A193829 -id:A193829 - cp's OEIS frontend

A356223 Position of n-th appearance of 2n in the sequence of prime gaps (A001223). If 2n does not appear at least n times, set a(n) = -1.

2, 6, 15, 79, 68, 121, 162, 445, 416, 971, 836, 987, 2888, 1891, 1650, 5637, 5518, 4834, 9237, 8152, 10045, 21550, 20248, 20179, 29914, 36070, 24237, 53355, 52873, 34206, 103134, 90190, 63755, 147861, 98103, 117467, 209102, 206423, 124954, 237847, 369223

Offset: 1

Gus Wiseman, Aug 04 2022

nonn

Prime gaps (A001223) are the differences between consecutive prime numbers. They begin: 1, 2, 2, 4, 2, 4, 2, 4, 6, ...

			We need the first 15 prime gaps (1, 2, 2, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6) before we reach the 3rd appearance of 6, so a(6) = 15.

The first appearances are at A038664, seconds A356221.
Diagonal of A356222.
A001223 lists the prime gaps.
A073491 lists numbers with gapless prime indices.
A356224 counts divisors with gapless prime indices, complement A356225.
A356226 = gapless interval lengths of prime indices, run-lengths A287170.
Cf. A000005, A028334, A029709, A060681, A119313, A137921, A193829, A274121.

nn=1000;
gaps=Differences[Array[Prime,nn]];
Table[Position[gaps,2*n][[n,1]],{n,Select[Range[nn],Length[Position[gaps,2*#]]>=#&]}]

A259366 Numbers for which the differences between consecutive divisors (ordered by size) are not distinct.

Original entry on oeis.org

6, 12, 15, 18, 20, 24, 30, 36, 40, 42, 45, 48, 54, 56, 60, 63, 66, 70, 72, 75, 78, 80, 84, 90, 91, 96, 99, 100, 102, 105, 108, 110, 112, 114, 120, 126, 130, 132, 135, 138, 140, 144, 150, 156, 160, 162, 165, 168, 174, 180, 182, 186, 189, 192, 195, 198, 200

Offset: 1

Reinhard Zumkeller, Jun 25 2015

nonn

			.    n |  A193829(n,*)      |  A027750(n,*)         |
.  ----+--------------------+-----------------------+------------
.   10 |  {1,3,5}           |  {1,2,5,10}           |
.   11 |  {10}              |  {1,11}               |
.   12 |  {1,1,1,2,6}       |  {1,2,3,4,6,12}       |  a(2) = 12
.   13 |  {12}              |  {1,13}               |
.   14 |  {1,5,7}           |  {1,2,7,14}           |
.   15 |  {2,2,10}          |  {1,3,5,15}           |  a(3) = 15
.   16 |  {1,2,4,8}         |  {1,2,4,8,16}         |
.   17 |  {16}              |  {1,17}               |
.   18 |  {1,1,3,3,9}       |  {1,2,3,6,9,18}       |  a(4) = 18
.   19 |  {18}              |  {1,19}               |
.   20 |  {1,2,1,5,10}      |  {1,2,4,5,10,20}      |  a(5) = 20
.   21 |  {2,4,14}          |  {1,3,7,21}           |
.   22 |  {1,9,11}          |  {1,2,11,22}          |
.   23 |  {22}              |  {1,23}               |
.   24 |  {1,1,1,2,2,4,12}  |  {1,2,3,4,6,8,12,24}  |  a(6) = 24
.   25 |  {4,20}            |  {1,5,25}             |            .

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

Cf. A193829, A027750, A060682, A000005, A060683 (complement), subsequence of A129512.

a259366 n = a259366_list !! (n-1)
a259366_list = filter (\x -> a060682 x < a000005' x - 1) [2..]

q[k_] := Module[{d = Differences[Divisors[k]]}, CountDistinct[d] < Length[d]]; Select[Range[200], q] (* Amiram Eldar, Jan 27 2025 *)

A060682(a(n)) < A000005(a(n)) - 1.

A328025 Irregular triangle read by rows where row n gives the differences between consecutive divisors of n in weakly decreasing order.

Original entry on oeis.org

1, 2, 2, 1, 4, 3, 1, 1, 6, 4, 2, 1, 6, 2, 5, 3, 1, 10, 6, 2, 1, 1, 1, 12, 7, 5, 1, 10, 2, 2, 8, 4, 2, 1, 16, 9, 3, 3, 1, 1, 18, 10, 5, 2, 1, 1, 14, 4, 2, 11, 9, 1, 22, 12, 4, 2, 2, 1, 1, 1, 20, 4, 13, 11, 1, 18, 6, 2, 14, 7, 3, 2, 1, 28, 15, 5, 4, 2, 1, 1, 1

Offset: 1

Gus Wiseman, Oct 02 2019

			Triangle begins:
   {}
   1
   2
   2  1
   4
   3  1  1
   6
   4  2  1
   6  2
   5  3  1
  10
   6  2  1  1  1
  12
   7  5  1
  10  2  2
   8  4  2  1
  16
   9  3  3  1  1
  18
  10  5  2  1  1
For example, the divisors of 18 are {1,2,3,6,9,18}, with differences {1,1,3,3,9}, so row 18 is {9,3,3,1,1}.

