A025427 -id:A025427 - cp's OEIS frontend

A345497 Numbers that are the sum of eight squares in ten or more ways.

70, 71, 73, 74, 77, 78, 79, 80, 82, 83, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131

Offset: 1

David Consiglio, Jr., Jun 20 2021

nonn

			71 = 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 8^2
   = 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 4^2 + 7^2
   = 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 4^2 + 5^2 + 5^2
   = 1^2 + 1^2 + 1^2 + 2^2 + 4^2 + 4^2 + 4^2 + 4^2
   = 1^2 + 1^2 + 1^2 + 3^2 + 3^2 + 3^2 + 4^2 + 5^2
   = 1^2 + 1^2 + 2^2 + 2^2 + 2^2 + 2^2 + 2^2 + 7^2
   = 1^2 + 1^2 + 2^2 + 2^2 + 2^2 + 4^2 + 4^2 + 5^2
   = 1^2 + 2^2 + 2^2 + 2^2 + 2^2 + 2^2 + 5^2 + 5^2
   = 1^2 + 2^2 + 2^2 + 2^2 + 2^2 + 3^2 + 3^2 + 6^2
   = 1^2 + 3^2 + 3^2 + 3^2 + 3^2 + 3^2 + 3^2 + 4^2
   = 2^2 + 2^2 + 2^2 + 3^2 + 3^2 + 3^2 + 4^2 + 4^2
so 71 is a term.

Sean A. Irvine, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A025427, A025429, A345487, A345496, A345540, A346803.

from itertools import combinations_with_replacement as cwr
from collections import defaultdict
keep = defaultdict(lambda: 0)
power_terms = [x**2 for x in range(1, 1000)]
for pos in cwr(power_terms, 8):
    tot = sum(pos)
    keep[tot] += 1
    rets = sorted([k for k, v in keep.items() if v >= 10])
    for x in range(len(rets)):
        print(rets[x])

def A345397(n): return (70, 71, 73, 74, 77, 78, 79, 80, 82, 83)[n-1] if n<11 else n+74 # Chai Wah Wu, May 09 2024

From Chai Wah Wu, May 09 2024: (Start)
All integers >= 85 are terms. Proof: since 594 can be written as the sum of 3 positive squares in 10 ways (see A025427) and any integer >= 34 can be written as a sum of 5 positive squares (see A025429), any integer >= 628 can be written as a sum of 8 positive squares in 10 or more ways. Integers from 85 to 627 are terms by inspection.
a(n) = 2*a(n-1) - a(n-2) for n > 12.
G.f.: x*(-x^11 + x^10 - x^9 + x^8 - 2*x^5 + 2*x^4 - x^3 + x^2 - 69*x + 70)/(x - 1)^2. (End)

A321457 Expansion of Product_{1 <= i <= j <= k} (1 + x^(i^2 + j^2 + k^2)).

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 1, 2, 0, 2, 2, 0, 3, 3, 1, 4, 4, 2, 5, 4, 3, 7, 7, 5, 9, 10, 7, 11, 14, 9, 15, 19, 12, 22, 23, 17, 30, 29, 23, 41, 37, 32, 54, 46, 45, 68, 59, 63, 85, 79, 85, 107, 103, 108, 136, 136, 139, 174, 177, 178, 222, 225, 226, 287, 282, 290

Offset: 0

Seiichi Manyama, Nov 10 2018

nonn

Cf. A025427, A321429, A321435, A321458, A321459.

G.f.: Product_{k>0} (1 + x^k)^A025427(k).

A321458 Expansion of Product_{1 <= i <= j <= k} (1 - x^(i^2 + j^2 + k^2)).

Original entry on oeis.org

1, 0, 0, -1, 0, 0, -1, 0, 0, 0, 0, -1, 0, 0, 0, 2, 0, 1, -1, -1, 2, 0, 0, 1, 0, 3, -3, -1, 1, -1, 4, -1, 1, 0, -5, 1, -1, 0, 0, -1, 5, -2, -7, -1, -1, 5, 0, 0, 2, -5, -2, -3, 1, 3, -3, 5, 1, -3, -8, -4, 16, 1, 4, -1, 2, 4, -17, 4, 5, 2, 10, -6, 10, -7, -23, 18, 3, 12

Offset: 0

Seiichi Manyama, Nov 10 2018

sign

Cf. A025427, A321432, A321436, A321457, A321459.

G.f.: Product_{k>0} (1 - x^k)^A025427(k).

