A273102 -id:A273102 - cp's OEIS frontend

A187202 The bottom entry in the difference table of the divisors of n.

1, 1, 2, 1, 4, 2, 6, 1, 4, 0, 10, 1, 12, -2, 8, 1, 16, 12, 18, -11, 8, -6, 22, -12, 16, -8, 8, -3, 28, 50, 30, 1, 8, -12, 28, -11, 36, -14, 8, -66, 40, 104, 42, 13, 24, -18, 46, -103, 36, -16, 8, 21, 52, 88, 36, 48, 8, -24, 58, -667, 60, -26, -8, 1, 40, 72

Offset: 1

Omar E. Pol, Aug 01 2011

Note that if n is prime then a(n) = n - 1.
Note that if n is a power of 2 then a(n) = 1.
a(A193671(n)) > 0; a(A187204(n)) = 0; a(A193672(n)) < 0. [Reinhard Zumkeller, Aug 02 2011]
First differs from A187203 at a(14). - Omar E. Pol, May 14 2016
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)

			a(18) = 12 because 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
with bottom entry a(18) = 12.
Note that A187203(18) = 4.

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

Cf. A000005, A007318, A027750, A187203, A273102.

a187202 = head . head . dropWhile ((> 1) . length) . iterate diff . divs
   where divs n = filter ((== 0) . mod n) [1..n]
         diff xs = zipWith (-) (tail xs) xs
-- Reinhard Zumkeller, Aug 02 2011

f:=proc(n) local k,d,lis; lis:=divisors(n); d:=nops(lis);
add( (-1)^k*binomial(d-1,k)*lis[d-k], k=0..d-1); end;
[seq(f(n),n=1..100)]; # N. J. A. Sloane, May 01 2016

Table[d = Divisors[n]; Differences[d, Length[d] - 1][[1]], {n, 100}] (* T. D. Noe, Aug 01 2011 *)

A187202(n)={ for(i=2,#n=divisors(n), n=vecextract(n,"^1")-vecextract(n,"^-1")); n[1]}  \\ M. F. Hasler, Aug 01 2011

a(n) = Sum_{k=0..d-1} (-1)^k*binomial(d-1,k)*D[d-k], where D is a sorted list of the d = A000005(n) divisors of n. - N. J. A. Sloane, May 01 2016

a(2^k) = 1.

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

A273103 Sum of the elements of the difference triangle of the divisors of n (including the divisors of n).

Original entry on oeis.org

1, 4, 6, 11, 10, 21, 14, 26, 25, 31, 22, 52, 26, 41, 54, 57, 34, 86, 38, 66, 72, 61, 46, 103, 71, 71, 90, 102, 58, 205, 62, 120, 108, 91, 134, 157, 74, 101, 126, 75, 82, 329, 86, 174, 218, 121, 94, 110, 141, 158, 162, 210, 106, 373, 202, 269, 180, 151, 118, -437, 122, 161, 250

Offset: 1

Omar E. Pol, May 15 2016

sign

a(n) is the sum of the n-th slice of the tetrahedron A273102.

First differs from A187215 at a(14).

			For n = 14 the divisors of 14 are 1, 2, 7, 14, and the difference triangle of the divisors is
1 . 2 . 7 . 14
. 1 . 5 . 7
. . 4 . 2
. . .-2
The sum of all elements of the triangle is 1 + 2 + 7 + 14 + 1 + 5 + 7 + 4 + 2 - 2 = 41, so a(14) = 41.
Note that A187215(14) = 45.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A027750, A125128, A187215, A273102.

Table[Total@ Flatten@ NestWhileList[Differences, Divisors@ n, Length@ # > 1 &], {n, 63}] (* Michael De Vlieger, May 17 2016 *)

a(n) = {my(d = divisors(n)); my(s = vecsum(d)); for (k=1, #d-1, d = vector(#d-1, n, d[n+1] - d[n]); s += vecsum(d);); s;} \\ Michel Marcus, May 16 2016

a(n) = 2n, if n is prime.

a(2^k) = A125128(k+1), k >= 0.

