A018820 -id:A018820 - cp's OEIS frontend

A309778 a(n) is the greatest integer such that, for every positive integer k <= a(n), n^2 can be written as the sum of k positive square integers.

1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 155, 1, 211, 1, 275, 1, 1, 2, 1, 1, 1, 1, 611, 662, 1, 1, 827, 886, 1, 1, 1, 1142, 1211, 1, 1355, 1, 1507, 2, 1667, 1, 1, 1, 2011, 1, 1, 1, 1, 2486, 2587, 2690, 2795, 1, 3011, 1, 1, 3350, 1, 3586, 3707, 1, 1, 1

Offset: 1

Bernard Schott, Aug 17 2019

nonn

The idea for this sequence comes from the 6th problem of the 2nd day of the 33rd International Mathematical Olympiad in Moscow, 1992 (see link).
There are four cases to examine and three possible values for a(n).
a(n) = 1 iff n is a nonhypotenuse number or iff n is in A004144.
a(n) >= 2 iff n is a hypotenuse number or iff n is in A009003.
a(n) = 2 iff n^2 is the sum of two positive squares but not the sum of three positive squares or iff n^2 is in A309779.
a(n) = n^2 - 14 iff n^2 is the sum of two and three positive squares or iff n^2 is in A231632.
Theorem: a square n^2 is the sum of k positive squares for all 1 <= k <= n^2 - 14 iff n^2 is the sum of 2 and 3 positive squares (proof in Kuczma). Consequently: A231632 = A018820.

			1 = 1^2, 4 = 2^2 and a(1) = a(2) = 1.
25 = 5^2 = 3^2 + 4^2 and a(5) = 2.
The first representations of 169 are 13^2 = 12^2 + 5^2 = 12^2 + 4^2 + 3^2 = 11^2 + 4^2 + 4^2 + 4^2 =  6^2 + 6^2 + 6^2 + 6^2 + 5^2  = 6^2 + 6^2 + 6^2 + 6^2 + 4^2 + 3^2 = ... and a(13) = 13^2 - 14 = 155.

Marcin E. Kuczma, International Mathematical Olympiads, 1986-1999, The Mathematical Association of America, 2003, pages 76-79.

IMO, 1992, Moscow, Second day. Problem 6

Cf. A018820, A004144, A009003, A231632, A309779.

A231632 Squares that are also sums of 2 and 3 nonzero squares.

Original entry on oeis.org

169, 225, 289, 625, 676, 841, 900, 1156, 1225, 1369, 1521, 1681, 2025, 2500, 2601, 2704, 2809, 3025, 3364, 3600, 3721, 4225, 4624, 4900, 5329, 5476, 5625, 6084, 6724, 7225, 7569, 7921, 8100, 8281, 9025, 9409, 10000, 10201, 10404, 10816, 11025, 11236, 11881, 12100, 12321, 12769, 13225, 13456, 13689, 14161

Offset: 1

Zak Seidov, Nov 12 2013

dead

All terms == {0, 1} (mod 4).
Intersection of A000290, A000404 and A000408.
A square n^2 is the sum of k positive squares for all 1 <= k <= n^2 - 14 iff n^2 is the sum of 2 and 3 positive squares (see A309778 and proof in Kuczma) . Consequently this is a duplicate of A018820. - Bernard Schott, Aug 17 2019

			169 = 13^2 = 5^2 + 12^2 = 3^2 + 4^2 + 12^2;
225 = 15^2 = 9^2 + 12^2 = 2^2 + 5^2 + 14^2.

Marcin E. Kuczma, International Mathematical Olympiads, 1986-1999, The Mathematical Association of America, 2003, pages 76-79.

Zak Seidov and Chai Wah Wu, Table of n, a(n) for n = 1..10000 n = 1..100 from Zak Seidov

Cf. A000290, A000404, A000408, A018820.

A341329 Numbers k such that k^2 is the sum of m nonzero squares for all 1 <= m <= k^2 - 14.

Original entry on oeis.org

13, 15, 17, 25, 26, 29, 30, 34, 35, 37, 39, 41, 45, 50, 51, 52, 53, 55, 58, 60, 61, 65, 68, 70, 73, 74, 75, 78, 82, 85, 87, 89, 90, 91, 95, 97, 100, 101, 102, 104, 105, 106, 109, 110, 111, 113, 115, 116, 117, 119, 120, 122, 123, 125, 130, 135, 136, 137

Offset: 1

Jianing Song, Feb 09 2021

Numbers k such that k^2 is in A018820. Note that k^2 is never the sum of k^2 - 13 positive squares.
A square k^2 is the sum of m positive squares for all 1 <= m <= k^2 - 14 if k^2 is the sum of 2 and 3 positive squares (see A309778 and proof in Kuczma).
Intersection of A009003 and A005767. Also A009003 \ A020714.
Numbers k not of the form 5*2^e such that k has at least one prime factor congruent to 1 modulo 4.
Has density 1 over all positive integers.

			13 is a term: 169 = 13^2 = 5^2 + 12^2 = 3^2 + 4^2 + 12^2 = 11^2 + 4^2 + 4^2 + 4^2 = 6^2 + 6^2 + 6^2 + 6^2 + 5^2 = 6^2 + 6^2 + 6^2 + 6^2 + 4^2 + 3^2 = ... = 3^2 + 2^2 + 2^2 + 1^2 + 1^2 + ... + 1^2 (sum of 155 positive squares, with 152 (1^2)'s), but 169 cannot be represented as the sum of 156 positive squares.

Marcin E. Kuczma, International Mathematical Olympiads, 1986-1999, The Mathematical Association of America, 2003, pages 76-79.

Jianing Song, Table of n, a(n) for n = 1..10000

Cf. A018820, A309778, A009003, A005767, A020714.

isA341329(n) = setsearch(Set(factor(n)[, 1]%4), 1) && !(n/5 == 2^valuation(n, 2))

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A309778 a(n) is the greatest integer such that, for every positive integer k <= a(n), n^2 can be written as the sum of k positive square integers.

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A231632 Squares that are also sums of 2 and 3 nonzero squares.

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A341329 Numbers k such that k^2 is the sum of m nonzero squares for all 1 <= m <= k^2 - 14.

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