A187202 -id:A187202 - cp's OEIS frontend

A273136 Difference table of the divisors of the positive integers (with every table read by columns).

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

Offset: 1

Omar E. Pol, Jun 26 2016

This is an irregular tetrahedron in which T(n,j,k) is the k-th element of the j-th column 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 columns sums give A273263.
If n is a power of 2 the subsequence lists the elements of the difference table of the divisors of n in nondecreasing order, for example if n = 8 the finite sequence of columns is [1, 1, 1, 1], [2, 2, 2], [4, 4], [8].
First differs from A273137 at a(86).

			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 columns gives the finite subsequence [1, 1, 0, 2, -4, 12], [2, 1, 2, -2, 8], [3, 3, 0, 6], [6, 3, 6], [9, 9], [18].

Cf. A000005, A000217, A027750, A184389, A187202, A272210, A273102, A273103, A273135, A273137, A273263.

Table[Transpose@ Map[Function[w, PadRight[w, Length@ #]], NestWhileList[Differences, #, Length@ # > 1 &]] &@ Divisors@ n, {n, 15}] /. 0 -> {} // Flatten (* Michael De Vlieger, Jun 26 2016 *)

A273131 Numbers n such that the bottom entry of the difference table of the divisors of n divides n.

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 14, 16, 24, 32, 64, 128, 152, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 262144, 524288, 1048576, 2097152, 4194304, 8388608, 16777216, 33554432, 67108864, 134217728, 268435456, 536870912, 1073741824, 2147483648

Offset: 1

Omar E. Pol, May 16 2016

nonn

All powers of 2 are in the sequence because the bottom entries of their difference triangles are always 1's.

Besides 6, 12, 14, 24 and 152, are there any other non-powers of 2 in this sequence? - David A. Corneth, May 19 2016

			For n = 14 the difference triangle of the divisors of 14 is
1 . 2 . 7 . 14
. 1 . 5 . 7
. . 4 . 2
. . .-2
The bottom entry is -2 and -2 divides 14, so 14 is in the sequence.

Lars Blomberg, Table of n, a(n) for n = 1..40

Cf. A000079, A027750, A187202, A273102, A273103, A273109.

Select[Range[10^6], Function[k, If[k == {0}, False, Divisible[#, First@ k]]]@ NestWhile[Differences, Divisors@ #, Length@ # > 1 &] &] (* Michael De Vlieger, May 17 2016 *)

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]);); vd[1] && ((n % vd[1]) == 0);} \\ Michel Marcus, May 16 2016

is(n) = my(d=divisors(n),s=sum(i=1,#d,binomial(#d-1,i-1)*(-1)^i*d[i]));if(s!=0,n%s==0) \\ David A. Corneth, May 19 2016

def is_A273131(n):
    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[len(D)-1, 0].divides(n)
print([n for n in range(1, 6000) if is_A273131(n)])
# Peter Luschny, May 18 2016

a(12) = 128 and a(14)-a(25) from Michel Marcus, May 16 2016
a(26)-a(28) from David A. Corneth, May 19 2016
a(29)-a(37) from Lars Blomberg, Oct 18 2016

A273133 a(n) = n minus the bottom entry of the difference table of the divisors of n.

Original entry on oeis.org

0, 1, 1, 3, 1, 4, 1, 7, 5, 10, 1, 11, 1, 16, 7, 15, 1, 6, 1, 31, 13, 28, 1, 36, 9, 34, 19, 31, 1, -20, 1, 31, 25, 46, 7, 47, 1, 52, 31, 106, 1, -62, 1, 31, 21, 64, 1, 151, 13, 66, 43, 31, 1, -34, 19, 8, 49, 82, 1, 727, 1, 88, 71, 63, 25, -6, 1, 31, 61, 148, 1, 12, 1, 106, 11, 31, 13, 22, 1, 439, 65, 118, 1, 1541

Offset: 1

Omar E. Pol, May 17 2016

sign

From David A. Corneth, May 20 2016: (Start)
The bottom 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, and the difference triangle of the divisors is:
1 . 2 . 3 . 6 . 9 . 18
. 1 . 1 . 3 . 3 . 9
. . 0 . 2 . 0 . 6
. . . 2 .-2 . 6
. . . .-4 . 8
. . . . . 12
The bottom entry is 12, so a(18) = 18 - 12 = 6.

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

Cf. A000005, A000040, A000225, A007318, A187202, A273102, A273103, A273109.

Array[# - First@ NestWhile[Differences, Divisors@ #, Length@ # > 1 &] &, 84] (* Michael De Vlieger, May 20 2016 *)

a(n) = my(d=divisors(n));n-sum(i=1,#d,binomial(#d-1,i-1)*(-1)^(#d-i)*d[i]) \\ David A. Corneth, May 20 2016

def A273133(n):
    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 n - T[len(D)-1, 0]
print([A273133(n) for n in range(1, 85)]) # Peter Luschny, May 18 2016

a(n) = n - A187202(n).
a(n) = 1, if n is prime.
a(2^k) = 2^k - 1 = A000225(k), k >= 0.

A273157 Numbers which have nonpositive entries in the difference table of their divisors (complement of A273130).

Original entry on oeis.org

6, 10, 12, 14, 15, 18, 20, 22, 24, 26, 28, 30, 34, 35, 36, 38, 40, 42, 44, 45, 46, 48, 50, 52, 54, 56, 58, 60, 62, 63, 66, 68, 70, 72, 74, 75, 76, 77, 78, 80, 82, 84, 86, 88, 90, 91, 92, 94, 96, 98, 99, 100, 102, 104, 105, 106, 108, 110, 112, 114, 116, 117

Offset: 1

Peter Luschny, May 16 2016

nonn

Primorial numbers (A002110) greater than 2 are in this sequence.

