A187202 -id:A187202 - cp's OEIS frontend

A242393 Records in A187202 by index.

1, 3, 5, 7, 11, 13, 17, 19, 23, 29, 30, 42, 90, 132, 150, 156, 168, 240, 360, 420, 756, 924, 960, 1260, 2160, 2520, 4620, 5040, 6720, 7560, 14280, 16380, 18480, 20160, 31680, 35280, 41580, 45360, 50400, 65520, 98280, 110880, 171360, 226800, 249480, 257040, 262080, 277200, 332640

Offset: 1

Robert G. Wilson v, May 12 2014

nonn

The first nine odd primes are present, and then only (very) abundant numbers.

			The final or last difference of 30 is 50 = A187202(30). The next higher value is 104 which occurs at 42, A187202(42) = 104.

Robert G. Wilson v, Table of n, a(n) for n = 1..106

Cf. A187202.

f[n_] := (dvr = Divisors@ n; Differences[dvr, Length@ dvr - 1][[1]]); k = 1; lst = {}; mx = 0; While[k < 100000001, If[ f@ k > mx, mx = f@ k; Print[{k, mx}]; AppendTo[lst, {k, mx}]]; k++]; Transpose[ lst][[1]]

A272374 Numbers n such that A187202(n) is <= 0.

Original entry on oeis.org

10, 14, 20, 22, 24, 26, 28, 34, 36, 38, 40, 46, 48, 50, 58, 60, 62, 63, 70, 74, 80, 82, 84, 86, 94, 96, 98, 99, 100, 105, 106, 117, 118, 120, 122, 134, 136, 138, 140, 142, 146, 152, 153, 154, 158, 160, 166, 170, 171, 174, 178, 180, 182, 184, 186, 189, 190, 192, 194, 196, 198, 200, 202, 206, 208, 214

Offset: 1

Robert G. Wilson v, Apr 28 2016

nonn

Odd terms: 63, 99, 105, 117, 153, 171, 189, ..., .

Indices n where A187202(n) =0 are 10, 171, 1947, 2619, 265105, ...- R. J. Mathar, May 06 2016

R. J. Mathar, Table of n, a(n) for n = 1..5000

Cf. A187202, A187204, A272375.

f[n_] := Block[{d = Divisors@ n}, Differences[d, Length@ d - 1][[1]]]; Select[ Range@ 215, f@# < 1&]

Edited by N. J. A. Sloane, May 01 2016

A272375 Negative records in A187202 by index.

Original entry on oeis.org

1, 10, 14, 20, 24, 38, 40, 48, 60, 84, 120, 180, 264, 300, 396, 432, 504, 540, 630, 720, 840, 1320, 2184, 2400, 2772, 3024, 3120, 3360, 3780, 3960, 5940, 6840, 7200, 8400, 9240, 10080, 12600, 15120, 21840, 22680, 25200, 27720, 30240, 40320, 52920, 55440, 83160, 128520, 131040, 138600, 166320, 196560, 221760

Offset: 1

Robert G. Wilson v, Apr 28 2016

nonn

Often a(n)/2 is a member of A242393.

Search limit: 8300000000.

Robert G. Wilson v, Table of n, a(n) for n = 1..123

Cf. A187202, A242393, A272374.

f[n_] := (dvr = Divisors@ n; Differences[dvr, Length@ dvr - 1][[1]]); lst = {}; k = 1; mx = 2; While[k < 1000001, a = f@ k; If[a < mx, mx = a; AppendTo[lst, k]]; k++]

A187203 The bottom entry in the absolute difference triangle of the divisors of n.

