A174905 -id:A174905 - cp's OEIS frontend

A129511 Numbers such that all differences between distinct divisors occur exactly once.

1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 16, 17, 19, 21, 22, 23, 25, 26, 27, 29, 31, 32, 33, 34, 35, 37, 38, 39, 41, 43, 44, 46, 47, 49, 50, 51, 52, 53, 55, 57, 58, 59, 61, 62, 64, 65, 67, 68, 69, 71, 73, 74, 76, 77, 79, 81, 82, 83, 85, 86, 87, 89, 92, 93, 94, 95, 97, 98, 101, 103

Offset: 1

Reinhard Zumkeller, Apr 19 2007

nonn

A129510(a(n)) = A066446(a(n)).

a(n) = A174905(n) for n<27, A174905(27)=37 whereas a(27)=35. - Reinhard Zumkeller, Apr 01 2010

			See example for a(33)=44 in A129510.

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

Cf. A027750, A066446, A129510, A129512 (complement), A174905.

import Data.List.Ordered (isect, union)
a129511 n = a129511_list !! (n-1)
a129511_list = filter (f [] . a027750_row') [1..] where
   f  [] = True
   f zs (d:ds) = null (dds `isect` zs) && f (dds `union` zs) ds
                 where dds = map (subtract d) ds
-- Reinhard Zumkeller, Jun 25 2015

Select[Range[103],CountDistinct[Differences /@ Subsets[Divisors[#], {2}]]==Length[Flatten[Differences /@ Subsets[Divisors[#], {2}]]]&] (* James C. McMahon, Jan 21 2025 *)

A298855 Squarefree semiprimes pq for which the symmetric representation of sigma(pq) has four parts, in increasing order.

Original entry on oeis.org

21, 33, 39, 51, 55, 57, 65, 69, 85, 87, 93, 95, 111, 115, 119, 123, 129, 133, 141, 145, 155, 159, 161, 177, 183, 185, 201, 203, 205, 213, 215, 217, 219, 235, 237, 249, 253, 259, 265, 267, 287, 291, 295, 301, 303, 305, 309, 319, 321, 327, 329, 335, 339, 341, 355, 365, 371, 377, 381, 393, 395

Offset: 1

Hartmut F. W. Hoft, Jan 27 2018

All numbers in this sequence are odd since the symmetric representation of 2*p, p prime > 3, has two parts each of size 3*(p+1)/2, and that for 6 has one part of size 12.
A number in this sequence has the form p*q, p and q prime, 3 <= p and 2*p < q, since in this case 2*p <= floor((sqrt(8*p*q + 1) - 1)/2) < q so that 1's in row p*q of A237048 occur only in positions 1, 2, p and 2*p.
This sequence is a subsequence of A046388, hence of A006881, as well as of A174905, A241008 and A280107.
The two central parts of the symmetric representation of sigma(p*q), each of size (p+q)/2, meet on the diagonal when q = 2*p + 1 since in this case 2*p = floor((sqrt(8*p*q + 1) - 1)/2). These triangular numbers p*(2p+1) form sequence A156592, except for its first element 10, and form a subsequence of the diagonal in the associated irregular triangle of this sequence given in the Example section. They also are a subsequence of A264104. A function to compute the coordinates on the diagonal where the two central parts meet is defined in sequence A240542.
Except for missing 10 the intersection of this sequence and A298856 equals A156592.

			21=3*7 is the smallest number in the sequence since 2*3<7.
1081=23*(2*23+1) is in the sequence; its central parts meet at 751 on the diagonal.
The semiprimes p*q can be arranged as an irregular triangle with rows and columns labeled by the respective odd primes:
  q\p|   3    5    7   11   13   17   19   23
  ---+---------------------------------------
   7 |  21
  11 |  33   55
  13 |  39   65
  17 |  51   85  119
  19 |  57   95  133
  23 |  69  115  161  253
  29 |  87  145  203  319  377
  31 |  93  155  217  341  403
  37 | 111  185  259  407  481  629
  41 | 123  205  287  451  533  697  779
  43 | 129  215  301  473  559  731  817
  47 | 141  235  329  517  611  799  893 1081

Cf. A001358, A005384, A005385, A006881, A046388, A068443, A156592, A174905, A237048, A237270, A237593, A240542, A241008, A264104, A280107, A298856.

