A335572 -id:A335572 - cp's OEIS frontend

A368312 Irregular triangle read by rows where row n lists the factor differences of n.

0, 1, 2, 0, 3, 4, 1, 5, 6, 2, 7, 0, 8, 3, 9, 10, 1, 4, 11, 12, 5, 13, 2, 14, 0, 6, 15, 16, 3, 7, 17, 18, 1, 8, 19, 4, 20, 9, 21, 22, 2, 5, 10, 23, 0, 24, 11, 25, 6, 26, 3, 12, 27, 28, 1, 7, 13, 29, 30, 4, 14, 31, 8, 32, 15, 33, 2, 34, 0, 5, 9, 16, 35, 36, 17, 37

Offset: 1

Kevin Ryde, Dec 21 2023

Factor differences of n are all abs(p-q) where n = p*q, for positive integers p,q.
p is each divisor of n which is >= sqrt(n), in ascending order (A161908), and the resulting differences p-q are distinct and in ascending order.
Row n has length A038548(n).
Row n begins with smallest difference T(n,1) = A056737(n) and this is 0 iff n is a perfect square.
Row n ends with n-1 and this is the sole entry iff n is 1 or prime.

			Triangle begins:
       k=1
  n=1:   0
  n=2:   1
  n=3:   2
  n=4:   0, 3
  n=5:   4
  n=6:   1, 5
  n=7:   6
  n=8:   2, 7
  n=9:   0, 8

Kevin Ryde, Table of n, a(n) for rows n=1..2500, flattened
Paul Erdős and Moshe Rosenfeld, The factor-difference set of integers, Acta Arithmetica, volume 79, number 4, 1997, pages 353-359.

Cf. A038548 (row lengths), A079667 (row sums), A068333 (row products).
Cf. A056737 (column k=1), A161908.
Cf. A335572 (factor sums).

row(n) = my(v=divisors(n)); (v-Vecrev(v))[#v\2+1..#v];

T(n,k) = d - n/d where d = A161908(n,k).

A368059 a(1)=3; for n>1, a(n) is the smallest positive integer not already used which has a factor sum in common with a(n-1).

Original entry on oeis.org

3, 4, 6, 10, 12, 7, 15, 16, 9, 5, 8, 14, 18, 20, 11, 27, 32, 17, 45, 13, 24, 21, 25, 48, 28, 30, 22, 36, 19, 51, 64, 33, 40, 42, 52, 60, 31, 87, 112, 57, 72, 26, 44, 23, 63, 39, 55, 108, 38, 54, 50, 56, 29, 81, 65, 77, 80, 41, 117, 85, 96, 34, 66, 46, 84, 43, 123, 160, 69, 88, 70, 78, 90, 62

Offset: 1

Neal Gersh Tolunsky, Dec 17 2023

nonn

A factor sum of x is any p+q where x=p*q, those sums being row x of A335572.
Is this an infinite sequence?
When every product of two integers with sum s has appeared in the sequence, that sum s is no longer a potential link between a(n) and a(n-1). If a number appears whose factor sums have all been exhausted, the sequence ends.

			For n=2, 3 can only be factored as 1*3, which has a sum of 4. The next term cannot be 1 or 2 as they do not have a factor sum of 4, but 4 = 2*2 does, so a(2) = 4.
For n=5, a(4)=10 has factor sums 7 and 11. The smallest unused number with one of those sums is a(5) = 12 = 3*4, sum of 7.

Neal Gersh Tolunsky, Table of n, a(n) for n = 1..10000
Thomas Scheuerle, MATLAB Program.
Thomas Scheuerle, Histogram of a(n)/n for the first 5000 values of this sequence.

Cf. A335572 (factor sums).

Cf. A368103 (with factor differences).

MATLAB
```
% See Scheuerle link.
```

A386303 Positive integers k such that the set {d+k/d : d|k} contains four consecutive integers.

Original entry on oeis.org

15120, 712800, 3341520, 10533600, 23284800, 85503600, 147026880, 171097920, 302702400, 477338400, 2058376320, 2633510880, 4204418400, 7342876800, 9673606800, 13035884400, 13734761040, 14895223200, 22388788800, 22647794400, 26108082000, 34183749600, 62246804400, 89169141600

Offset: 1

Giedrius Alkauskas, Jul 18 2025

nonn

a(n) is divisible by 720.
Subsequence of A072389 (with two consecutive instead of four).
Integers k with five consecutive integers in the set {d+k/d : d|k} seem not to exist.
As terms must be of the form k * (k + 1) * m * (m + 1) and divisible by 720 we can restrict the search based on g = gcd(k * (k + 1), 720) which is at least 2. We must have (720 / g) | m * (m + 1). - David A. Corneth, Jul 19 2025
If q is the number of divisors of a(n) then the first of these four divisors is generally d[q/2 + 1] at least for nonsquares. For three consecutive integers (cf. A386302) there is the exception 180180. - David A. Corneth, Jul 20 2025

			a(1)=15120=M is a term of this sequence since 105, 108, 112, 120 are divisors of M, and 120+M/120=246, 112+M/112=247, 108+M/108=248, 105+M/105=249. It is the first term since no smaller such positive integer exists.