Original entry on oeis.org

1, 1, 2, 1, 4, 2, 6, 1, 4, 0, 10, 1, 12, 2, 8, 1, 16, 4, 18, 1, 8, 6, 22, 2, 16, 8, 8, 3, 28, 4, 30, 1, 8, 12, 24, 1, 36, 14, 8, 0, 40, 4, 42, 3, 20, 18, 46, 1, 36, 0, 8, 3, 52, 8, 36, 0, 8, 24, 58, 3, 60, 26, 4, 1, 40, 12, 66, 3, 8, 2, 70, 4, 72, 32, 32, 3

Offset: 1

Omar E. Pol, Aug 01 2011

nonn

Note that if n is prime then a(n) = n - 1.
Where records occurs gives the odd noncomposite numbers (A006005).
First differs from A187202 at a(14).
It is important to note that at each step in the process, the absolute differences are taken, and not just at the end. This sequence is therefore not abs(A187202) as I mistakenly assumed at first. - Alonso del Arte, Aug 01 2011

			a(18) = 4 because the divisors of 18 are 1, 2, 3, 6, 9, 18, and the absolute difference triangle of the divisors is:
  1 . 2 . 3 . 6 . 9 . 18
  . 1 . 1 . 3 . 3 . 9
  . . 0 . 2 . 0 . 6
  . . . 2 . 2 . 6
  . . . . 0 . 4
  . . . . . 4
with bottom entry a(18) = 4.
Note that A187202(18) = 12.

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A006005, A027750, A187202, A187205, A187208.

a187203 = head . head . dropWhile ((> 1) . length) . iterate diff . divs
   where divs n = filter ((== 0) . mod n) [1..n]
         diff xs = map abs $ zipWith (-) (tail xs) xs
-- Reinhard Zumkeller, Aug 02 2011

Table[d = Divisors[n]; While[Length[d] > 1, d = Abs[Differences[d]]]; d[[1]], {n, 100}] (* T. D. Noe, Aug 01 2011 *)
Table[Nest[Abs[Differences[#]]&,Divisors[n],DivisorSigma[0,n]-1],{n,100}]//Flatten (* Harvey P. Dale, Nov 07 2022 *)

A187203(n)={ for(i=2,#n=divisors(n), n=abs(vecextract(n,"^1")-vecextract(n,"^-1"))); n[1]}  \\ M. F. Hasler, Aug 01 2011

Edited by Omar E. Pol, May 14 2016

A273102 Difference table of the divisors of the positive integers.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, May 15 2016

This is an irregular tetrahedron T(n,j,k) read by rows in which the slice n lists the elements of the rows of the difference triangle of the divisors of n (including the divisors of n).
The first row of the slice n is also the n-th row of the triangle A027750.
The bottom entry of the slice n is A187202(n).
The sum of the elements of the slice n is A273103(n).
For another version see A273104, from which differs at a(92).
From David A. Corneth, May 20 2016: (Start)
Each element of the difference table of the divisors of n can be expressed in terms of the divisors of n and use of Pascal's triangle. Suppose a, b, c, d and e are the divisors of n. Then the difference table is as follows (rotated for ease of reading):
a
. . b-a
b . . . . c-2b+a
. . c-b . . . . . d-3c+3b-a
c . . . . d-2c+b . . . . . . e-4d+6c-4b+a
. . d-c . . . . . e-3d+3c-b
d . . . . e-2d+c
. . e-d
e
From here we can see Pascal's triangle occurring. Induction can be used to show that it's the case in general.
(End)

			For n = 18 the divisors of 18 are 1, 2, 3, 6, 9, 18, so the difference triangle of the divisors of 18 is
  1 . 2 . 3 . 6 . 9 . 18
    1 . 1 . 3 . 3 . 9
      0 . 2 . 0 . 6
        2 .-2 . 6
         -4 . 8
           12
and the 18th slice is
  1, 2, 3, 6, 9, 18;
  1, 1, 3, 3, 9;
  0, 2, 0, 6;
  2,-2, 6;
  -4, 8;
  12;
The tetrahedron begins:
  1;
  1, 2;
  1;
  1, 3;
  2;
  1, 2, 4;
  1, 2;
  1;
  ...
This is also an irregular triangle T(n,r) read by rows in which row n lists the difference triangle of the divisors of n flattened. Row lengths are the terms of A184389. Row sums give A273103.
Triangle begins:
  1;
  1, 2, 1;
  1, 3, 2;
  1, 2, 4, 1, 2, 1;
  ...

