A088723 -id:A088723 - cp's OEIS frontend

A137921 Number of divisors d of n such that d+1 is not a divisor of n.

1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 2, 3, 2, 3, 4, 4, 2, 4, 2, 4, 4, 3, 2, 5, 3, 3, 4, 5, 2, 5, 2, 5, 4, 3, 4, 6, 2, 3, 4, 6, 2, 5, 2, 5, 6, 3, 2, 7, 3, 5, 4, 5, 2, 6, 4, 6, 4, 3, 2, 7, 2, 3, 6, 6, 4, 6, 2, 5, 4, 7, 2, 8, 2, 3, 6, 5, 4, 6, 2, 8, 5, 3, 2, 8, 4, 3, 4, 7, 2, 8, 4, 5, 4, 3, 4, 9, 2, 5, 6, 7, 2, 6, 2, 7, 8

Offset: 1

Reinhard Zumkeller, Feb 23 2008

a(n) = number of "divisor islands" of n. A divisor island is any set of consecutive divisors of a number where no pairs of consecutive divisors in the set are separated by 2 or more. - Leroy Quet, Feb 07 2010

			The divisors of 30 are 1,2,3,5,6,10,15,30. The divisor islands are (1,2,3), (5,6), (10), (15), (30). (Note that the differences between consecutive divisors 5-3, 10-6, 15-10 and 30-15 are all > 1.) There are 5 such islands, so a(30)=5.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Divisor Function.

Bisections: A099774, A174199.
First appearance of n is at position A173569(n).
Numbers whose divisors have no non-singleton runs are A005408.
The longest run of divisors of n has length A055874(n).
The number of successive pairs of divisors of n is A129308(n).
Cf. A000005, A001620, A027750, A060680, A088723, A088725, A181063, A199970, A328165, A328166, A328448, A328450.

a137921 n = length $ filter (> 0) $
   map ((mod n) . (+ 1)) [d | d <- [1..n], mod n d == 0]
-- Reinhard Zumkeller, Nov 23 2011

with(numtheory): disl := proc (b) local ct, j: ct := 1: for j to nops(b)-1 do if 2 <= b[j+1]-b[j] then ct := ct+1 else end if end do: ct end proc: seq(disl(divisors(n)), n = 1 .. 120); # Emeric Deutsch, Feb 12 2010

f[n_] := Length@ Split[ Divisors@n, #2 - #1 == 1 &]; Array[f, 105] (* f(n) from Bobby R. Treat *) (* Robert G. Wilson v, Feb 22 2010 *)
Table[Count[Differences[Divisors[n]],?(#>1&)]+1,{n,110}] (* _Harvey P. Dale, Jun 05 2012 *)
a[n_] := DivisorSum[n, Boole[!Divisible[n, #+1]]&]; Array[a, 100] (* Jean-François Alcover, Dec 01 2015 *)

a(n)=my(d,s=0);if(n%2,numdiv(n),d=divisors(n);for(i=1,#d,if(n%(d[i]+1),s++));s)

a(n)=sumdiv(n,d,(n%(d+1)!=0)); \\ Joerg Arndt, Jan 06 2015

from sympy import divisors
def A137921(n):
    return len([d for d in divisors(n,generator=True) if n % (d+1)])
# Chai Wah Wu, Jan 05 2015

a(n) <= A000005(n), with equality iff n is odd; a(A137922(n)) = 2.
a(n) = A000005(n) - A129308(n). - Michel Marcus, Jan 06 2015
a(n) = A001222(A328166(n)). - Gus Wiseman, Oct 16 2019
Sum_{k=1..n} a(k) ~ n * (log(n) + 2*gamma - 2), where gamma is Euler's constant (A001620). - Amiram Eldar, Jan 18 2024

Corrected and edited by Charles R Greathouse IV, Apr 19 2010

Edited by N. J. A. Sloane, Aug 10 2010

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

Reinhard Zumkeller, Oct 12 2003

nonn

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

Ray Chandler, Table of n, a(n) for n=0..57

Cf. A088722, A088723, A088724, A088725.

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

Extended by Ray Chandler, May 29 2008

A113502 A number n is included if at least one of its divisors > 1 is a triangular number (i.e., is of the form m(m+1)/2, m >= 2).

Original entry on oeis.org

3, 6, 9, 10, 12, 15, 18, 20, 21, 24, 27, 28, 30, 33, 36, 39, 40, 42, 45, 48, 50, 51, 54, 55, 56, 57, 60, 63, 66, 69, 70, 72, 75, 78, 80, 81, 84, 87, 90, 91, 93, 96, 99, 100, 102, 105, 108, 110, 111, 112, 114, 117, 120, 123, 126, 129, 130, 132, 135, 136, 138, 140, 141

Offset: 1

Leroy Quet, Jan 10 2006

nonn

A number n is in the sequence iff it is not a "triangle-free" positive integer.