(* Function a237270[] is defined in A237270 *)
a006881Q[n_] := Module[{f=FactorInteger[n]}, Length[f]==2 && AllTrue[Last[Transpose[f]], #==1&]]
a298855[m_, n_] := Select[Range[m, n], a006881Q[#] && Length[a237270[#]]==4 &]
a298855[1, 400] (* data *)
(* column for prime p through number n *)
stalk[n_, p_] := Select[a298855[1, n], First[First[FactorInteger[#]]]==p&]

A379461 Irregular triangle read by rows in which row n lists the divisors m of n such that there is a divisor d of n with d < m < 2*d, or 0 if such divisors do not exist.

Original entry on oeis.org

0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 3, 4, 6, 0, 0, 5, 0, 0, 3, 9, 0, 5, 0, 0, 0, 3, 4, 6, 8, 12, 0, 0, 0, 7, 0, 3, 5, 6, 10, 15, 0, 0, 0, 0, 7, 3, 4, 6, 9, 12, 18, 0, 0, 0, 5, 8, 10, 0, 3, 7, 21, 0, 0, 5, 9, 15, 0, 0, 3, 4, 6, 8, 12, 16, 24, 0, 0, 0, 0, 0, 3, 9, 27, 0

Offset: 1

Omar E. Pol, Dec 23 2024

The number of positive terms in row n is A174903(n).
The indices of the rows that contain a zero give A174905.
The indices of the rows that contain positive integers give A005279.
The positive integers in the n-th row are the missing divisors of n in the n-th row of A379374.
The odd integers in the n-th row are the missing odd divisors of n in the n-th row of A379288.

			Triangle begins:
  0;
  0;
  0;
  0;
  0;
  3;
  0;
  0;
  0;
  0;
  0;
  3, 4, 6;
  0;
  0;
  5;
  0;
  0;
  3, 9;
  0;
  5;
  ...
From _Omar E. Pol_, Apr 19 2025: (Start)
For n = 12 there are three divisors m of 12 such that there is a divisor d of 12 with d < m < 2*d. Those divisors are 3, 4 and 6 as shown below:
   d  <  m  <  2*d
--------------------
   1            2
   2     3      4
   3     4      6
   4     6      8
   6           12
  12           24
.
So the 12th row of the triangle is [3, 4, 6]. (End)

Cf. A005279, A027750, A174903, A174905, A182469, A237271, A379288, A379374, A379379, A379384, A383209.

row[n_] := Module[{d = Partition[Divisors[n], 2, 1], e}, e = Select[d, #[[2]] < 2*#[[1]] &][[;; , 2]]; If[e == {}, {0}, e]]; Table[row[n], {n, 1, 55}] // Flatten (* Amiram Eldar, Dec 23 2024 *)

More terms from Amiram Eldar, Dec 23 2024

Name changed by Omar E. Pol, Feb 05 2025

A347273 Number of positive widths in the symmetric representation of sigma(n).

Original entry on oeis.org

1, 3, 4, 7, 6, 11, 8, 15, 13, 18, 12, 23, 14, 24, 23, 31, 18, 35, 20, 39, 32, 36, 24, 47, 31, 42, 40, 55, 30, 59, 32, 63, 48, 54, 45, 71, 38, 60, 56, 79, 42, 83, 44, 84, 73, 72, 48, 95, 57, 93, 72, 98, 54, 107, 72, 111

Offset: 1

Omar E. Pol, Aug 29 2021

a(n) is also the number of columns that contain ON cells in the ziggurat diagram of n. Both diagrams can be unified in a three-dimensional version.
a(n) is also the number of nonzero terms in the n-th row of A249351.
The number of widths in the symmetric representation of sigma(n) is equal to 2*n - 1 = A005408(n-1).
The sum of the positive widths (also the sum of all widths) of the symmetric representation of sigma(n) equals A000203(n).
Indices where a(n) = 2*n - 1 give A174973 and also A238443.
a(p) = p + 1, if p is prime.
a(n) = 2*n - 1, if and only if A237271(n) = 1.
a(n) = A000203(n) if n is a member of A174905.
For the definition of "width" see A249351.

