)); a(22) is the smallest exception. Other exceptions include {471240, 1113840, 3341520, 2827440, 16576560, 28274400, ...}. A000720(A053669(a(22))) - A000720(A006530(a(22))) = 1, but the first instance of 2 for this function is a(35) = 16576560. This is evident by mapping A054841 across a(n). Are there a finite number of terms of a(n) that are also in A002182? Are there a finite number of terms of a(n) that have A006530(a(n)) < A053669(a(n)); are they becoming less frequent as n increases? - Michael De Vlieger, May 20 2017
In other words, a(n) is the least integer with exactly n divisors that are oblong (A002378). - Bernard Schott, Jul 30 2022

			a(3) = 12 because the divisors of 12 are {1, 2, 3, 4, 6, 12} with three pairs of consecutive divisors (1, 2), (2, 3) and (3, 4).

Cf. A000005, A002378, A027750, A129308, A130317.

Essentially the same as A088726.

with(numtheory):
for n from 0 to 60 do:
ii:=0:
  for k from 1 to 10^8 while(ii=0) do:
    d0:=divisors(k):n0:=nops(d0):c0:=0:
      for i from 1 to n0-1 do:
        if d0[i+1]=d0[i]+1
         then
          c0:=c0+1:
          else
         fi:
       od:
       if c0=n
       then
     ii:=1:printf(“%d %d \n”,n,k):
     else
     fi:
   od:
  od:

Function[s, Function[t, ReplacePart[t, Map[#1 -> #2 & @@ # &, Transpose@{1 + Keys@ s, Values[s][[All, 1]]}]]]@ ConstantArray[0, Max@ Keys@ s]]@ KeySort@ PositionIndex@ Table[DivisorSum[n, 1 &, Divisible[n, # + 1] &], {n, 2 * 10^6}] (* Michael De Vlieger, May 20 2017, Version 10 *)

isok(n,k) = {dk = divisors(k); ddk = vector(#dk-1, j, dk[j+1] - dk[j]); #select(x->x==1, ddk) == n;}
a(n) = {my(k=1); while (!isok(n, k), k++); k;} \\ Michel Marcus, May 20 2017

a(n) = 2*A130317(n) for n >= 1. - Bernard Schott, Jul 30 2022

A195307 Where records occur in A129308 and also in A195155.

Original entry on oeis.org

1, 2, 6, 12, 60, 180, 360, 420, 840, 1260, 2520, 5040, 13860, 27720, 55440, 83160, 166320, 277200, 360360, 720720, 1081080, 2162160, 2827440, 4324320, 6126120, 12252240, 24504480, 36756720, 73513440, 147026880, 183783600, 232792560, 367567200, 465585120, 698377680

Offset: 1

Omar E. Pol, Oct 16 2011

nonn

Observation: a(n) ending at 0, if 5 <= n <= 24 and possibly more.
From David A. Corneth, Apr 14 2021: (Start)
Conjecture: for each term k > 1 in the sequence there exists prime p such that k/p is in the sequence.
From the first 35 terms only a(23) = 2827440 is not in A025487.
In the list of conjectured terms, if actual terms <= 10^16 are 97-smooth and have the following property: a(n+1) = a(n) + k*gcd(a(n), a(n-1), ..., a(n-20)) setting a(n) = 1 for n < 1 then those terms are actual terms.
The conjectured terms are 41-smooth and satisfy a(n+1) = a(n) + k*gcd(a(n), a(n-1), ..., a(n-13)). (End)
From Bernard Schott, Jul 30 2022: (Start)
Equivalently, integers whose number of oblong divisors (A129308) sets a new record.
Corresponding records of number of oblong divisors are 0, 1, 2, 3, 5, 6, 7, 8, 9, 10, 12, 13, 14, ... (End)

			a(4) = 12 is in the sequence because A129308(12) = 3 is larger than any earlier value in A129308. - _Bernard Schott_, Jul 30 2022

David A. Corneth, Conjectured terms <= 10^16 (a(1)..a(35) are certain)

Cf. A002378, A007862, A088726, A093687, A097212, A129308, A181826, A195155, A287142.

More terms a(6)-a(24) from Alois P. Heinz, Oct 16 2011

a(25)-a(35) from David A. Corneth, Apr 14 2021

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

Extensions

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

Formula

Extensions

A130317 Smallest number having exactly n triangular divisors.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

Extensions

A088724 Numbers having exactly one divisor d>1 such that also d+1 is a divisor.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A132895 Even numbers for which all divisors, with the exception of 1 and 2, are isolated. A positive divisor, k, of n is isolated if neither (k-1) nor (k+1) divides n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Python

Formula

Extensions

A287142 Least k such that the number of pairs of consecutive divisors of k equals n.

Views