Giedrius Alkauskas, Consecutive integers in the set S_n={d+n/d: d|n}
David A. Corneth, 360 x 360 pixels image where white pixels with coordinate (k, m) have 720 | (k * (k + 1) * m * (m + 1))
David A. Corneth, PARI program

Cf. A002496, A027862, A072389, A335572, A386302.

M:=2*10^10:
Ki:={}:
Vi:=floor(sqrt(2*M)):
Ski:=floor((19*M)^(1/4)/2):
for F from 1 to Vi-4 do
  for y from 1 to min(floor((Vi-F)/2),Ski) do
     G:=F+2*y+1:
     if issqr(2*F^2-G^2+2) and issqr(3*F^2-2*G^2+6) then
       x:=(F+G-1)/2:
       n:=x*(x+1)*y*(y+1):
       Ki:=Ki union {n}:
     end if:
  end do:
end do:
Ki;

More terms from David A. Corneth, Jul 19 2025

A352796 Numbers m such that {d + m/d : d | m } does not contain consecutive integers.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 73, 74

Offset: 1

José María Grau Ribas, Jun 08 2022

nonn

Conjecture: Complement of A072389.

Michel Marcus, Table of n, a(n) for n = 1..10000

Cf. A072389, A335572.

S[n_]:=Divisors[n]+n/Divisors[n]//Union; Test[n_]:= {aux=S[n];Union[ {False},Table[aux[[i+1]]-aux[[i]] ==1,{i,Length[aux]-1}]]}[[1]]   //Last; Select[Range[1000],Test[#]&]

isok(m) = my(list=List()); fordiv(m, d, listput(list, d+m/d)); my(w = Set(vector(#list-1, k, list[k+1]-list[k]))); #select(x->(x==1), w) == 0; \\ Michel Marcus, Jun 09 2022

from sympy import divisors
def ok(n):
    s = sorted(set(d + n//d for d in divisors(n)))
    return 1 not in set(s[i+1]-s[i] for i in range(len(s)-1))
print([k for k in range(1, 75) if ok(k)]) # Michael S. Branicky, Jul 10 2022

A386302 Positive integers k such that the set {d+k/d : d|k} contains three consecutive integers.

Original entry on oeis.org

144, 180, 1260, 1440, 2520, 5040, 5544, 7200, 14040, 15120, 25200, 31680, 33660, 37800, 46800, 59400, 62244, 65520, 70560, 83160, 107100, 110880, 115920, 166320, 169344, 176400, 180180, 183600, 190944, 221760, 277200, 287280, 297540

Offset: 1

Giedrius Alkauskas, Jul 17 2025

nonn

Terms are divisible by 36.

Subsequence of A072389 (with two consecutive rather than three).

			a(1)=144, since 144/12+12=24, 144/9+9=25, 144/8+8=26, and no smaller integer with such property exists.

Giedrius Alkauskas, Consecutive integers in the set S_n={d+n/d: d|n}

Cf. A002496, A027862, A072389, A335572, A386303.

M:=300000:
Ki:={}:
Vi:=floor(sqrt(2*M)):
Ski:=floor((19*M)^(1/4)/2):
for F from 1 to Vi-4 do
  for y from 1 to min(floor((Vi-F)/2),Ski) do
     G:=F+2*y+1:
     if issqr(2*F^2-G^2+2) then
       x:=(F+G-1)/2;
       n:=x*(x+1)*y*(y+1):
       Ki:=Ki union {n}:
     end if:
  end do:
end do:
Ki;

isok(m, nb=3) = nb--; my(v = Set(apply(x->x+m/x, divisors(m)))); if (#v >= nb, select(x->(x==nb), vector(#v-nb, k, v[k+nb]-v[k]))); \\ Michel Marcus, Jul 18 2025

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A368312 Irregular triangle read by rows where row n lists the factor differences of n.

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A368059 a(1)=3; for n>1, a(n) is the smallest positive integer not already used which has a factor sum in common with a(n-1).

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A386303 Positive integers k such that the set {d+k/d : d|k} contains four consecutive integers.

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A352796 Numbers m such that {d + m/d : d | m } does not contain consecutive integers.

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Mathematica

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A386302 Positive integers k such that the set {d+k/d : d|k} contains three consecutive integers.

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Programs

Maple

PARI