A056737 -id:A056737 - cp's OEIS frontend

A076388 a(n) = minimum of y-x such that x <= y, x*y = n and gcd(x,y)=1.

0, 1, 2, 3, 4, 1, 6, 7, 8, 3, 10, 1, 12, 5, 2, 15, 16, 7, 18, 1, 4, 9, 22, 5, 24, 11, 26, 3, 28, 1, 30, 31, 8, 15, 2, 5, 36, 17, 10, 3, 40, 1, 42, 7, 4, 21, 46, 13, 48, 23, 14, 9, 52, 25, 6, 1, 16, 27, 58, 7, 60, 29, 2, 63, 8, 5, 66, 13, 20, 3, 70, 1, 72, 35, 22, 15, 4, 7, 78, 11, 80, 39

Offset: 1

T. D. Noe, Oct 11 2002

If n is a prime or a power of a prime, a(n) = n-1. Similar to A056737, which does not have the gcd(x,y)=1 condition.

			a(12) = 1 because of the possible (x,y) pairs, (1,12), (2,6), (3,4), the pair (3,4) yields the minimum difference and satisfies gcd(x,y)=1.

Antti Karttunen, Table of n, a(n) for n = 1..10000 (terms 2..10000 from Ivan Neretin)
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000

Cf. A034699, A052128, A056737, A354933.

Differs from |A354988(n)| for the first time at n=60, where a(60) = 7, while A354988(60) = -11.

nMax = 100; Table[dvs = Divisors[n]; i = 1; j = 1; While[n/dvs[[i]] > dvs[[i]], If[GCD[n/dvs[[i]], dvs[[i]]] == 1, j = i]; i++ ]; n/dvs[[j]] - dvs[[j]], {n, 2, nMax}]

A076388(n) = fordiv(n,d,if((d>=(n/d)) && 1==gcd(d,n/d), return(d-(n/d)))); \\ Antti Karttunen, Jun 16 2022

a(n) = A354933(n) - A052128(n). - Corrected by Antti Karttunen, Jun 16 2022

Definition formally changed from x < y to x <= y, to accommodate the prepended term a(1)=0 - Antti Karttunen, Jun 16 2022

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

Original entry on oeis.org

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).

A082119 Smallest positive difference between d and n/d for any divisor d of n.

Original entry on oeis.org

1, 2, 3, 4, 1, 6, 2, 8, 3, 10, 1, 12, 5, 2, 6, 16, 3, 18, 1, 4, 9, 22, 2, 24, 11, 6, 3, 28, 1, 30, 4, 8, 15, 2, 5, 36, 17, 10, 3, 40, 1, 42, 7, 4, 21, 46, 2, 48, 5, 14, 9, 52, 3, 6, 1, 16, 27, 58, 4, 60, 29, 2, 12, 8, 5, 66, 13, 20, 3, 70, 1, 72, 35, 10, 15, 4, 7, 78, 2, 24, 39, 82, 5

Offset: 2

Ralf Stephan, Apr 04 2003

a(n) = A056737(n) for nonsquare n. a(p) = p-1 for prime p.

Alois P. Heinz, Table of n, a(n) for n = 2..10000

Cf. A082120, A000037 (nonsquares), A056737.

with(numtheory):
a:= n-> min(seq((h-> `if`(h>0, h, NULL))(d-n/d), d=divisors(n))):
seq(a(n), n=2..100);  # Alois P. Heinz, Nov 12 2014

a[n_] := Min[Table[If[# > 0, #, Nothing]&[d-n/d], {d, Divisors[n]}]];
Table[a[n], {n, 2, 100}] (* Jean-François Alcover, Jan 13 2025, after Alois P. Heinz *)

a(n) = my(v=divisors(n), r=sqrt(n), t=0); for(k=1, length(v), if(v[k]>=r, t=k; break)); if(v[t]^2==n, u=t, u=t-1);  if(v[t]-v[u]<1, u=u-1; t=t+1); v[t]-v[u];

A139693 a(n) is the smallest positive integer m where, for k divides m, minimum(|k - m/k|) = n.