Cf. A000040, A005408, A174905, A174973, A196020, A235791, A236106, A237270, A237271, A237591, A237593, A238443, A249351 (widths), A253258, A347361.

a(n) = A005408(n-1) - A347361(n).

A352696 a(n) = k if the binary representation of k has a 1 (0) exactly where a 1 in the n-th row of A237048 occurs at an odd (even) position, reading from left to right.

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 2, 1, 5, 2, 2, 3, 2, 2, 11, 1, 2, 6, 2, 3, 10, 2, 2, 3, 5, 2, 10, 3, 2, 13, 2, 1, 10, 2, 11, 6, 2, 2, 10, 3, 2, 13, 2, 2, 45, 2, 2, 3, 5, 5, 10, 2, 2, 13, 10, 3, 10, 2, 2, 14, 2, 2, 43, 1, 10, 13, 2, 2, 10, 11, 2, 7, 2, 2, 44, 2, 11, 12, 2, 3, 21, 2, 2, 14, 10

Offset: 1

Hartmut F. W. Hoft, Mar 29 2022

nonn

The phrase "symmetric representation of sigma(n)" is abbreviated below as SRS(n).
Every number in this sequence is a nondiving number and therefore in A061854. Number 22 with binary pattern 10110 is the smallest nondiving number in A061854, but not in this sequence since a number n with 5 odd divisors must have the form n = 2^m * p^4 for some prime p and some m>=0, and the pattern 10110 of odd/even positions of 1's in a row of A237048 requires 1's at positions 1 < 2^(m+1) < p < p^2 < 2^(m+1) * p <= row(n), a contradiction.
a(2^n) = 1 for all n>=0. The single part of SRS(2^n) has width 1, see A238443.
a(2^m * p) = 3 for odd primes p < 2^(m+1) with m >= 1. SRS(2^m * p) consists of a single part whose 2 subparts have sizes 2*T(n, 1) - 1 = 2^m * p - 1 and  2*T(n, p) - 1 = 2^m - p where T(n, k) = ceiling((n+1)/k -(k+1)/2), see A235791. The numbers 2^m * p are a subsequence of A174973 = A238443.
a(p^k) = A000975(k+1) for all odd primes p and k >= 0. Number a(p^k) in binary has k+1 digits with 1's and 0's alternating. SRS(p^k) has k+1 parts all of width 1 and of the symmetric sizes T(p^k, p^i) - T(p^k, 2*p^i) = (p^(k-i) + p^i)/2, for 0 <= i <= k. The numbers p^k are a subsequence of A174905, the odd primes p form the 1st column in the irregular triangle of A239929 and the numbers p^2 form the 1st column in the irregular triangle of A247687.