Same as A193829 with rows sorted in weakly decreasing order.
Same as A328027 with rows reversed.
Row sums are A001477.
Row lengths are A000005.
First column is A060681.
Heinz numbers of rows are A328023.
Cf. A027750, A060680, A060682, A060683, A070211, A129308, A328024.

Table[Sort[Differences[Divisors[n]],Greater],{n,30}]

A328027 Irregular triangle read by rows where row n lists, in weakly increasing order, the differences between consecutive divisors of n.

Original entry on oeis.org

1, 2, 1, 2, 4, 1, 1, 3, 6, 1, 2, 4, 2, 6, 1, 3, 5, 10, 1, 1, 1, 2, 6, 12, 1, 5, 7, 2, 2, 10, 1, 2, 4, 8, 16, 1, 1, 3, 3, 9, 18, 1, 1, 2, 5, 10, 2, 4, 14, 1, 9, 11, 22, 1, 1, 1, 2, 2, 4, 12, 4, 20, 1, 11, 13, 2, 6, 18, 1, 2, 3, 7, 14, 28, 1, 1, 1, 2, 4, 5, 15

Offset: 1

Gus Wiseman, Oct 02 2019

			Triangle begins:
   {}
   1
   2
   1  2
   4
   1  1  3
   6
   1  2  4
   2  6
   1  3  5
  10
   1  1  1  2  6
  12
   1  5  7
   2  2 10
   1  2  4  8
  16
   1  1  3  3  9
  18
   1  1  2  5 10
   2  4 14
   1  9 11
  22
   1  1  1  2  2  4 12
For example, the divisors of 18 are {1,2,3,6,9,18}, with differences {1,1,3,3,9}, which is row 18.

Same as A193829 with rows sorted in increasing order.
Same as A328025 with rows reversed.
Row sums are A001477.
Row lengths are A000005.
First column is A060680.
Cf. A027750, A060681, A060682, A060683, A070211, A129308, A328024.

Table[Sort[Differences[Divisors[n]]],{n,30}]

A057449 Product of differences between consecutive positive divisors of n.

Original entry on oeis.org

1, 1, 2, 2, 4, 3, 6, 8, 12, 15, 10, 12, 12, 35, 40, 64, 16, 81, 18, 100, 112, 99, 22, 192, 80, 143, 216, 588, 28, 600, 30, 1024, 352, 255, 224, 1944, 36, 323, 520, 2400, 40, 3087, 42, 3388, 2880, 483, 46, 12288, 252, 5625, 952, 6084, 52, 19683, 1056, 14112, 1216

Offset: 1

Leroy Quet, Sep 25 2000

Row products of triangle A193829, if n >= 2. - Omar E. Pol, Aug 31 2011

When p is prime a(p) = p-1. - Michel Marcus, May 15 2014

			Since positive divisors of 12 are 1, 2, 3, 4, 6 and 12, a(12) = (2-1) *(3-2) *(4-3) *(6-4) *(12-6) = 1 *1 *1 *2 *6 = 12.

Alois P. Heinz, Table of n, a(n) for n = 1..10000

a057449 = product . a193829_row  -- Reinhard Zumkeller, Jun 23 2013

Times@@Differences[Divisors[#]]&/@Range[80]  (* Harvey P. Dale, Apr 20 2011 *)

a(n) = {my(d = divisors(n)); prod(i=1, #d-1, d[i+1] - d[i]);} \\ Michel Marcus, May 15 2014

a(n) = Product_{k=1..A000005(n)-1} A027750(n,k+1)-A027750(n,k). - Reinhard Zumkeller, Jun 23 2013

A330492 a(n) = sum of second differences of the sorted divisors of n.

Original entry on oeis.org

0, 0, 1, 0, 2, 0, 3, 4, 4, 0, 5, 0, 6, 8, 7, 0, 8, 0, 9, 12, 10, 0, 11, 16, 12, 16, 13, 0, 14, 0, 15, 20, 16, 24, 17, 0, 18, 24, 19, 0, 20, 0, 21, 28, 22, 0, 23, 36, 24, 32, 25, 0, 26, 40, 27, 36, 28, 0, 29, 0, 30, 40, 31, 48, 32, 0, 33, 44, 34, 0, 35, 0, 36

Offset: 2

Michel Lagneau, Dec 16 2019

nonn

The sums of the first differences of the divisors of n are given by the sequence b(n) = n - 1.