Cf. A007182, A027750, A184389, A187202, A187204, A273103, A273104, A273109.

Table[Drop[FixedPointList[Differences, Divisors@ n], -2], {n, 15}] // Flatten (* Michael De Vlieger, May 16 2016 *)

def A273102_DTD(n): # DTD = Difference Table of Divisors
    D = divisors(n)
    T = matrix(ZZ, len(D))
    for (m, d) in enumerate(D):
        T[0, m] = d
        for k in range(m-1, -1, -1) :
            T[m-k, k] = T[m-k-1, k+1] - T[m-k-1, k]
    return [T.row(k)[:len(D)-k] for k in range(len(D))]
# Keeps the rows of the DTD, for instance
# A273102_DTD(18)[1] = 1,1,3,3,9 (see the example above).
for n in range(1,19): print(A273102_DTD(n)) # Peter Luschny, May 18 2016

A200154 T(n,k) = number of 0..k arrays x(0..n-1) of n elements with zero (n-1)-st difference.

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 1, 4, 5, 4, 1, 5, 8, 9, 2, 1, 6, 13, 22, 15, 8, 1, 7, 18, 41, 40, 39, 2, 1, 8, 25, 66, 103, 112, 45, 16, 1, 9, 32, 107, 202, 275, 182, 129, 6, 1, 10, 41, 158, 381, 730, 685, 688, 149, 32, 1, 11, 50, 219, 636, 1589, 2036, 2525, 844, 243, 2, 1, 12, 61, 304, 1033, 3000, 5153, 7488, 5221, 2090, 369, 64, 1

Offset: 1

R. H. Hardin, Nov 13 2011

Table starts
 1    1     1     1      1      1       1       1        1        1
 3    4     5     6      7      8       9      10       11       12
 5    8    13    18     25     32      41      50       61       72
 9   22    41    66    107    158     219     304      403      516
15   40   103   202    381    636    1033    1550     2287     3212
39  112   275   730   1589   3000    5181    8350    13871    21588
45  182   685  2036   5153  11370   23035   43284    76523   129052
129  688  2525  7488  18809  52166  121921  253768   484977   867086
149  844  5221 19262  68813 194818  514113 1171190  2531421  5019770
243 2090 13897 62772 256859 841122 2347671 6169890 14503751 31169760
T(n,k) is the number of integer lattice points in k*C(n) where C(n) is a certain polytope with vertices having rational entries (the intersection of [0,1]^n with a hyperplane).  Thus row n is an Ehrhart quasi-polynomial of degree n-1. - Robert Israel, Dec 12 2019

			Some solutions for n=7, k=6:
  5  6  5  3  6  0  0  5  4  1  2  2  0  2  1  2
  3  1  5  1  6  5  4  0  2  5  2  0  2  0  4  0
  3  3  6  5  6  1  6  2  0  1  1  4  3  4  6  2
  3  2  3  6  5  1  3  6  0  2  1  6  3  3  6  3
  2  0  2  5  5  3  2  6  1  6  2  5  3  1  5  2
  1  1  6  5  6  2  6  1  2  6  3  3  4  3  4  1
  4  1  1  3  1  2  0  1  5  0  3  1  6  1  2  4

R. H. Hardin, Table of n, a(n) for n = 1..321

Row 3 is A000982(n+1).