More terms from Michel Marcus, May 16 2016

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 *)

A272210 Difference table of the divisors of the positive integers (with every table read by antidiagonals upwards).

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, May 18 2016

This is an irregular tetrahedron in which T(n,j,k) is the k-th element of the j-th antidiagonal (read upwards) of the difference table 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 diagonals are also the divisors of the powers of 2 from 1 to n, for example if n = 8 the finite sequence of diagonals is [1], [1, 2], [1, 2, 4], [1, 2, 4, 8].
First differs from A273132 at a(89).

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

Cf. A000005, A000217, A027750, A161700, A184389, A187202, A273102, A273103, A273109, A273135, A273132, A273136, A273261, A273262, A273263.

Table[Table[#[[m - k + 1, k]], {m, Length@ #}, {k, m}] &@ NestWhileList[Differences, Divisors@ n, Length@ # > 1 &], {n, 15}] // Flatten (* Michael De Vlieger, Jun 29 2016 *)

A273104 Absolute 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 absolute 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 A187203(n).
The sum of the elements of the slice n is A187215(n).
For another version see A273102 from which differs at a(92).

			For n = 18 the divisors of 18 are 1, 2, 3, 6, 9, 18, so the absolute 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
. . . . 0 . 4
. . . . . 4
and the 18th slice is
1, 2, 3, 6, 9, 18;
1, 1, 3, 3, 9;
0, 2, 0, 6;
2, 2, 6;
0, 4;
4;
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 absolute difference triangle of the divisors of n flattened.
Row lengths are the terms of A184389. Row sums give A187215.
Triangle begins:
1;
1, 2, 1;
1, 3, 2;
1, 2, 4, 1, 2, 1;
...

Cf. A027750, A184389, A187202-A187205, A187207-A187209, A187215, A273102.

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

A273130 Numbers which have only positive entries in the difference table of their divisors.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 21, 23, 25, 27, 29, 31, 32, 33, 37, 39, 41, 43, 47, 49, 51, 53, 55, 57, 59, 61, 64, 65, 67, 69, 71, 73, 79, 81, 83, 85, 87, 89, 93, 95, 97, 101, 103, 107, 109, 111, 113, 115, 119, 121, 123, 125, 127, 128, 129, 131, 133

Offset: 1

Peter Luschny, May 16 2016

nonn

Primes and powers of primes are in the sequence.

			85 is in the sequence because the difference table of the divisors of 85 has only entries greater than 0:
[1, 5, 17, 85]
[4, 12, 68]
[8, 56]
[48]

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A014567, A187202, A273102, A273103, A273109, A273157 (complement).

Select[Range@ 1000, {} == NestWhile[ Differences, Divisors @ #, # != {} && Min[#] > 0 &] &] (* Giovanni Resta, May 16 2016 *)

has(v)=if(#v<2, v[1]>0, if(vecmin(v)<1, 0, has(vector(#v-1,i,v[i+1]-v[i]))))
is(n)=has(divisors(n)) \\ Charles R Greathouse IV, May 16 2016

def sf(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 False
    return True
print([z for z in range(1,100) if sf(z)])

A273262 Irregular triangle read by rows: T(n,k) = sum of the elements of the k-th antidiagonal of the difference table of the divisors of n.

Original entry on oeis.org

1, 1, 3, 1, 5, 1, 3, 7, 1, 9, 1, 3, 4, 13, 1, 13, 1, 3, 7, 15, 1, 5, 19, 1, 3, 10, 17, 1, 21, 1, 3, 4, 5, 11, 28, 1, 25, 1, 3, 16, 21, 1, 5, 7, 41, 1, 3, 7, 15, 31, 1, 33, 1, 3, 4, 13, 6, 59, 1, 37, 1, 3, 7, 3, 31, 21, 1, 5, 13, 53, 1, 3, 28, 29, 1, 45, 1, 3, 4, 5, 11, 4, 36, 39, 1, 9, 61, 1, 3, 34, 33, 1, 5, 19, 65

Offset: 1

Omar E. Pol, May 20 2016

If n is prime then row n contains only two terms: 1 and 2*n-1.
Row 2^k gives the first k+1 positive terms of A000225, k >= 0.
Note that this sequence contains negative terms.
First differs from A274532 at a(41).