Original entry on oeis.org

1, 2, 3, 10, 5, 14, 7, 44, 33, 22, 11, 26, 13, 68, 51, 34, 17, 38, 19, 92, 69, 46, 23, 174, 145, 116, 87, 58, 29, 62, 31, 222, 185, 148, 111, 74, 37, 164, 123, 82, 41, 86, 43, 188, 141, 94, 47, 318, 265, 212, 159, 106, 53, 354, 295, 236, 177, 118, 59, 122, 61, 402, 335

Offset: 0

Leroy Quet, Apr 29 2008

nonn

a(n) is the minimum positive integer m where A056737(m) = n.

Robert G. Wilson v, Table of n, a(n) for n = 0..10000

Cf. A056737, A139694.

f(n) = {n=divisors(n); n[(2+#n)\2]-n[(1+#n)\2]} \\ A056737
a(n) = my(k=1); while(f(k) != n, k++); k; \\ Michel Marcus, Mar 20 2019

More terms from Max Alekseyev, Oct 28 2008

A369110 a(n) is the number of distinct elements appearing in the sequence formed by recursively applying A063655 when starting from n.

Original entry on oeis.org

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

Offset: 1

Adnan Baysal, Jan 13 2024

A063655(n) gives the smallest semiperimeter of an integral rectangle with area n, which is the same thing as the minimum sum of two positive integers whose product is n. In this sequence, A063655 is applied recursively until a cycle is found. Then the number of distinct elements appearing in this process is given as a(n). Note that it's conjectured that a cycle will be found at some point.
Conjecture: The cycle part of each sequence generated by the recursion is one of (4), (5, 6), or (6, 5). Confirmed through 1 millionth term.
The conjecture is true. Proof: The conjecture holds for n <= 6. Suppose n >= 7 and the conjecture holds for lower values of n. If n is composite, then A063655(n) <= n/2+2. If n is prime, then A063655(n) = n+1 is even and A063655(A063655(n)) <= (n+1)/2+2. In both cases, n reaches a lower number and the conjecture holds for n. - Jason Yuen, Mar 30 2024

			n = 1 can be factored as 1*1 with minimum sum 2 (similarly, A063655(1) = 2). Then 2 = 1*2, so minimum sum is 3 = A063655(2). 3 = 1*3 which means the next number in the recursion is 4 = A063655(3). 4 = 2*2 which gives the same number 4 = A063655(4), hence this recursion will create a cycle at this point. Starting from n = 1 (including 1), we generated these numbers: (1, 2, 3, 4, 4, 4, ...). Therefore, a(1) = 4. a(2), a(3), and a(4) are trivially deduced from this example.

Jason Yuen, Table of n, a(n) for n = 1..10000

Cf. A063655, A056737.

from sympy import divisors
def A369110(n):
    c = {n}
    while n<4 or n>5:
        c.add(n:=(d:=divisors(n))[((l:=len(d))-1)>>1]+d[l>>1])
    if n==5:
        c.add(6)
    return len(c) # Chai Wah Wu, Apr 25 2024

A324921 Index of first occurrence of n in A324920.

Original entry on oeis.org

0, 1, 2, 3, 10, 11, 26, 83, 178, 179, 362, 1835, 9188, 42709, 162466, 358487, 1877938, 11596979, 57866702, 94418279, 1888365980

Offset: 0

Robert G. Wilson v, Mar 20 2019

Cf. A056737, A324920.

g[n_] := Block[{d = Divisors@ n}, len = Length@ d; If[ OddQ@ len, 0, d[[1 + len/2]] - d[[len/2]]]]; f[n_] := Length@ NestWhileList[g, n, # > 0 &] - 1; t[_] := -1; k = 0; While[k < 1000000001, a = f@ k; If[ t[a] == -1, t[a] = k]; k++]; t@# & /@ Range[0, 17]

a056737(n)=n=divisors(n); n[(2+#n)\2]-n[(1+#n)\2] \\ after M. F. Hasler in A056737
a324920(n) = my(x=n, i=0); while(x!=0, i++; x=a056737(x)); i
a(n) = for(k=0, oo, if(a324920(k)==n, return(k))) \\ Felix Fröhlich, Mar 20 2019

a(18)-a(20) from Daniel Suteu, Mar 20 2019

A369793 a(n) is the number of occurrences of n in A063655.