A321459 Expansion of Product_{1 <= i <= j <= k} 1/(1 - x^(i^2 + j^2 + k^2)).

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 2, 0, 0, 3, 0, 1, 5, 0, 2, 6, 0, 4, 10, 1, 6, 13, 3, 10, 19, 5, 15, 26, 10, 23, 37, 16, 32, 52, 26, 48, 74, 38, 70, 100, 59, 101, 138, 85, 144, 189, 126, 204, 256, 180, 284, 345, 262, 392, 468, 372, 537, 629, 522, 730, 847, 723, 992, 1134, 998, 1336

Offset: 0

Seiichi Manyama, Nov 10 2018

nonn

Cf. A025427, A321433, A321437, A321457, A321458.

G.f.: Product_{k>0} 1/(1 - x^k)^A025427(k).

A306396 Consider the numbers in A024796, numbers expressible in more than one way as i^2 + j^2 + k^2, where 1 <= i <= j <= k; sequence number of ways these numbers can be expressed.

Original entry on oeis.org

2, 2, 2, 2, 2, 3, 2, 2, 2, 3, 2, 2, 2, 2, 3, 2, 3, 3, 2, 2, 2, 3, 3, 2, 2, 2, 2, 3, 2, 3, 2, 2, 2, 2, 2, 2, 3, 4, 3, 2, 4, 2, 2, 2, 2, 4, 2, 3, 3, 2, 4, 2, 2, 2, 4, 3, 2, 2, 3, 2, 4, 3, 3, 2, 2, 3, 2, 3, 3, 3, 2, 4, 5, 2, 2, 4, 4, 2, 2, 5, 6, 2, 2, 2, 2, 3, 2, 2, 3, 3, 3, 3, 2, 3, 2, 2, 5, 3, 4, 2, 3, 2, 3, 3, 4, 3, 4, 2, 4, 2, 4, 4, 4, 3, 2, 4, 2, 3, 5, 2, 5, 4, 2

Offset: 1

A. Timothy Royappa, Feb 12 2019

nonn

Number of accidental degeneracies in the quantum mechanical 3-D "particle-in-a-box" model.

			The fourth term in A024796 is 41, which can be expressed in two ways as the sum of three nonzero squares (1^2 + 2^2 + 6^2 or 3^2 + 4^2 + 4^2), so a(4) = 2.

Jinyuan Wang, Table of n, a(n) for n = 1..1000

Cf. A024796, A025427.

r[n_] := Length@ IntegerPartitions[n, {3}, Range[Sqrt[n]]^2]; Select[ Array[r, 300], # > 1 &] (* Giovanni Resta, Feb 21 2020 *)

a(n) = A025427(A024796(n)).

Offset changed to 1 by Jinyuan Wang, Feb 20 2020

A152829 Numbers k whose squares can be written in exactly one way as a sum of three squares: k^2 = a^2 + b^2 + c^2 with 1 <= a <= b <= c.

Original entry on oeis.org

3, 6, 7, 12, 13, 14, 24, 26, 28, 48, 52, 56, 96, 104, 112, 192, 208, 224, 384, 416, 448, 768, 832, 896, 1536, 1664, 1792, 3072, 3328, 3584, 6144, 6656, 7168, 12288, 13312, 14336, 24576, 26624, 28672, 49152, 53248, 57344, 98304, 106496, 114688, 196608, 212992, 229376

Offset: 1

Peter Pein (petsie(AT)dordos.net), Dec 13 2008

nonn

Numbers k such that k^2 is in A025321. - Joerg Arndt, Mar 22 2022

2k is a term iff k is also a term, so the conjecture from Colin Barker (see Formula) is true iff 3, 7, and 13 are the only odd terms. - Jon E. Schoenfield, Mar 22 2022

			9 is not in this sequence because 9^2 = 1^2 + 4^2 + 8^2 = 3^2 + 6^2 + 6^2 = 4^2 + 4^2 + 7^2.
7 is in this sequence because 7^2 = 2^2 + 3^2 + 6^2 is the only way to write 7^2 as a sum of three squares.

M. A. Fajjal, Posting in sci.math.symbolic

Cf. A025321.