Let the set {D(i)} = {d(i + 1) - d(i)} where the d(i) are the divisors of an integer m listed in ascending order with i = 1, 2 , ..., tau(n)-1. The sequence is given by a(n) = Sum_{k = 1..tau(n)-2} (D(k + 1) - D(k)).

			a(12) = 5 because the divisors of 12 are {1, 2, 3, 4, 6, 12} and {D(i)} = {d(i+1)-d(i)} ={1, 1, 1, 2, 6}, Sum_{D(i), i = 1..4} {D(i+1)-D(i)} = 0 + 0 + 1 + 4 = 5.

Antti Karttunen, Table of n, a(n) for n = 2..8191
Antti Karttunen, Data supplement: n, a(n) computed for n = 2..65537

Cf. A000005, A027750, A036263, A193829.

with(numtheory):nn:=100:
for n from 2 to nn do:
d:=divisors(n):n0:=nops(d):T:=array(1..n0-1,[0$n0-1]):
  for j from 1 to n0-1 do:
   T[j]:=d[j+1]-d[j]:
  od:
   s:=sum(‘T[i+1]-T[i] ’,‘i’=1..n0-2): printf(`%d, `,s):
od:
*** alternative program using the formula ***
with(numtheory):nn:=100:
for n from 2 to nn do:
d:=divisors(n):t:=tau(n):s:=d[t]-d[t-1]+d[1]-d[2] :
  printf(`%d, `,s):
od:

Array[Total@ Differences[Divisors@ #, 2] &, 73, 2] (* Michael De Vlieger, Dec 16 2019 *)

a(n) = my(d=divisors(n)); d[#d] - d[#d-1] + d[1] - d[2]; \\ Michel Marcus, Feb 05 2020

from sympy import primefactors
def a(n): p = primefactors(n)[0]; return (n//p - 1) * (p - 1)
print([a(n) for n in range(2, 75)]) # Michael S. Branicky, Apr 04 2021

a(n) = d(tau(n)) - d(tau(n) - 1) + d(1) - d(2) where d(i) are the divisors of n.
a(prime(n)) = 0 and a(2k) = k-1, k = 1, 2, ...
a(p^2) = (p-1)^2 if p prime, with the generalization a(p^m) = (p-1)(p^(m-1) - 1).
a(n) = (n/p-1)*(p-1), where p is the least prime factor of n. - Nathaniel Gregg, Apr 04 2021

A337806 Numbers that are a divisor of the product of the differences of its consecutive divisors.

Original entry on oeis.org

8, 12, 16, 20, 24, 27, 28, 30, 32, 36, 40, 44, 45, 48, 52, 56, 60, 63, 64, 68, 70, 72, 76, 80, 81, 84, 88, 90, 92, 96, 99, 100, 104, 105, 108, 112, 116, 117, 120, 124, 125, 126, 128, 132, 135, 136, 140, 144, 148, 150, 152, 153, 154, 156, 160, 164, 165, 168, 171, 172, 175, 176, 180, 182, 184

Offset: 1

Scott R. Shannon, Sep 23 2020

nonn

			8 is a term as the divisors of 8 are 1,2,4,8, the differences of which are 1,2,4, and 1*2*4 = 8 which is a multiple of 8.
20 is a term as the divisors of 20 are 1,2,4,5,10,20, the differences of which are 1,2,1,5,10, and 1*2*1*5*10 = 100 which is a multiple of 20.
27 is a term as the divisors of 27 are 1,3,9,27, the differences of which are 2,6,18, and 2*6*18 = 216 which is a multiple of 27.
99 is a term as the divisors of 99 are 1,3,9,11,33,99, the difference of which are 2,6,2,22,66, and 2*6*2*22*66 = 34848 which is a multiple of 99.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A193829, A027750. Contains A337877.

filter:= proc(n) local R,p;
  R:= sort(convert(numtheory:-divisors(n),list));
  p:= convert(R[2..-1]-R[1..-2],`*`);
  p mod n = 0
end proc:
select(filter, [$2..1000]); # Robert Israel, Sep 27 2020

Select[Range[2, 200], Divisible[Times @@ Differences @ Divisors[#], #] &] (* Amiram Eldar, Sep 23 2020 *)

isok(k) = my(d=divisors(k)); (#d > 1) && (vecprod(vector(#d-1, k, d[k+1]-d[k])) % k) == 0; \\ Michel Marcus, Sep 23 2020

A356221 Position of second appearance of 2n in the sequence of prime gaps A001223; if 2n does not appear at least twice, a(n) = -1.