			Sequence values for the first 4 powers of 3: {a(1), a(3), a(9), a(27)} = {1, 2, 5, 10} = {1, 10, 101, 1010}.
Table for a(1..16), a(27) and a(28) together with their lists of the base-2 representation, of the odd/even positions of 1's in the n-th row of A237048, and of the sizes of the parts in SRS(n):
n  a(n) odd/even   A237048         A237270
1   1   {1}        {1}             {1}
2   1   {1}        {1}             {3}
3   2   {1,0}      {1,1}           {2,2}
4   1   {1}        {1,0}           {7}
5   2   {1,0}      {1,1}           {3,3}
6   3   {1,1}      {1,0,1}         {12}
7   2   {1,0}      {1,1,0}         {4,4}
8   1   {1}        {1,0,0}         {15}
9   5   {1,0,1}    {1,1,1}         {5,3,5}
10  2   {1,0}      {1,0,0,1}       {9,9}
11  2   {1,0}      {1,1,0,0}       {6,6}
12  3   {1,1}      {1,0,1,0}       {28}
13  2   {1,0}      {1,1,0,0}       {7,7}
14  2   {1,0}      {1,0,0,1}       {12,12}
15 11   {1,0,1,1}  {1,1,1,0,1}     {8,8,8}
16  1   {1}        {1,0,0,0,0}     {31}
...
27 10   {1,0,1,0}  {1,1,1,0,0,1}   {14,6,6,14}
28  3   {1,1}      {1,0,0,0,0,0,1} {56}
...

Cf. A000975, A061854, A174905, A174973, A235791, A237048, A237270, A237591, A237593, A238443, A239929, A247687, A352030.

(* function a237048[ ] is defined in A237048 *)
b237048[n_] := Fold[2#1+Mod[#2, 2]&, 0, Flatten[Position[a237048[n], 1]]]
a352696[n_] := Map[b237048, Range[n]]
a352696[85]

A368087 Numbers of the form 2^k * p^s with k>=0, s>=0, p>2 prime and 2^(k+1) < p.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 16, 17, 19, 22, 23, 25, 26, 27, 29, 31, 32, 34, 37, 38, 41, 43, 44, 46, 47, 49, 50, 52, 53, 58, 59, 61, 62, 64, 67, 68, 71, 73, 74, 76, 79, 81, 82, 83, 86, 89, 92, 94, 97, 98, 101, 103, 106, 107, 109, 113, 116, 118, 121, 122, 124, 125, 127, 128

Offset: 1

Hartmut F. W. Hoft, Dec 11 2023

nonn

This sequence is a subsequence of A174905 = A241008 union A241010. The symmetric representation of sigma (cf. A237593) for a number m in this sequence consists of s+1 parts, the number of odd divisors of m, each part having width 1.

			14 = 2*7 is a term since 4 < 7.
44 = 4*11 is a term since 8 < 11.

Cf. A005279, A174905, A235791, A237048, A237270, A237593, A241008, A241010, A249223, A341969.

propQ[n_] := Module[{fL=FactorInteger[n]}, Length[fL]==1||(Length[fL]==2&&fL[[1, 1]]==2&&fL[[1, 1]]^(fL[[1, 2]]+1)

A375611 Numbers k whose symmetric representation of sigma(k) has at least a part with maximum width 2.

Original entry on oeis.org

6, 12, 15, 18, 20, 24, 28, 30, 35, 36, 40, 42, 45, 48, 54, 56, 63, 66, 70, 75, 77, 78, 80, 88, 91, 96, 99, 100, 102, 104, 105, 108, 110, 112, 114, 117, 130, 132, 135, 138, 143, 150, 153, 154, 156, 160, 162, 165, 170, 174, 175, 176, 182, 186, 187, 189, 190, 192, 195, 196, 200

Offset: 1

Hartmut F. W. Hoft, Aug 21 2024

nonn

Number m = 2^k * q, k >= 0 and q odd, is in this sequence precisely when for any divisor s <= A003056(m) of q there is at most one divisor t of q satisfying s < t <= min(2^(k+1) * s, A003056(m)), and at least one such pair s < t of successive odd divisors exists. Equivalently, row m of the triangle in A249223 contains at least one 2, but no number larger than 2.

			a(4) = 18 has width pattern 1 2 1 2 1 in its symmetric representation of sigma consisting of a single part, and row 18 in the triangle of A249223 is 1 1 2 1 1.
a(9) = 35 has width pattern 1 0 1 2 1 0 1 in its symmetric representation of sigma consisting of 3 parts, and row 35 in the triangle of A249223 is 1 0 0 0 1 1 2.
Irregular triangle of rows a(n) in triangle of A341970, i.e. of positions of 1's in triangle of A237048, and for the corresponding widths to the diagonal in triangle of A341969:
a(n)| row in A341970      left half of row in A341969
6   | 1   3               1   2
12  | 1   3               1   2
15  | 1   2   3   5       1   0   1   2
18  | 1   3   4           1   2   1
20  | 1   5               1   2
24  | 1   3               1   2
28  | 1   7               1   2
30  | 1   3   4   5       1   2   1   2
35  | 1   2   5   7       1   0   1   2
36  | 1   3   8           1   2   1
...