Multiples of A226863. - Charles R Greathouse IV, Jul 29 2016

			12 is included because its divisors are 1, 2, 3, 4, 6 and 12, two of which (3 and 6) are triangular numbers > 1.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A112886, A000217, A013929, A226863.

v={};Do[If[b=Select[Divisors[n], #>1 && IntegerQ[(1+8#)^(1/2)]&]; b!={}, AppendTo[v, n]], {n, 200}]; v (* Farideh Firoozbakht, Jan 12 2006 *)
Select[Range[200],AnyTrue[Rest[Divisors[#]],OddQ[Sqrt[8#+1]]&]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Dec 13 2017 *)

is(n)=fordiv(n,d, if(ispolygonal(d,3) && d>1, return(1))); 0 \\ Charles R Greathouse IV, Jul 29 2016

a(n) = A088723(n)/2. - Ray Chandler, May 29 2008

More terms from Farideh Firoozbakht, Jan 12 2006

A094519 Numbers having at least one pair (x,y) of divisors with x

Original entry on oeis.org

6, 12, 18, 20, 24, 30, 36, 40, 42, 48, 54, 56, 60, 66, 70, 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

Offset: 1

Reinhard Zumkeller, May 06 2004

nonn

If m is in the sequence then so is k*m for k > 0. Furthermore, all terms are even. - David A. Corneth, Aug 31 2019
If (x,y) = (1,m) with m > 1, then oblong numbers m*(m+1) >= 6 belong to this sequence, and each oblong number >= 6 is a primitive term of the subsequence {k*m*(m+1), k >= 1}. Examples: with pair (1,2), we get multiples of 6 (see A008588); with (1,3) we get multiples of 12 (see A008594); with (1,4) we get multiples of 20 (see A008602); with (1,7) we get multiples of 56. - Bernard Schott, Aug 31 2019
The numbers of terms that do not exceed 10^k, for k = 1, 2, ..., are 1, 22, 230, 2317, 23201, 232209, 2322920, 23232166, 232332309, 2323370184, ... . Apparently, the asymptotic density of this sequence exists and equals 0.23233... . - Amiram Eldar, Apr 20 2025

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

Cf. A094518.
Complement of A094520.
A superset of A088723. - R. J. Mathar, Sep 16 2007
Subsequences: A002378 \ {0, 2}, A008588 \ {0}, A008602 \ {0}.

aQ[n_] := AnyTrue[Total /@ Subsets[Divisors[n], {2}], Divisible[n, #] &]; Select[Range[250], aQ] (* Amiram Eldar, Aug 31 2019 *)

is(n) = {my(d = divisors(n)); for(i = 1, #d - 2, for(j = i + 1, #d - 1, if(n % (d[i] + d[j]) == 0, return(1) ) ) ); 0 } \\ David A. Corneth, Aug 31 2019

from itertools import count, islice
from sympy import divisors
def A094519_gen(): # generator of terms
    for n in count(1):
        for i in range(1,len(d:=divisors(n))):
            di = d[i]
            for j in range(i):
                if n % (di+d[j]) == 0:
                    yield n
                    break
            else:
                continue
            break
A094519_list = list(islice(A094519_gen(),20)) # Chai Wah Wu, Dec 26 2021

A094518(a(n)) > 0.

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

A228870 Numbers n such that 2 * (1^n + 2^n + 3^n + ... + n^n) is not 0 (mod n).

Original entry on oeis.org

6, 12, 18, 20, 24, 30, 36, 40, 42, 48, 54, 60, 66, 72, 78, 80, 84, 90, 96, 100, 102, 108, 110, 114, 120, 126, 132, 138, 140, 144, 150, 156, 160, 162, 168, 174, 180, 186, 192, 198, 200, 204, 210, 216, 220, 222, 228, 234, 240, 246, 252, 258, 260, 264, 270, 272

Offset: 1

T. D. Noe, Sep 06 2013

nonn

These are the numbers not appearing in A228869; the even numbers not in A226872.
Also, positive integers n such that there exists an odd prime divisor p of n such that (p-1) also divides n (cf. A124240). - Max Alekseyev, Sep 07 2013
This sequence agrees with A088723 for many terms, but they are different.
If n is in the sequence, then so are the multiples of n. See A280187 for primitive members of this sequence. - Charles R Greathouse IV, Dec 28 2016

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A228869, A226872.