			30 is in this sequence because the difference table of the divisors of 30 is:
[1, 2, 3, 5, 6, 10, 15, 30]
[1, 1, 2, 1, 4, 5, 15]
[0, 1, -1, 3, 1, 10]
[1, -2, 4, -2, 9]
[-3, 6, -6, 11]
[9, -12, 17]
[-21, 29]
[50]

Cf. A069059, A187202, A273102, A273103, A273109, A273130 (complement).

def nsf(z):
    D = divisors(z)
    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]
            if T[m-k, k] <= 0: return True
    return False
print([n for n in range(1, 100) if nsf(n)])

A331573 The bottom entry in the forward difference table of the Euler totient function phi for 1..n.

Original entry on oeis.org

1, 0, 1, -2, 5, -14, 39, -102, 247, -558, 1197, -2494, 5167, -10850, 23311, -51132, 113333, -250694, 547871, -1175998, 2475153, -5117486, 10439895, -21142030, 42777735, -86960284, 178221401, -368541508, 767762191, -1606535062, 3365499467, -7038925364, 14671422797, -30450115592

Offset: 1

Robert G. Wilson v, Jan 20 2020

sign

a(2n) is a nonpositive even number while a(2n-1) is an odd positive number.

Abs(a(n)) < abs(a(n+1)) for 1 < n < 8000.

			a(8) = -102 because:
1     1     2     2     4     2     6     4  (first 8 terms of A000010)
   0     1     0     2    -2     4    -2     (first 7 terms of A057000)
      1    -1     2    -4     6     6
        -2     3    -6    10   -12
            5    -9    16   -22
             -14    25   -38
                 39   -63
                  -102
The first principal right descending diagonal is this sequence.

Cf. A187202, A000010, A057000.

Cf. A002088.

f[n_] := Differences[ Array[ EulerPhi, n], n -1][[1]]; Array[f, 34] (* or *)
nmx = 34; Join[ {1}, Differences[ Array[ EulerPhi, nmx], #][[1]] & /@ Range[nmx - 1]]

a(n) = Sum_{k=1..n} (-1)^(n-k)*binomial(n-1,k-1)*phi(k). - Ridouane Oudra, Aug 21 2021

a(n) = Sum_{k=1..n} (-1)^(n-k)*binomial(n,k)*A002088(k). - Ridouane Oudra, Oct 02 2022

A373921 The last entry in the difference table for {the n-th row of A177028 arranged in increasing order}.

Original entry on oeis.org

3, 4, 5, 3, 7, 8, 5, 7, 11, 7, 13, 14, 6, 12, 17, 11, 19, 20, 8, 17, 23, 15, 21, 26, 17, 19, 29, 19, 31, 32, 21, 27, 30, 6, 37, 38, 25, 32, 41, 27, 43, 44, 12, 37, 47, 31, 45, 50, 20, 42, 53, 35, 44, 56, 37, 47, 59, 39, 61, 62, 41, 44, 57, 12, 67, 68, 45, 49, 71, 47, 73, 74, 32

Offset: 3

Robert G. Wilson v, Jun 22 2024

Inspired by A342772 and A187202.
The n-th row of A177028 are all integers k for which n is a k-gonal number.
As an example: row 10 of A177028 contain 3 and 10, because 10 is a 10-gonal number but also a triangular number.
-3n/2 < a(n) <= n.
a(n) = n if n is an odd prime (A065091), an odd composite number in A274967, or even numbers in A274968.
a(n) = 0: 231, tested up to 150000.
a(n) < 0: 441, 540, 561, 1089, 1128, 1296, 1521, 1701, 1716, 1881, 2016, 2211, 2541, 2556, 2601, ..., .
a(n) is negative less than 1% of the time.

			a(15) = 6, because the 15th row of A177028 is {3,6,15} -> {3,9} -> {6};
a(36) = 6, because the 36th row of A177028 is {3,4,13,36} -{1,9,23} - {8,14} -> {6};
a(225) = 37, because the 225th row of A177028 is {4,8,24,76,225} -> {4,16,52,149} -> {12,36,97} -> {24,61} -> {37};
a(561) = -82, because the 561st row of A177028 is {3,6,12,39,188,561} -> {3,6,27,149,373} -> {3,21,122,224} -> {18,101,102}, {83,1} -> {-82}; etc.

Robert G. Wilson v, Table of n, a(n) for n = 3..150000

Cf. A063778, A177025, A177028, A176774, A176948, A274967, A342550, A342805.

planeFigurateQ[n_, r_] := IntegerQ[((r -4) + Sqrt[(r -4)^2 + 8n (r -2)])/(2 (r -2))]; a[n_] := Block[{pg = Select[ Range[3, n], planeFigurateQ[n, #] &]}, Differences[pg, Length@ pg - 1][[1]]]; Array[a, 73, 3]

cp's OEIS Frontend

A273136 Difference table of the divisors of the positive integers (with every table read by columns).

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A273131 Numbers n such that the bottom entry of the difference table of the divisors of n divides n.

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A273133 a(n) = n minus the bottom entry of the difference table of the divisors of n.

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A273157 Numbers which have nonpositive entries in the difference table of their divisors (complement of A273130).

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A331573 The bottom entry in the forward difference table of the Euler totient function phi for 1..n.

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Formula

A373921 The last entry in the difference table for {the n-th row of A177028 arranged in increasing order}.

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