Cf. A187202 (for 3rd PARI function).

pad(d, n) = while(#d != n, d = concat([0], d)); d;
mydigits(i,n) = if (n<2, vector(i), digits(i,n));
bedt(n) = {for(i=2, #n=n, n=vecextract(n, "^1")-vecextract(n, "^-1")); n[1];}
T(n, k) = {k++; my(nbok = 0); for (i=0, k^n-1, d = pad(mydigits(i,k), n); if (bedt(d) == 0, nbok++);); nbok;} \\ Michel Marcus, Apr 08 2017

A187204 Numbers n such that the bottom entry in the difference table of the divisors of n is 0.

Original entry on oeis.org

10, 171, 1947, 2619, 265105, 478834027, 974622397, 11373118351

Offset: 1

Omar E. Pol, Aug 01 2011

Numbers n such that A187202(n) = 0.
11373118351 and 1756410942451 are also in the sequence (not necessarily the next two terms). - Donovan Johnson, Aug 05 2011
For every integer m, does there exist a prime p such that abs(A187202(r * m)) > abs(A187202(q * m)) and sign(A187202(r * m)) = sign(A187202(q * m)), and q >= p is prime and prime r > q? - David A. Corneth, Apr 08 2017
No other terms up to 3*10^9. - Michel Marcus, Apr 09 2017
a(9) > 6*10^10. 138662735650982521 and 168248347462416481 are also terms. - Giovanni Resta, Apr 12 2017

			10 has divisors 1, 2, 5, 10. The third difference of these numbers is 0.  This is the only possible number having 2 prime factors of the form p*q. The other terms have factorization 171 = 3^2*19, 1947 = 3*11*59, 2619 = 3^3*97, and 265105 = 5*37*1433.

Cf. A027750, A187202, A187203, A193671, A193672.

import Data.List (elemIndices)
a187204 n = a187204_list !! (n-1)
a187204_list = map (+ 1) $ elemIndices 0 $ map a187202 [1..]
-- Reinhard Zumkeller, Aug 02 2011

t = {}; Do[d = Divisors[n]; If[Differences[d, Length[d]-1] == {0}, AppendTo[t, n]], {n, 10^4}]; t (* T. D. Noe, Aug 01 2011 *)

is(n) = my(d=divisors(n)); !sum(i=1, #d, binomial(#d-1,i-1)*d[i]*(-1)^i) \\ David A. Corneth, Apr 08 2017

Suggested by T. D. Noe in the "history" of A187203.
a(6)-a(7) from Donovan Johnson, Aug 03 2011
a(8) from Giovanni Resta, Apr 11 2017

A187205 Numbers such that the last of the absolute differences of divisors is 0.

Original entry on oeis.org

10, 40, 50, 56, 104, 130, 136, 160, 170, 171, 224, 230, 232, 250, 290, 310, 312, 370, 392, 410, 430, 459, 470, 520, 530, 560, 590, 610, 624, 640, 648, 670, 710, 730, 790, 830, 890, 896, 970, 1000, 1010, 1030, 1070, 1088, 1090, 1130, 1160, 1216, 1218, 1221

Offset: 1

Omar E. Pol, Aug 01 2011

nonn

Numbers n such that A187203(n) = 0.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A027750, A187202, A187203, A187204.

import Data.List (elemIndices)
a187205 n = a187205_list !! (n-1)
a187205_list = map (+ 1) $ elemIndices 0 $ map a187203 [1..]

A273109 Numbers n such that in the difference triangle of the divisors of n (including the divisors of n) the diagonal from the bottom entry to n gives the divisors of n.

Original entry on oeis.org

1, 2, 4, 8, 12, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 262144, 524288, 1048576, 2097152, 4194304, 8388608, 16777216, 33554432, 67108864, 134217728, 268435456, 536870912, 1073741824, 2147483648, 4294967296, 8589934592

Offset: 1

Omar E. Pol, May 15 2016

nonn

Is this also the union of 12 and the powers of 2?

All powers of 2 are in the sequence.