			Triangle begins:
1;
1, 3;
1, 5;
1, 3, 7;
1, 9;
1, 3, 4, 13;
1, 13;
1, 3, 7, 15;
1, 5, 19;
1, 3, 10, 17;
1, 21;
1, 3, 4, 5, 11, 28;
1, 25;
1, 3, 16, 21;
1, 5, 7, 41;
1, 3, 7, 15, 31;
1, 33;
1, 3, 4, 13, 6, 59;
1, 37;
1, 3, 7, 3, 31, 21;
1, 5, 13, 53;
1, 3, 28, 29;
1, 45;
1, 3, 4, 5, 11, 4, 36, 39;
1, 9, 61;
1, 3, 34, 33;
1, 5, 19, 65;
...
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 antidiagonal sums give [1, 3, 4, 13, 6, 59] which is also the 18th row of the irregular triangle.

Row lengths give A000005. Column 1 is A000012. Right border gives A161700. Row sums give A273103.

Cf. A000225, A161700, A187202, A272210, A273102, A273135, A273261, A273263, A274532.

Table[Map[Total, Table[#[[m - k + 1, k]], {m, Length@ #}, {k, m}], {1}] &@ NestWhileList[Differences, Divisors@ n, Length@ # > 1 &], {n, 27}] (* Michael De Vlieger, Jun 26 2016 *)

row(n) = {my(d = divisors(n)); my(nd = #d); my(m = matrix(#d, #d)); for (j=1, nd, m[1,j] = d[j];); for (i=2, nd, for (j=1, nd - i +1, m[i,j] = m[i-1,j+1] - m[i-1,j];);); vector(nd, i, sum(k=0, i-1, m[i-k, k+1]));}
tabf(nn) = for (n=1, nn, print(row(n)););
lista(nn) = for (n=1, nn, v = row(n); for (j=1, #v, print1(v[j], ", "));); \\ Michel Marcus, Jun 25 2016

A273263 Irregular triangle read by rows: T(n,k) is the sum of the elements of the k-th column of the difference table of the divisors of n.

Original entry on oeis.org

1, 2, 2, 3, 3, 3, 4, 4, 5, 5, 4, 5, 6, 6, 7, 7, 4, 6, 8, 8, 7, 9, 9, 4, 7, 10, 10, 11, 11, 4, 6, 8, 10, 12, 12, 13, 13, 4, 9, 14, 14, 11, 13, 15, 15, 5, 8, 12, 16, 16, 17, 17, 12, 11, 12, 15, 18, 18, 19, 19, -3, 4, 10, 15, 20, 20, 13, 17, 21, 21, 4, 13, 22, 22, 23, 23, -4, 3, 8, 12, 16, 20, 24, 24, 21, 25, 25

Offset: 1

Omar E. Pol, May 22 2016

If n is prime then row n is [n, n].
It appears that the last two terms of the n-th row are [n, n], n > 1.
First differs from A274533 at a(38).

			Triangle begins:
   1;
   2,  2;
   3,  3;
   3,  4,  4;
   5,  5;
   4,  5,  6,  6;
   7,  7;
   4,  6,  8,  8;
   7,  9,  9;
   4,  7, 10, 10;
  11, 11;
   4,  6,  8, 10, 12, 12;
  13, 13;
   4,  9, 14, 14;
  11, 13, 15, 15;
   5,  8, 12, 16, 16;
  17, 17;
  12, 11, 12, 15, 18, 18;
  19, 19;
  -3,  4, 10, 15, 20, 20;
  13, 17, 21, 21;
   4, 13, 22, 22;
  23, 23;
  -4,  3,  8, 12, 16, 20, 24, 24;
  21, 25, 25;
   4, 15, 26, 26;
  ...
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 column sums give [12, 11, 12, 15, 18, 18] which is also the 18th row of the irregular triangle.

Row lengths give A000005. Right border gives A000027. Column 1 is A161857. Row sums give A273103.

