A141490 -id:A141490 - cp's OEIS frontend

A180466 The number of representations of n as a minimal number of squares, A002828(n).

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 3, 3, 1, 2, 1, 2, 3, 1, 2, 4, 1, 2, 3, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 1, 3, 1, 1, 1, 2, 1, 2, 3, 4, 1, 1, 1, 1, 3, 1, 2, 3, 1, 1, 1, 3, 1

Offset: 1

T. D. Noe, Jan 19 2011

nonn

By Lagrange's four square theorem, the minimal number of squares required to represent a number is 4 or less. See A141490 for the numbers k that have n minimal representations.

			27 has the following representations as the sum of 4 or fewer squares: 1+1+25, 9+9+9, and 1+1+9+16. The minimal number of squares is 3 and there are 2 such representations.  Hence a(27)=2.

T. D. Noe, Table of n, a(n) for n = 1..1000

Table[r=PowersRepresentations[n,4,2]; Sort[Tally[4-Count[#,0]& /@ r]][[1,2]], {n,100}]

A258258 Least number k having exactly n representations as a sum of the minimal number of triangular numbers, A000217.

Original entry on oeis.org

1, 16, 40, 75, 52, 82, 166, 178, 147, 217, 334, 247, 481, 634, 457, 516, 921, 646, 1047, 1132, 822, 787, 2110, 1351, 1537, 1542, 1402, 1192, 1666, 1696, 2137, 1759, 1876, 2271, 1792, 2712, 2587, 3216, 3909, 2782, 3007, 2956, 4242, 3397, 3682, 4039, 3607, 3601

Offset: 1

Martin Renner, May 24 2015

nonn

Fermat's polygonal number theorem states that every positive integer is a sum of at most n n-gonal numbers. The triangular case was proved in 1796 by Gauss (Eureka theorem), stating that every positive integer is the sum of at most three triangular numbers. This sequence is based on this representation as a sum of the minimal number of triangular numbers.

			a(2) = 16 = 1 + 15 = 6 + 10 is the smallest number with two representations.
a(3) = 40 = 1 + 3 + 36 = 6 + 6 + 28 = 10 + 15 + 15 is the smallest number with three representations.
a(4) = 75 = 3 + 6 + 66 = 3 + 36 + 36 = 10 + 10 + 55 = 15 + 15 + 45 is the smallest number with four representations.

Cf. A141490, A258257.

cp's OEIS Frontend

A180466 The number of representations of n as a minimal number of squares, A002828(n).

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