Column 2 of A253258.
Subsequence of A005279.
Some subsequences are A352030, A370205, A370206, A370209.
Cf. A003056, A174905, A235791, A237593, A249223, A249351, A250068, A341969, A341970.

eP[n_] := If[EvenQ[n], FactorInteger[n][[1, 2]], 0]+1
sDiv[n_] := Module[{d=Select[Divisors[n], OddQ]}, Select[Union[d, d*2^eP[n]], #<=row[n]&]]
mW2Q[n_] := Max[FoldWhileList[#1+If[OddQ[#2], 1, -1]&, sDiv[n], #1<=2&]]==2
a375611[m_, n_] := Select[Range[m, n], mW2Q]
a375611[1, 200]

A381137 Lexicographically earliest sequence of distinct positive integers such that no 3 terms are in harmonic progression.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 12, 13, 14, 16, 17, 19, 21, 22, 23, 25, 26, 27, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 41, 43, 44, 46, 47, 48, 49, 50, 51, 52, 53, 55, 57, 58, 59, 60, 61, 62, 63, 64, 65, 67, 68, 69, 70, 71, 73, 74, 76, 79, 81, 82, 83, 85, 86

Offset: 1

Neal Gersh Tolunsky, Feb 15 2025

nonn

A harmonic progression is a sequence of values whose reciprocals are in arithmetic progression. Equivalently, if (a, b, c) is a harmonic progression, then b is the harmonic mean of a and c.
a(n) is the smallest integer greater than a(n-1) which does not form a 3-term harmonic progression with 2 previously occurring terms.
Every prime occurs in the sequence.

			6 is not a term in the sequence because it would form a harmonic progression with 2 and 3, which occurred earlier. The progression (1/6, 1/3, 1/2) has common difference 1/6.

Neal Gersh Tolunsky, Table of n, a(n) for n = 1..10000

Analogous sequences: A003278 (for arithmetic progressions), A000452 (for geometric progressions).

Cf. A005279, A174905.

from itertools import count
def A381137_generator():
    a_list = []
    forbidden = set()
    a = 0
    while 1:
        a = next(k for k in count(a+1) if k not in forbidden)
        yield a
        forbidden.update(a*b//m for b in a_list if (m:=2*b-a) > 0 and a*b%m == 0)
        a_list.append(a) # Pontus von Brömssen, Mar 04 2025

cp's OEIS Frontend

A129511 Numbers such that all differences between distinct divisors occur exactly once.

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A298855 Squarefree semiprimes p*q for which the symmetric representation of sigma(p*q) has four parts, in increasing order.

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A379461 Irregular triangle read by rows in which row n lists the divisors m of n such that there is a divisor d of n with d < m < 2*d, or 0 if such divisors do not exist.

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A347273 Number of positive widths in the symmetric representation of sigma(n).

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A352696 a(n) = k if the binary representation of k has a 1 (0) exactly where a 1 in the n-th row of A237048 occurs at an odd (even) position, reading from left to right.

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A368087 Numbers of the form 2^k * p^s with k>=0, s>=0, p>2 prime and 2^(k+1) < p.

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A375611 Numbers k whose symmetric representation of sigma(k) has at least a part with maximum width 2.

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A381137 Lexicographically earliest sequence of distinct positive integers such that no 3 terms are in harmonic progression.

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A298855 Squarefree semiprimes pq for which the symmetric representation of sigma(pq) has four parts, in increasing order.