A352244 -id:A352244 - cp's OEIS frontend

A352242 Regular triangle T(n,k) = (n-k)*(n^3-k^3) for n>=2 and 1 <= k <= n-1, read by rows.

7, 52, 19, 189, 112, 37, 496, 351, 196, 61, 1075, 832, 567, 304, 91, 2052, 1675, 1264, 837, 436, 127, 3577, 3024, 2425, 1792, 1161, 592, 169, 5824, 5047, 4212, 3325, 2416, 1539, 772, 217, 8991, 7936, 6811, 5616, 4375, 3136, 1971, 976, 271, 13300, 11907, 10432, 8869, 7236, 5575, 3952, 2457, 1204, 331

Offset: 2

Michel Marcus, Mar 09 2022

			Triangle begins:
     7;
    52,   19;
   189,  112,   37;
   496,  351,  196,  61;
  1075,  832,  567, 304,  91;
  2052, 1675, 1264, 837, 436, 127;
  ...

Michael De Vlieger, Table of n, a(n) for n = 2..11176 (rows n = 2..150, flattened)

Cf. A003215 (right diagonal), A138849 (left border).
Cf. A352243, A352244.
Cf. A055461, A132111.

Table[(n - k) (n^3 - k^3), {n, 2, 11}, {k, n - 1}] // Flatten (* Michael De Vlieger, Mar 09 2022 *)

row(n) = vector(n-1, k, (n-k)*(n^3-k^3));

T(n,k) = A055461(n,k)*A132111(n,k). - R. J. Mathar, Feb 10 2025

A352243 Positive integers of the form (x-y)*(x^3-y^3).

Original entry on oeis.org

7, 19, 37, 52, 61, 91, 112, 127, 169, 189, 196, 217, 271, 304, 331, 351, 397, 436, 469, 496, 547, 567, 592, 631, 721, 772, 817, 832, 837, 919, 976, 1027, 1075, 1141, 1161, 1204, 1261, 1264, 1387, 1456, 1519, 1539, 1657, 1675, 1732, 1792, 1801, 1951, 1971, 2032, 2052

Offset: 1

Michel Marcus, Mar 09 2022

nonn

Integers that are in the A352242 triangle.

			7, 19, 37, 52 and 61 are respectively A352242(1,1), A352242(2,2), A352242(3,3), A352242(2,1) and A352242(4,4).

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A352242, A352244.

N:= 10^4: # for terms <= N
S:= {}:
for y from 1 while 3*y^2 + 3*y + 1 <= N do
  for x from y+1 do
    v:= (x-y)*(x^3-y^3);
    if v > N then break fi;
    S:= S union {v};
od; od:
sort(convert(S,list)); # Robert Israel, May 16 2024

row(n) = vector(n-1, k, (n-k)*(n^3-k^3));
lista(nn) = {my(list = List(), n=2); while (3*n*(n-1)+1 <= nn, my(rown = row(n)); for (k=1, #rown, if (rown[k] <= nn, listput(list, rown[k]))); n++;); Set(Vec(list));}

A373001 Positive integers that can be expressed in at least three ways as (x-y)*(x^3-y^3).

Original entry on oeis.org

1641000816, 1773440487, 2476486656, 20129719792, 26256013056, 28375047792, 39623786496, 106509692016, 132921066096, 143648679447, 200595419136, 218247135232, 322075516672, 420096208896, 454000764672, 600908378112, 631190070000, 633980583936, 877482798192, 1025625510000, 1108400304375

Offset: 1

David A. Corneth, May 19 2024

nonn

			1773440487 is here via 1773440487 = (2706 - 2697) * (2706^3 - 2697^3) = (417 - 354) * (417^3 - 354^3) = (211 - 22) * (211^3 - 22^3).

Chai Wah Wu, Table of n, a(n) for n = 1..352

Cf. A352244.

is(n) = {
	if(valuation(n, 3) == 1, return(0));
	my(f = factor(n), cf = f, q, c, dc);
	cf[,2]>>=1;
	c = factorback(cf);
	dc = divisors(c);
	for(i = 1, #dc,
		dc2 = dc[i]^2;
		dk = n/dc2;
		if(dk > dc2 && (dk - dc2)%3 == 0,
			D = dc2 + 4*(dk - dc2)/3;
			if(issquare(D, &sD) && denominator((-dc[i] + sD)/2) == 1,
				q++
			)
		)	
	);
	q >= 3
}

cp's OEIS Frontend

A352242 Regular triangle T(n,k) = (n-k)*(n^3-k^3) for n>=2 and 1 <= k <= n-1, read by rows.

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A352243 Positive integers of the form (x-y)*(x^3-y^3).

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A373001 Positive integers that can be expressed in at least three ways as (x-y)*(x^3-y^3).

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