Original entry on oeis.org

3, 6, 11, 72, 42, 47, 62, 295, 180, 259, 297, 327, 446, 462, 650, 1315, 1059, 1532, 4052, 2344, 3732, 3861, 8805, 7234, 4754, 2810, 4231, 14124, 5949, 9834, 17200, 10229, 19724, 25248, 15927, 30765, 42673, 28593, 24554, 50523, 44227, 44390, 29040, 89715, 47350

Offset: 1

Gus Wiseman, Aug 02 2022

nonn

Prime gaps (A001223) are the differences between consecutive prime numbers. They begin: 1, 2, 2, 4, 2, 4, 2, 4, 6, ...

The position of the first (instead of second) appearance of 2n is A038664.
Column k = 2 of A356222.
The position of the n-th appearance of 2n is A356223.
A001223 lists the prime gaps, reduced A028334.
A073491 lists numbers with gapless prime indices.
A274121 counts appearances of the n-th prime gap in those prior.
A356226 gives the lengths of maximal gapless intervals of prime indices.
Cf. A029709, A066205, A137921, A193829, A287170, A328335, A328457, A356224, A356225.

nn=1000;
gaps=Differences[Array[Prime,nn]];
mnrm[s_]:=If[Min@@s==1,mnrm[DeleteCases[s-1,0]]+1,0];
Table[Position[gaps,2*n][[2,1]],{n,mnrm[Select[Range[nn],Length[Position[gaps,2*#]]>=2&]]}]

A356222 Array read by antidiagonals upwards where A(n,k) is the position of the k-th appearance of 2n in the sequence of prime gaps A001223. If A001223 does not contain 2n at least k times, set A(n,k) = -1.

Original entry on oeis.org

2, 4, 3, 9, 6, 5, 24, 11, 8, 7, 34, 72, 15, 12, 10, 46, 42, 77, 16, 14, 13, 30, 47, 53, 79, 18, 19, 17, 282, 62, 91, 61, 87, 21, 22, 20, 99, 295, 66, 97, 68, 92, 23, 25, 26, 154, 180, 319, 137, 114, 80, 94, 32, 27, 28, 189, 259, 205, 331, 146, 121, 82, 124, 36, 29, 33

Offset: 1

Gus Wiseman, Aug 04 2022

Prime gaps (A001223) are the differences between consecutive prime numbers. They begin: 1, 2, 2, 4, 2, 4, 2, 4, 6, ...

This is a permutation of the positive integers > 1.

			Array begins:
        k=1 k=2 k=3 k=4 k=5 k=6 k=7 k=8 k=9
  n=1:   2   3   5   7  10  13  17  20  26
  n=2:   4   6   8  12  14  19  22  25  27
  n=3:   9  11  15  16  18  21  23  32  36
  n=4:  24  72  77  79  87  92  94 124 126
  n=5:  34  42  53  61  68  80  82 101 106
  n=6:  46  47  91  97 114 121 139 168 197
  n=7:  30  62  66 137 146 150 162 223 250
  n=8: 282 295 319 331 335 378 409 445 476
  n=9:  99 180 205 221 274 293 326 368 416
For example, the positions in A001223 of appearances of 2*3 begin: 9, 11, 15, 16, 18, 21, 23, ..., which is row n = 3 (A320701).

The row containing n is A028334(n).
Row n = 1 is A029707.
Row n = 2 is A029709.
Column k = 1 is A038664.
The column containing n is A274121(n).
Column k = 2 is A356221.
The diagonal A(n,n) is A356223.
A001223 lists the prime gaps.
A073491 lists numbers with gapless prime indices.
A356224 counts even divisors with gapless prime indices, complement A356225.
Cf. A066205, A119313, A193829, A287170, A328457, A356226, A356232.

gapa=Differences[Array[Prime,10000]];
Table[Position[gapa,2*(k-n+1)][[n,1]],{k,6},{n,k}]

cp's OEIS Frontend

A356223 Position of n-th appearance of 2n in the sequence of prime gaps (A001223). If 2n does not appear at least n times, set a(n) = -1.

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A259366 Numbers for which the differences between consecutive divisors (ordered by size) are not distinct.

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A328025 Irregular triangle read by rows where row n gives the differences between consecutive divisors of n in weakly decreasing order.

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A328027 Irregular triangle read by rows where row n lists, in weakly increasing order, the differences between consecutive divisors of n.

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A057449 Product of differences between consecutive positive divisors of n.

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A330492 a(n) = sum of second differences of the sorted divisors of n.

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A337806 Numbers that are a divisor of the product of the differences of its consecutive divisors.

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A356221 Position of second appearance of 2n in the sequence of prime gaps A001223; if 2n does not appear at least twice, a(n) = -1.

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