Original entry on oeis.org

0, 1, 1, 2, 1, 3, 2, 3, 3, 3, 2, 5, 3, 5, 5, 5, 4, 5, 5, 7, 6, 7, 5, 8, 6, 7, 7, 7, 6, 10, 7, 9, 8, 9, 9, 10, 8, 10, 9, 11, 8, 13, 9, 13, 11, 12, 11, 14, 13, 11, 12, 15, 10, 15, 13, 15, 13, 14, 12, 15, 12, 18, 16, 15, 15, 17, 13, 17

Offset: 1

Adnan Baysal, Feb 07 2024

nonn

Construct a directed graph whose vertex set is the set of all positive integers, and a directed edge from k to n belongs to this graph iff A063655(k) = n. a(n) is the in-degree of the vertex n in this graph. As conjectured in A369110, it is also conjectured here that the only cycles in this graph are from 4 to itself and between 5 and 6.

			a(1) = 0 since 1 does not exist in A063655. This is also clear from the definition of A063655, because there is no integral rectangle with semiperimeter 1.
a(2) = 1 because there is only one integral rectangle of area 1 with a minimal semiperimeter 2, which is the 1 X 1 square. So 2 appears only once in A063655, which means a(2) = 1.
a(4) = 2, because only A063655(3) and A063655(4) have the value 4. For any n > 4, A063655(n) > 4, because A063655(n) > 2 * sqrt(n) > 2 * sqrt(4) = 4. Hence, 4 cannot appear in the rest of A063655.

Cf. A063655, A369110, A056737.

a=1156;Table[Count[Table[2*Median[Divisors[m]], {m,a}] ,n],{n,Floor[2*Sqrt[a]]}] (* James C. McMahon, Mar 12 2024 *)

from sympy import divisors
def A369793(n): return sum(1 for m in range(1,(n**2>>2)+1) if (d:=divisors(m))[((l:=len(d))-1)>>1]+d[l>>1]==n) # Chai Wah Wu, Mar 25 2024

A147763 a(n) = smallest k such that n = d(d-b)(d+c), where b, c, d >= 0 and k = b+c.

Original entry on oeis.org

0, 1, 2, 1, 4, 2, 6, 0, 2, 4, 10, 1, 12, 6, 4, 2, 16, 1, 18, 3, 6, 10, 22, 2, 4, 12, 0, 5, 28, 3, 30, 2, 10, 16, 6, 1, 36, 18, 12, 3, 40, 5, 42, 9, 2, 22, 46, 1, 6, 3, 16, 11, 52, 3, 10, 5, 18, 28, 58, 2, 60, 30, 4, 0, 12, 9, 66, 15, 22, 5, 70, 3, 72, 36, 2, 17, 10, 11, 78, 1, 6, 40, 82, 4, 16

Offset: 1

Samuel Zbarsky (sa_zbarsky(AT)yahoo.com), Nov 11 2008

			n = 9 = 3*1*3 = 3*(3-2)*(3+0) with d = 3, b = 2, c = 0. So a(9) = k = 2+0 = 2, since there is no solution with k = 1.

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A056737.

[ Min(v) where v is [ b+c: d, y, z in Divisors(n) | d*(d-b)*(d+c) eq n and d ge 0 and b ge 0 and c ge 0 where b is d-y where c is z-d ]: n in [1..85] ]; // Klaus Brockhaus, Nov 19 2008

first(n) = {my(res = vector(n, i, i), t = 0); for(i = 1, n, for(j = i, n \ i, for(k = j, n \ (i * j), res[i * j * k] = min(res[i * j * k], k - i)))); res} \\ David A. Corneth, May 12 2018

a(n) = n-1 for n prime; a(n) = 0 for n a 3rd power. - Klaus Brockhaus, Nov 19 2008

Definition and example edited, and extended beyond a(42) by Klaus Brockhaus, Nov 19 2008

A243981 Minimum range of sets of natural numbers with a product of n.

