cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A273201 Integers which are prime power-like but not prime powers.

Original entry on oeis.org

21, 33, 39, 65, 85, 95, 115, 133, 145, 155, 161, 185, 203, 205, 215, 217, 235, 259, 261, 265, 279, 287, 295, 301, 305, 329, 335, 341, 355, 365, 371, 395, 407, 413, 415, 427, 445, 451, 469, 473, 481, 485, 497
Offset: 1

Views

Author

Peter Luschny, May 17 2016

Keywords

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).
We define an integer n to be prime power-like if and only if DTD(n) is positive and monotone. All prime powers (in the sense of A246655 (but not in the sense of A000961)) are prime power-like integers. Sequence A273200 provides the prime power-like integers. This 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.

Examples

			95 is in this sequence because the DTD of 95 has positive entries and nondecreasing rows and columns:
[ 1   5  19  95]
[ 4  14  76]
[10  62]
[52]

Crossrefs

Cf. A000040, A246655, A273102, A273130, A273199, A273200.

Programs

  • Mathematica
    pplikeQ[n_] := Module[{T, DTD, DTD2}, If[n == 1 || PrimePowerQ[n], 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[500], pplikeQ] (* Jean-François Alcover, Jun 28 2019 *)
  • Sage
    # uses[is_prime_power_like from A273200]
# Compare script in A273200.
def is_A273201(n):
    return not is_prime_power(n) and is_prime_power_like(n)
print(list(filter(is_A273201, range(1, 500))))

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

Original entry on oeis.org

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

Views

Author

Peter Luschny, May 17 2016

Keywords

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)])

A273195 The determinant of the difference table of the divisors vanishes for these numbers.

Original entry on oeis.org

10, 28, 50, 99, 110, 130, 170, 171, 190, 196, 222, 230, 250, 290, 310, 370, 410, 430, 470, 476, 530, 532, 550, 590, 610, 644, 650, 670, 710, 730, 790, 812, 830, 850, 868, 890, 950, 970
Offset: 1

Views

Author

Peter Luschny, May 18 2016

Keywords

Comments

Prime power-like numbers (A273200) have nonvanishing determinants.

Examples

			50 is in this sequence because the determinant of DTD(50) = 0.
[  1  2  5 10 25 50]
[  1  3  5 15 25  0]
[  2  2 10 10  0  0]
[  0  8  0  0  0  0]
[  8 -8  0  0  0  0]
[-16  0  0  0  0  0]

Crossrefs

Cf. A273200.

Programs

  • Mathematica
    selQ[n_] := Module[{d = Divisors[n], ld}, ld = Length[d]; Det @ Table[ PadRight[ Differences[d, k], ld], {k, 0, ld-1}] == 0];
Select[Range[1000], selQ] (* Jean-François Alcover, Jul 15 2019 *)
  • Sage
    def is_A273195(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.det() == 0
print([n for n in range(1, 1000) if is_A273195(n)])
Showing 1-3 of 3 results.

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