A309778 -id:A309778 - cp's OEIS frontend

A018820 Numbers k that are the sum of m nonzero squares for all 1 <= m <= k - 14.

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

Offset: 1

David W. Wilson

nonn

Intersection of A000290, A000404 and A000408. - Zak Seidov, Nov 12 2013
A square k^2 is the sum of m positive squares for all 1 <= m <= k^2 - 14 iff k^2 is the sum of 2 and 3 positive squares (see A309778 and proof in Kuczma). - Bernard Schott, Aug 17 2019
Note that k is never the sum of k - 13 positive squares. - Jianing Song, Feb 09 2021

			169 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. - _Jianing Song_, Feb 09 2021

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)
Index entries for sequences related to sums of squares

Cf. A000290, A000404, A000408, A309778, A341329.

isA018820(n) = issquare(n) && isA341329(sqrtint(n)) \\ Jianing Song, Feb 09 2021, see A341329 for its program

a(n) = A341329(n)^2. - Jianing Song, Feb 09 2021

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.

A309779 Squares that can be expressed as the sum of two positive squares but not as the sum of three positive squares.

Original entry on oeis.org

25, 100, 400, 1600, 6400, 25600, 102400, 409600, 1638400, 6553600, 26214400, 104857600, 419430400, 1677721600, 6710886400, 26843545600, 107374182400, 429496729600, 1717986918400, 6871947673600, 27487790694400, 109951162777600, 439804651110400, 1759218604441600

Offset: 1

Bernard Schott, Aug 17 2019

This sequence comes from the study of A309778, exactly, A309778(n) = 2 iff n^2 belongs to this sequence here.
According to Draxl link, a(n) is a term of this sequence iff a(n) = 5^2 * 4^(n-1) with n >= 1.
This sequence is a subsequence of A219222 whose terms are all of the form b_0 * 4^k with b_0 in A051952, hence, the only primitive term of this sequence here is 25.

			25 = 5^2 = 3^2 + 4^2,
100 = 10^2 = 6^2 + 8^2,
5^2 * 4^(n-1) = (5 * 2^(n-1))^2 = (3 * 2^(n-1))^2 + (4 * 2^(n-1))^2, but these terms are not the sum of three positive squares.

H.-P. Baltes, Peter K. J. Draxl, and Eberhard R. Hilf, Quadratsummen und gewisse Randwertprobleme der Mathematischen Physik, Publications of the Small Systems Group Oldenburg, preprint, 1973.
H.-P. Baltes, Peter K. J. Draxl, and Eberhard R. Hilf, Quadratsummen und gewisse Randwertprobleme der Mathematischen Physik, Journ. Reine Angewandte Mathematik, Vol. 268/269, 1974, 410-417.
P. K. J. Draxl, Sommes de deux carrés qui ne sont pas sommes de trois carrés., Mémoires de la SMF, tome 37 (1974), p. 53-53.
Index entries for linear recurrences with constant coefficients, signature (4).

Intersection of A000290 and A219222.

Cf. A000302, A000378, A000408, A051952, A134422, A309778.

Array[25*4^(# - 1) &, 24] (* Michael De Vlieger, Aug 19 2019 *)

a(n) = 25 * 4^(n-1); \\ Jinyuan Wang, Aug 18 2019

a(n) = 5^2 * 4^(n-1) with n >= 1.
a(n) = 4*a(n-1) for n > 1. G.f.: 25*x/(1 - 4*x). - Chai Wah Wu, Aug 29 2019
a(n) = 25 * A000302(n-1). - Alois P. Heinz, Aug 29 2019
E.g.f.: 25*(exp(4*x) - 1)/4. - Stefano Spezia, Oct 28 2023

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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A018820 Numbers k that are the sum of m nonzero squares for all 1 <= m <= k - 14.

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

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A309779 Squares that can be expressed as the sum of two positive squares but not as the sum of three positive 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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