A273199 Integers which have a positive but not monotone difference table of their divisors.

51, 55, 57, 69, 87, 93, 111, 119, 123, 129, 141, 159, 177, 183, 201, 207, 213, 219, 237, 249, 253, 267, 275, 291, 303, 309, 319, 321, 327, 333, 339, 369, 377, 381, 393, 403, 411, 417, 447, 453, 471, 489, 501, 519, 537, 543, 573, 579, 591, 597

Offset: 1

Author

Peter Luschny, May 17 2016

Keywords

• nonn

Comments

For an integer n>0 and not the unity we define DTD(n) to be the difference table of the divisors of n. We say that DTD(n) is positive if all entries in the table are positive and we call DTD(n) monotone if all rows and all columns of the table are nondecreasing (reading from left to right and from top to bottom).

Examples

			159 is in this sequence because the DTD of 159 has only positive entries but not all columns are nondecreasing:
[  1   3   53 159]
[  2  50  106]
[ 48  56]
[  8]
		

Crossrefs

Cf. A000040, A246655, A273102, A273130, A273200, A273201.

Programs

Sage

def is_A273199(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]
            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(0,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 True
    return False
print([n for n in range(1,600) if is_A273199(n)])