#include 
#include 
int main (int argc, char *argv[]) {
    long n,n2,a,a2,b,b2,c,c2; int s = 0; n=atol(argv[1]); n2=n*n;
    for (a=1; a 3sq.txt
# gives the terms less than 1000

Guessed o.g.f.: x*(x^4 + 6*x^3 + 7*x^2 + 6*x + 3)/(1 - 2*x^3).
{k: A025427(k^2)=1}. - R. J. Mathar, Dec 15 2008
Conjecture: a(n) = 2*a(n-3) for n > 5. - Colin Barker, Mar 12 2012

a(25)-a(36) (from comment) verified and added by Donovan Johnson, Nov 08 2013

a(37)-a(48) from Jon E. Schoenfield, Mar 22 2022

A307219 a(n) is the number of partitions of (prime(n)^2 + 2)/3 into 3 squares.

Original entry on oeis.org

1, 1, 2, 2, 1, 2, 2, 5, 6, 2, 6, 3, 6, 5, 14, 8, 6, 5, 6, 15, 10, 6, 14, 24, 14, 6, 12, 12, 6, 16, 19, 18, 18, 36, 18, 10, 16, 20, 20, 12, 28, 18, 8, 24, 38, 27, 40, 20, 17, 30, 52, 18, 22, 26, 29, 21, 42, 31, 26, 26, 18, 44, 38, 40, 46, 26, 30, 44, 38, 36, 52, 28, 27, 38, 103, 22, 38, 78

Offset: 3

Marius A. Burtea, Mar 29 2019

nonn

If p >= 5 is a prime number it can be written as p = 6m-1 or p = 6m+1. The identities ((6m-1)^2 + 2)/3 = (2m)^2 + (2m)^2 + (2m-1)^2 and ((6m+1)^2 + 2)/3 = (2m)^2 + (2m)^2 + (2m+1)^2 show that the number (p^2 + 2)/3 can be written as a sum of 3 squares of integers in at least one way.

			For n = 3, p = prime(3) = 5, (5^2+2)/3 = 9 = 2^2 + 2^2 + 1^2, so a(3) = 1.
For n = 9, p = prime(9) = 23, (23^2+2)/3 = 177 = 13^2 + 2^2 + 2^2 = 8^2 + 8^2 + 7^2, so a(9) = 2.
For n = 17, p = prime(17) = 59, (59^2+2)/3 = 1161 = 34^2 + 2^2 + 1^2 = 33^2 + 6^2 + 6^2 = 24^2 + 11^2 + 4^2 = 31^2 + 14^2 + 2^2 = 31^2 + 10^2 + 10^2 = 30^2 + 15^2 + 6^2 = 29^2 + 16^2 + 8^2 = 28^2 + 19^2 + 4^2 = 28^2 + 16^2 + 11^2 = 26^2 + 22^2 + 1^2 = 26^2 + 17^2 + 14^2 = 24^2 + 24^2 + 3^2 = 24^2 + 21^2 + 12^2 = 20^2 + 20^2 + 19^2, so a(17) = 14.

Ion Cucurezeanu, Pătrate și cuburi perfecte de numere întregi (Squares and perfect cubes of integer numbers), Ed. Gil., Zalău, 2007, ch. 1, p. 21, pr. 166.
Laurențiu Panaitopol, Dinu Șerbănescu, Number theory and combinatorial problems for juniors, Ed. Gil, Zalău, (2003), ch. 1, p. 5, pr. 4. (in Romanian).

Cf. A000040, A000378, A024795, A025427.

[#RestrictedPartitions(Floor((p*p+2)/3),3,{d*d:d in [1..p]}): p in PrimesInInterval(5,500) ];

a(n)={k=(prime(n+2)^2+2)/3;sum(a=1, k, sum(b=1, a, sum(c=1, b, a^2+b^2+c^2==k)));} \\ Jinyuan Wang, Mar 30 2019

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A345497 Numbers that are the sum of eight squares in ten or more ways.

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A321457 Expansion of Product_{1 <= i <= j <= k} (1 + x^(i^2 + j^2 + k^2)).

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A321458 Expansion of Product_{1 <= i <= j <= k} (1 - x^(i^2 + j^2 + k^2)).

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A321459 Expansion of Product_{1 <= i <= j <= k} 1/(1 - x^(i^2 + j^2 + k^2)).

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A306396 Consider the numbers in A024796, numbers expressible in more than one way as i^2 + j^2 + k^2, where 1 <= i <= j <= k; sequence number of ways these numbers can be expressed.

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A152829 Numbers k whose squares can be written in exactly one way as a sum of three squares: k^2 = a^2 + b^2 + c^2 with 1 <= a <= b <= c.

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A307219 a(n) is the number of partitions of (prime(n)^2 + 2)/3 into 3 squares.

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