Select[Range[100], Mod[2*Sum[PowerMod[k, #, #], {k, #}], #] > 0 &]

is(n)=my(f=factor(n)[,1]); for(i=1,#f, if(n%(f[i]-1)==0 && f[i]>2, return(1))); 0 \\ Charles R Greathouse IV, Dec 28 2016

A328458 Maximum run-length of the nontrivial divisors (greater than 1 and less than n) of n.

Original entry on oeis.org

1, 0, 0, 1, 0, 2, 0, 1, 1, 1, 0, 3, 0, 1, 1, 1, 0, 2, 0, 2, 1, 1, 0, 3, 1, 1, 1, 1, 0, 2, 0, 1, 1, 1, 1, 3, 0, 1, 1, 2, 0, 2, 0, 1, 1, 1, 0, 3, 1, 1, 1, 1, 0, 2, 1, 2, 1, 1, 0, 5, 0, 1, 1, 1, 1, 2, 0, 1, 1, 1, 0, 3, 0, 1, 1, 1, 1, 2, 0, 2, 1, 1, 0, 3, 1, 1, 1, 1, 0, 2, 1, 1, 1, 1, 1, 3, 0, 1, 1, 2, 0, 2, 0, 1, 1

Offset: 1

Gus Wiseman, Oct 17 2019

nonn

By convention, a(1) = 1, and a(p) = 0 for p prime.

			The non-singleton runs of the nontrivial divisors of 1260 are: {2,3,4,5,6,7} {9,10} {14,15} {20,21} {35,36}, so a(1260) = 6.

Antti Karttunen, Table of n, a(n) for n = 1..100000

Positions of first appearances are A328459.
Positions of 0's and 1's are A088723.
The version that looks at all divisors is A055874.
The number of successive pairs of divisors > 1 of n is A088722(n).
The Heinz number of the multiset of run-lengths of divisors of n is A328166(n).
Cf. A033676, A060681, A060775, A070824, A088725, A129308, A181063, A199970, A328165, A163870, A328194, A328448, A328449.

Table[Switch[n,1,1,?PrimeQ,0,,Max@@Length/@Split[DeleteCases[Divisors[n],1|n],#2==#1+1&]],{n,100}]

A328458(n) = if(1==n,n,my(rl=0,pd=0,m=0); fordiv(n, d, if(1(1+pd), m = max(m,rl); rl=0); pd=d; rl++)); max(m,rl)); \\ Antti Karttunen, Feb 23 2023

Data section extended up to a(105) by Antti Karttunen, Feb 23 2023

A291022 Even numbers m such that every odd divisor > 1 of m is the sum of two divisors.

Original entry on oeis.org

6, 12, 18, 20, 24, 30, 36, 40, 42, 48, 54, 80, 96, 100, 108, 140, 150, 156, 160, 162, 192, 198, 200, 220, 264, 272, 280, 294, 312, 320, 324, 342, 384, 396, 400, 440, 486, 500, 510, 520, 528, 544, 546, 560, 624, 640, 684, 702, 714, 750, 768, 798, 800, 880, 912

Offset: 1

Michel Lagneau, Aug 31 2017

nonn

The numbers of the form p*2^q (6, 12, 20, ...) where p belongs to the set {3, 5, 17, 257, 65537} (A019434: Fermat primes or primes of the form 2^(2^k) + 1, for some k >= 0) are in the sequence.

The sequence is included in A088723 (Numbers having at least one divisor d>1 such that also d+1 is a divisor).

			42 is in the sequence because the divisors are {1, 2, 3, 6, 7, 14, 21, 42} and 3 = 2 + 1, 7 = 6 + 1 and 21 = 14 + 7.

Cf. A019434, A088723.

with(numtheory):EV:=array(1..100):OD:=array(1..100):nn:=5*10^4:
for n from 2 by 2 to nn do:
  d:=divisors(n):n1:=nops(d):k0:=0:k1:=0:it:=0:
   for i from 1 to n1 do:
    if irem(d[i],2)=0
     then
     k0:=k0+1:EV[k0]:=d[i]:
     else
     k1:=k1+1:OD[k1]:=d[i]:
    fi:
   od:
     for j from 2 to k1 do:
       for k from 1 to k1 do:
         for l from 1 to k0 do:
          if OD[j]=OD[k]+EV[l]
         then
          it:=it+1:
          else
         fi:
       od:
      od:
     od:
     if it>0 and it = k1-1
      then
      printf(`%d, `,n):
      else
     fi:
  od:

Showing 1-10 of 10 results.

cp's OEIS Frontend

A137921 Number of divisors d of n such that d+1 is not a divisor of n.

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

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

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

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A113502 A number n is included if at least one of its divisors > 1 is a triangular number (i.e., is of the form m(m+1)/2, m >= 2).

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A094519 Numbers having at least one pair (x,y) of divisors with x

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

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