Cf. A187202, A273102, A273135, A272210, A273136, A273261, A273262, A274533.

Table[Total /@ Transpose@ Map[Function[w, PadRight[w, Length@ #]], NestWhileList[Differences, #, Length@ # > 1 &]] &@ Divisors@ n, {n, 25}] // Flatten (* Michael De Vlieger, Jun 26 2016 *)

row(n) = {my(d = divisors(n)); my(nd = #d); my(m = matrix(#d, #d)); for (j=1, nd, m[1,j] = d[j];); for (i=2, nd, for (j=1, nd - i +1, m[i,j] = m[i-1,j+1] - m[i-1,j];);); vector(nd, j, sum(i=1, nd, m[i, j]));}
tabf(nn) = for (n=1, nn, print(row(n)););
lista(nn) = for (n=1, nn, v = row(n); for (j=1, #v, print1(v[j], ", "));); \\ Michel Marcus, Jun 25 2016

A273200 Prime power-like integers.

Original entry on oeis.org

2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 21, 23, 25, 27, 29, 31, 32, 33, 37, 39, 41, 43, 47, 49, 53, 59, 61, 64, 65, 67, 71, 73, 79, 81, 83, 85, 89, 95, 97, 101, 103, 107, 109, 113, 115, 121, 125, 127, 128, 131, 133, 137, 139, 145, 149, 151, 155, 157, 161, 163, 167, 169

Offset: 1

Peter Luschny, May 17 2016

nonn

Let DTD(n) denote the difference table of the divisors of n. The DTDs of prime powers (in the sense of A246655) have only positive entries and the rows and columns of their DTD are nondecreasing.
We define an integer n>0 and not the unity to be prime power-like if and only if DTD(n) has only positive entries and nondecreasing rows and columns (read from left to right and from top to bottom).
This sequence lists the prime power-like integers and sequence A273201 lists the integers which are prime power-like but not prime powers. Thus we have the inclusions A000040 < A246655 < A273200 and the union A273200 = A273201 U A246655.
Integers which have a positive but not monotone DTD are listed in A273199. Integers with a positive DTD are listed in A273130.

			125 is in this sequence because it is a prime power and has the DTD:
[  1    5    25  125]
[  4   20   100]
[ 16   80]
[ 64]
161 is in this sequence because the DTD of 161 has only positive entries and nondecreasing rows and columns:
[   1    7   23  161]
[   6   16  138]
[  10  122]
[ 112]

Cf. A000040, A246655, A273102, A273130, A273199, A273201.

pplikeQ[n_] := Module[{T, DTD, DTD2}, If[n == 1, Return[False]]; T = Divisors[n]; DTD = Table[Differences[T, k], {k, 0, Length[T] - 1}]; If[AnyTrue[Flatten[DTD], NonPositive], Return[False]]; DTD2 = Transpose[PadRight[#, Length[T], Infinity]& /@ DTD]; AllTrue[DTD, OrderedQ] && AllTrue[DTD2, OrderedQ]];
Select[Range[200], pplikeQ] (* Jean-François Alcover, Jun 28 2019 *)

def is_prime_power_like(n):
    if n == 1: return False
    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]
            if T[m-k, k] <= 0: return False
    non_decreasing = lambda L: all(x<=y for x, y in zip(L, L[1:]))
    b = True
    for k in range(len(D)-1):
        b &= non_decreasing(T.row(k)[:len(D)-k])
        b &= non_decreasing(T.column(k)[:len(D)-k])
        if not b: return False
    return b
[n for n in range(1, 170) if is_prime_power_like(n)]

cp's OEIS Frontend

A187202 The bottom entry in the difference table of the divisors of n.

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A273103 Sum of the elements of the difference triangle of the divisors of n (including the divisors of n).

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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.

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

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A272210 Difference table of the divisors of the positive integers (with every table read by antidiagonals upwards).

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A273104 Absolute difference table of the divisors of the positive integers.

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A273130 Numbers which have only positive entries in the difference table of their divisors.

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A273262 Irregular triangle read by rows: T(n,k) = sum of the elements of the k-th antidiagonal of the difference table of the divisors of n.