A374287 -id:A374287 - cp's OEIS frontend

A374417 a(n) is the smallest number which can be represented as the sum of n distinct positive cubes in exactly 2 ways, or -1 if no such number exists.

-1, 1729, 1009, 1036, 1161, 1504, 1899, 2512, 3024, 4355, 6552, 9296, 11648, 14392, 19305, 25137, 30997, 35757, 44092, 53353, 64001, 76168, 88669, 104625, 122201, 144153, 167401, 191772, 216161, 245952, 278757, 312993, 352297, 393822, 434295, 489167, 541081, 605656, 671446

Offset: 1

Ilya Gutkovskiy, Jul 08 2024

sign

			a(2) = 1729 = 1^3 + 12^3 = 9^3 + 10^3.
a(3) = 1009 = 1^3 + 2^3 + 10^3 = 4^3 + 6^3 + 9^3.

David A. Corneth, Table of n, a(n) for n = 1..675
Index entries for sequences related to sums of cubes

Cf. A000537, A011541, A350270, A374287.

G:= mul(1+t*x^(i^3), i=1..35):
R:= -1:
for m from 2 do
  C:= expand(coeff(G,t,m)):
  C2:= convert(select(s -> subs(x=1,s)=2, C),list);
  v:= min(map(degree,C2));
  if v >= 36^3 + add(i^3,i=1..m-1) then break fi;
  R:= R,v;
od:
R; # Robert Israel, Jul 08 2024

a(15)-a(27) from Robert Israel, Jul 08 2024

a(28)-a(39) from Michael S. Branicky, Jul 10 2024

A374579 a(n) is the smallest number which can be represented as the sum of n distinct nonzero fourth powers in exactly 2 ways, or -1 if no such number exists.

Original entry on oeis.org

-1, 635318657, 6578, 6834, 7459, 8755, 18755, 23412, 43893, 53893, 92309, 123335, 182566, 232647, 298183, 393848, 521849, 681849, 872105, 1066586, 1343612, 1623453, 1955229, 2412205, 2811596, 3424941, 4132222

Offset: 1

Ilya Gutkovskiy, Jul 12 2024

			a(2) = 635318657 = 59^4 + 158^4 = 133^4 + 134^4.
a(3) = 6578 = 1^4 + 2^4 + 9^4 = 3^4 + 7^4 + 8^4.

Cf. A000538, A003824, A360306, A374287, A374417.

a(18)-a(27) from Michael S. Branicky, Jul 13 2024

A377729 a(n) is the smallest number which can be represented as the sum of n distinct nonzero n-gonal numbers in exactly 2 ways.

Original entry on oeis.org

19, 90, 162, 299, 509, 816, 1248, 1837, 2619, 3634, 4926, 6543, 8537, 10964, 13884, 17361, 21463, 26262, 31834, 38259, 45621, 54008, 63512, 74229, 86259, 99706, 114678, 131287, 149649, 169884, 192116, 216473, 243087, 272094, 303634, 337851, 374893, 414912, 458064, 504509

Offset: 3

Ilya Gutkovskiy, Nov 05 2024

nonn

From David A. Corneth, Nov 06 2024: (Start)
a(n) <= (n^4 - 2*n^3 + 38*n^2 - 85*n + 72)/6 for n >= 5. Proof:
A polygonal number is of the form P(m, n) = m/2 * ((n - 2) * m - n + 4).
We have P(n - 5, n) + P(n - 4, n) + P(n, n) = P(n - 6, n) + P(n - 2, n) + P(n - 1, n) = (3*n^3 - 18*n^2 + 21*n) / 2.
This lets us find the upper bound on a(n) by making two lists from 1 through n + 3. From one of them we remove n-2, n-1 and n + 3 and from the other we remove n-3, n+1 and n+2. The sum for remaining polygonal numbers is the same giving an upper bound on a(n) which turns out to be (n^4 - 2*n^3 + 38*n^2 - 85*n + 72)/6 (End)

			a(3) = 19 = 1 + 3 + 15 = 3 + 6 + 10.
a(4) = 90 = 1^2 + 2^2 + 6^2 + 7^2 = 1^2 + 3^2 + 4^2 + 8^2.

Eric Weisstein's World of Mathematics, Polygonal Number

Cf. A006484, A025377, A057145, A350405, A374144, A374256, A374287.

From David A. Corneth, Nov 06 2024: (Start)
a(n) >= A006484(n).
Conjecture: a(n) = (n^4 - 2*n^3 + 38*n^2 - 85*n + 72)/6 for n >= 5. (End)
Conjectured g.f.: x^3*(19 - 5*x - 98*x^2 + 199*x^3 - 171*x^4 + 72*x^5 - 12*x^6) / (1 - x)^5.

a(12)-a(36) from Michael S. Branicky, Nov 06 2024

More terms from David A. Corneth, Nov 10 2024

cp's OEIS Frontend

A374417 a(n) is the smallest number which can be represented as the sum of n distinct positive cubes in exactly 2 ways, or -1 if no such number exists.

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A374579 a(n) is the smallest number which can be represented as the sum of n distinct nonzero fourth powers in exactly 2 ways, or -1 if no such number exists.

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A377729 a(n) is the smallest number which can be represented as the sum of n distinct nonzero n-gonal numbers in exactly 2 ways.

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