			For n = 12 the difference triangle of the divisors of 12 is
1 . 2 . 3 . 4 . 6 . 12
. 1 . 1 . 1 . 2 . 6
. . 0 . 0 . 1 . 4
. . . 0 . 1 . 3
. . . . 1 . 2
. . . . . 1
The bottom entry is 1, and the diagonal from the bottom entry to 12 is [1, 2, 3, 4, 6, 12] hence the diagonal gives the divisors of 12, so 12 is in the sequence.
Note that for n = 12 and the powers of 2 the descending diagonals, from left to right, are symmetrics, for example: the first diagonal is 1, 1, 0, 0, 1, 1.

Cf. A000079, A027750, A187202, A273102, A273103.

aQ[n_] := Module[{d=Divisors[n]}, nd = Length[d]; vd = d; ans = True; Do[ vd = Differences[vd]; If[Max[vd] != d[[nd-k]], ans=False; Break[]], {k,1,nd-1}]; ans]; Select[Range[100000], aQ] (* Amiram Eldar, Feb 23 2019 *)

isok(n) = {my(d = divisors(n)); my(nd = #d); my(vd = d); for (k=1, nd-1, vd = vector(#vd-1, j, vd[j+1] - vd[j]); if (vecmax(vd) != d[nd-k], return (0));); return (1);} \\ Michel Marcus, May 16 2016

a(12)-a(21) from Michel Marcus, May 16 2016

a(22)-a(35) from Amiram Eldar, Feb 23 2019

A273135 Difference table of the divisors of the positive integers (with every table read by antidiagonals downwards).

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, May 18 2016

This is an irregular tetrahedron T(n,j,k) in which the slice n lists the elements of the j-th antidiagonal of the difference triangle of the divisors of n.
The first row of the slice n is also the n-th row of the triangle A027750.
The bottom entry of the slice n is A187202(n).
The number of elements in the n-th slice is A000217(A000005(n)) = A184389(n).
The sum of the elements of the n-th slice is A273103(n).
The antidiagonal sums give A273262.
If n is a power of 2 the antidiagonals are also the divisors of the powers of 2 from 1 to n in decreasing order, for example if n = 8 the finite sequence of antidiagonals is [1], [2, 1], [4, 2, 1], [8, 4, 2, 1].
First differs from A272121 at a(92).

			The tables of the first nine positive integers are
  1; 1, 2; 1, 3; 1, 2, 4; 1, 5; 1, 2, 3, 6; 1, 7; 1, 2, 4, 8; 1, 3, 9;
     1;    2;    1, 2;    4;    1, 1, 3;    6;    1, 2, 4;    2, 6;
                 1;             0, 2;             1, 2;       4;
                                2;                1;
For n = 18 the difference table of the divisors of 18 is
  1,  2, 3, 6, 9, 18;
  1,  1, 3, 3, 9;
  0,  2, 0, 6;
  2, -2, 6;
 -4,  8;
 12;
This table read by antidiagonals downwards gives the finite subsequence [1], [2, 1], [3, 1, 0], [6, 3, 2, 2], [9, 3, 0, -2, -4], [18, 9, 6, 6, 8, 12].

Cf. A000005, A000217, A027750, A161700, A184389, A187202, A272210, A272121, A273102, A273103, A273262.

Table[Table[#[[m - k + 1, k]], {m, Length@ #}, {k, m, 1, -1}] &@ NestWhileList[Differences, Divisors@ n, Length@ # > 1 &], {n, 15}] // Flatten (* Michael De Vlieger, Jun 26 2016 *)

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A242393 Records in A187202 by index.

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A272374 Numbers n such that A187202(n) is <= 0.

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A272375 Negative records in A187202 by index.

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A187203 The bottom entry in the absolute difference triangle of the divisors of n.

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A273102 Difference table of the divisors of the positive integers.

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A200154 T(n,k) = number of 0..k arrays x(0..n-1) of n elements with zero (n-1)-st difference.

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A187204 Numbers n such that the bottom entry in the difference table of the divisors of n is 0.

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A187205 Numbers such that the last of the absolute differences of divisors is 0.

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