A295702 -id:A295702 - cp's OEIS frontend

A295795 Largest number with exactly n representations as a sum of seven positive squares.

20, 44, 38, 68, 65, 60, 83, 89, 92, 81, 107, 99, 116, 110, 108, 90, 131, 128, 140, 134, 132, 125, 155, 143, 127, 164, 158, 148, 149, 144, 163, 179, 167, 151, 176, 185, 172, 188, 173, 203, 180, 177, 174, 195

Offset: 0

Robert Price, Nov 27 2017

It appears that a(44) does not exist.

E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, New York, 1985, p. 86, Theorem 1.

D. H. Lehmer, On the Partition of Numbers into Squares, The American Mathematical Monthly, Vol. 55, No. 8, October 1948, pp. 476-481.

Cf. A025422, A025431, A295669, A295702.

A374486 Numbers k such that Taxicab(2,j,k) exists for large j.

Original entry on oeis.org

1, 2, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 18, 19, 20, 21, 22, 24, 25, 26, 27, 28, 29, 30, 31, 33, 37, 39, 40, 42, 44, 48, 50, 51, 52, 53, 56, 59, 62, 66, 68, 70, 72, 74, 77, 79, 87, 91, 92, 96, 97, 103, 108, 112, 115, 117, 120, 121, 124, 130, 131, 138, 148, 149, 161, 164, 176, 184, 185, 194, 200

Offset: 1

Oliver Lippard and Brennan G. Benfield, Aug 04 2024

nonn

Here Taxicab(2,j,k) denotes the smallest number (if it exists) that is the sum of j perfect squares in exactly k ways. For sufficiently large N, Taxicab(2,j,k) either always exists for j > N or always does not exist for j > N.
Conjecture: Infinitely many positive integers are in this sequence, and infinitely many positive integers are not in this sequence.
Conjecture: This sequence grows exponentially. Computationally it appears to have asymptotic a(n) = 1.03691*exp(0.594473*n^(1/2)).

			For k = 3, Taxicab(2,j,3) does not exist for all j > 9, hence 3 is not a member of the sequence.

E. Grosswald. Representations of Integers as Sums of Squares. Springer New York, NY, 1985.

Oliver Lippard, Table of n, a(n) for n = 1..372
B. Benfield, O. Lippard, and A. Roy, End Behavior of Ramanujan's Taxicab Numbers, arXiv:2404.08190 [math.NT], 2024.

Cf. A025416, A080673, A295702, A295795

cp's OEIS Frontend

A295795 Largest number with exactly n representations as a sum of seven positive squares.

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