Original entry on oeis.org

1, 2, 0, 4, 1, 6, 0, 0, 3, 10, 1, 12, 5, 2, 0, 16, 1, 18, 1, 4, 9, 22, 1, 0, 11, 0, 3, 28, 1, 30, 0, 8, 15, 2, 0, 36, 17, 10, 3, 40, 1, 42, 7, 2, 21, 46, 1, 0, 3, 14, 9, 52, 1, 6, 1, 16, 27, 58, 2, 60, 29, 2, 0, 8, 5, 66, 13, 20, 3, 70, 1, 72, 35, 2, 15, 4, 7, 78, 1, 0, 39, 82, 4, 12, 41, 26, 3, 88, 1, 6, 19, 28, 45, 14, 1, 96, 5, 2, 0

Offset: 2

Paul Richards, Nov 11 2014

The minimum difference between the largest and smallest values in the sets of positive integers with a product of n, excluding the singleton set {n}.

			For 45 the sets are {1,45}, {3,15}, {5,9}, {3,3,5} with differences of 44, 12, 4 and 2 respectively.  2 is the minimum and so a(45) = 2.

Alois P. Heinz, Table of n, a(n) for n = 2..10000

Cf. A056737, A060794, A076388, A082119.

a(n) <= A046665(n) for all composite n, a(p) = p - 1 for primes p. - Charlie Neder, Jan 13 2019

A285119 Min(|d(k+1-i) - d(i)|, for i = 1..k), where d(1)..d(k) are the divisors of n^3.

Original entry on oeis.org

0, 2, 6, 0, 20, 6, 42, 16, 0, 15, 110, 12, 156, 7, 30, 0, 272, 9, 342, 20, 84, 33, 506, 20, 0, 65, 162, 84, 812, 30, 930, 128, 176, 153, 70, 0, 1332, 209, 182, 6, 1640, 42, 1806, 110, 132, 345, 2162, 96, 0, 250, 170, 78, 2756, 162, 330, 56, 152, 609, 3422

Offset: 1

Clark Kimberling, Apr 11 2017

a(n) = 0 if and only if n is a square (A000290).

			3^3 = 27 has divisors 1,3,9,27, so that k=4 and d(k+1-i) - d(i) ranges through {-26,-6,6,27}, so that a(3) = 6.

Clark Kimberling, Table of n, a(n) for n = 1..10000

Cf. A002378, A000578, A056737.

f[n_] := f[n] = n^3; Table[Divisors[f[n]] - Reverse[Divisors[f[n]]], {n, 1, 10}]
Table[Min[Abs[Divisors[f[n]] - Reverse[Divisors[f[n]]]]], {n, 1, 100}]

a(n) = my(d=divisors(n^3)); vecmin(apply(abs, d - Vecrev(d))); \\ Michel Marcus, Feb 22 2021

a(n) = A056737(A000578(n)).

cp's OEIS Frontend

A076388 a(n) = minimum of y-x such that x <= y, x*y = n and gcd(x,y)=1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A082119 Smallest positive difference between d and n/d for any divisor d of n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

A139693 a(n) is the smallest positive integer m where, for k divides m, minimum(|k - m/k|) = n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Extensions

A369110 a(n) is the number of distinct elements appearing in the sequence formed by recursively applying A063655 when starting from n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Python

A324921 Index of first occurrence of n in A324920.

Views

Author

Keywords

Crossrefs

Programs

Mathematica

PARI

Extensions

A369793 a(n) is the number of occurrences of n in A063655.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Python

A147763 a(n) = smallest k such that n = d*(d-b)*(d+c), where b, c, d >= 0 and k = b+c.

Views

Author

Keywords

Examples

A147763 a(n) = smallest k such that n = d(d-b)(d+c), where b, c, d >= 0 and k = b+c.