A080673 -id:A080673 - cp's OEIS frontend

A025429 Number of partitions of n into 5 nonzero squares.

0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 3, 0, 1, 3, 1, 3, 2, 1, 3, 2, 1, 3, 3, 2, 3, 2, 2, 2, 2, 3, 3, 5, 2, 2, 5, 1, 3, 5, 1, 5, 4, 2, 5, 3, 2, 5, 5, 3, 4, 4, 4, 3, 5, 4, 4, 7, 3, 5, 6, 2, 4, 7, 4, 7, 6, 3, 7, 4, 3, 8, 6, 5, 7, 5, 5, 4, 6, 7, 6, 9, 5, 6, 8, 2, 8

Offset: 0

David W. Wilson

a(33) is the last zero in this sequence, cf. the link to Mathematics Stack Exchange and also A080673(n) for the largest index k with a(k)=n. - M. F. Hasler, May 30 2014

First occurrence of k beginning with 0: 0, 5, 20, 29, 62, 53, 80, 77, 91, 101, ..., (A080654). - Robert G. Wilson v, May 30 2014

M. F. Hasler, Table of n, a(n) for n = 0..10000
H. v. Eitzen, in reply to user James47, What is the largest integer with only one representation as a sum of five nonzero squares? on Mathematics Stack Exchange, May 2014
Index entries for sequences related to sums of squares

Column k=5 of A243148.

b:= proc(n, i, t) option remember; `if`(n=0, `if`(t=0, 1, 0),
      `if`(i<1 or t<1, 0, b(n, i-1, t)+
      `if`(i^2>n, 0, b(n-i^2, i, t-1))))
    end:
a:= n-> b(n, isqrt(n), 5):
seq(a(n), n=0..120);  # Alois P. Heinz, May 30 2014

f[n_] := Block[{c = Range@ Sqrt@ n^2}, Length@ IntegerPartitions[n, {5}, c]]; Array[f, 105, 0] (* Robert G. Wilson v, May 30 2014 *)
b[n_, i_, t_] := b[n, i, t] = If[n==0, If[t==0, 1, 0], If[i<1 || t<1, 0, b[n, i-1, t] + If[i^2>n, 0, b[n-i^2, i, t-1]]]]; a[n_] := b[n, Sqrt[n] // Floor, 5]; Table[a[n], {n, 0, 120}] (* Jean-François Alcover, Oct 12 2015, after Alois P. Heinz *)

A025429(n)=sum(d=sqrtint(max(n, 5)\5), sqrtint(max(n-4, 0)), nn=n-d^2; sum(a=sqrtint(max(nn-d^2, 4)\4), min(sqrtint(max(nn-3, 0)),d), sum(b=sqrtint((nn-a^2)\3-1)+1, min(sqrtint(nn-a^2-2), a), sum(c=sqrtint((t=nn-a^2-b^2)\2-1)+1, min(sqrtint(t-1), b), issquare(t-c^2) )))) \\ M. F. Hasler, May 30 2014

a(n) = [x^n y^5] Product_{k>=1} 1/(1 - y*x^(k^2)). - Ilya Gutkovskiy, Apr 19 2019

a(n) = Sum_{l=1..floor(n/5)} Sum_{k=l..floor((n-1)/4)} Sum_{j=k..floor((n-k-l)/3)} Sum_{i=j..floor((n-j-k-l)/2)} A010052(i) * A010052(j) * A010052(k) * A010052(l) * A010052(n-i-j-k-l). - Wesley Ivan Hurt, Apr 19 2019

A080654 Smallest number with exactly n representations as a sum of five positive squares or 0 if no such number exists (cf. A025429).

Original entry on oeis.org

5, 20, 29, 62, 53, 80, 77, 91, 101, 107, 128, 133, 131, 139, 166, 163, 181, 187, 179, 219, 203, 214, 227, 238, 211, 262, 275, 251, 291, 277, 314, 298, 259, 299, 326, 307, 399, 334, 374, 346, 347, 355, 373, 331, 411, 391, 430, 371, 445, 421, 394, 486, 379, 406

Offset: 1

Rainer Rosenthal, Mar 01 2003

It seems as if 33 is the largest number with no such representation. 60 seems to be the largest one with exactly one representation.

More generally, see A080673 for the largest number with n such representations. - M. F. Hasler, Mar 04 2016

			a(4) = 62 because there are exactly four representations as a sum of 5 squares: 62 = 1+4+4+4+49 = 1+4+16+16+25 = 4+4+4+25+25 = 4+4+9+9+36.

Hagen von Eitzen, Table of n, a(n) for n = 1..78107

Cf. A080673, A025429.

f[k_] := f[k] = Length[Select[PowersRepresentations[k, 5, 2], #[[1]] > 0 &]]; a[n_] := (k = 1; While[f[k++] != n]; k-1); Array[a, 54] (* Jean-François Alcover, Apr 26 2011 *)
f[n_] := f[n] = Block[{c = Range@ Sqrt@ n^2}, Length@ IntegerPartitions[n, {5}, c]]; t = Array[f, 50000, 0]; Table[ Position[t, n, 1, 1], {n, 190}] - 1 (* Robert G. Wilson v, Jun 01 2014 *)

More terms from Reinhard Zumkeller, Apr 26 2004

Definition adjusted to cope with otherwise undefined values and b-file extended by Hagen von Eitzen, Jun 05 2014

A243077 Numbers missing from A025429 (number of partitions of n into 5 nonzero squares).

Original entry on oeis.org

188, 259, 304, 308, 372, 394, 483, 497, 594, 634, 685, 705, 722, 729, 740, 750, 756, 766, 780, 782, 834, 850, 864, 884, 885, 888, 922, 925, 946, 955, 956, 1016, 1026, 1048, 1069, 1070, 1071, 1080, 1082, 1108, 1134, 1140, 1154, 1159, 1160, 1161, 1187, 1198, 1266, 1268

Offset: 1

M. F. Hasler, May 30 2014 and Hagen von Eitzen, Jun 05 2014

nonn

For these indices, A080654 and A080673 are ill-defined. Stated another way, a(n) is the n-th index where A080654 and A080673 default to 0.

			a(1)=188 because all the values 0, 1, ..., 187 appear somewhere in A025429, but 188 doesn't.

Hagen von Eitzen, Table of n, a(n) for n = 1..29477
"James47", What is the largest integer with only one representation as a sum of five nonzero squares?, Math.StackExchange, 2014.

Cf. A025429, A080654, A080673.

Edited by N. J. A. Sloane, Jun 06 2014

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

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A025429 Number of partitions of n into 5 nonzero squares.

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A080654 Smallest number with exactly n representations as a sum of five positive squares or 0 if no such number exists (cf. A025429).

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A243077 Numbers missing from A025429 (number of partitions of n into 5 nonzero squares).

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

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