A004215 -id:A004215 - cp's OEIS frontend

A167615 Total number of positive integers below 10^n with 4 positive squares in their representation as sum of squares.

1, 15, 165, 1665, 16664, 166664, 1666663, 16666663, 166666661, 1666666662, 16666666661, 166666666660, 1666666666661, 16666666666660, 166666666666659, 1666666666666660, 16666666666666658, 166666666666666657, 1666666666666666660, 16666666666666666656

Offset: 1

Martin Renner, Jan 18 2011

nonn

A049416(n) + A180416(n) + A180425(n) + a(n) = A002283(n).

			a(1) = 1 since 7 is the only natural number below 10 which is the sum of 4 but no fewer nonzero squares.

Eric Weisstein's World of Mathematics, Lagrange's Four-Square Theorem.
Eric Weisstein's World of Mathematics, Sum of Squares Function.

Cf. A004215.

a:=proc(n)
  local f,s,k;
  f:=(x,y)->ceil(10^y/2^(2*x+3)-7/8):
  s:=0:
  for k from 0 by 1 while not f(k,n)=0 do
    s:=s+f(k,n);
  od:
  return(s);
end;

a[n_] := Module[{f, s = 0, k}, f[x_, y_] := Ceiling[10^y/2^(2x+3) - 7/8]; For[k = 0, f[k, n] != 0, k++, s += f[k, n]]; Return[s]];
Array[a, 20] (* Jean-François Alcover, Oct 31 2020, after Maple *)

a(n) = Sum_{i=0..k} ceiling(10^n/2^(2*i+3) - 7/8) with minimal k for which ceiling(10^n/2^(2*k+3) - 7/8) = 0.

A261904 Largest x such that n can be written as n = x^2 + y^2 + z^2 with x >= y >= z >= 0, or -1 if no such x exists.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Sep 08 2015

sign

a(n) = -1 iff n is in A004215, a(n) >= 0 iff n is in A000378.

Somehow maximizing x seems like the right thing to do (since it is natural to try a greedy algorithm first). If we minimize x we get A261915.

			Tabls showing initial values of n,x,y,z:
0 0 0 0
1 1 0 0
2 1 1 0
3 1 1 1
4 2 0 0
5 2 1 0
6 2 1 1
7 -1 -1 -1
8 2 2 0
9 3 0 0
10 3 1 0
11 3 1 1
12 2 2 2
13 3 2 0
14 3 2 1
15 -1 -1 -1
16 4 0 0
17 4 1 0
18 4 1 1
19 3 3 1
20 4 2 0
...

David Consiglio, Jr., Table of n, a(n) for n = 0..10000
David Consiglio, Jr., Python Program
Index entries for sequences related to sums of squares

Cf. A000378, A004215, A005875, A261915.

Analogs for 4 squares: A178786 and A122921.

More terms from David Consiglio, Jr., Sep 08 2015

A261915 Smallest x such that n can be written as n = x^2 + y^2 + z^2 with x >= y >= z >= 0, or -1 if no such x exists.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Sep 11 2015

sign

a(n) = -1 iff n is in A004215, a(n) >= 0 iff n is in A000378.

If we maximize x we get A261904.

			Table showing initial values of n,x,y,z:
   0  0  0  0
   1  1  0  0
   2  1  1  0
   3  1  1  1
   4  2  0  0
   5  2  1  0
   6  2  1  1
   7 -1 -1 -1
   8  2  2  0
   9  2  2  1
  10  3  1  0
  11  3  1  1
  12  2  2  2
  13  3  2  0
  14  3  2  1
  15 -1 -1 -1
  16  4  0  0
  17  3  2  2
  18  3  3  0
  19  3  3  1
  20  4  2  0
  ...

David Consiglio, Jr., Table of n, a(n) for n = 0..10000
David Consiglio, Jr., Python Program
Index entries for sequences related to sums of squares

Cf. A000378, A004215, A005875, A261904.

Analogs for 4 squares: A178786 and A122921.

a(17) corrected, more terms from David Consiglio, Jr., Sep 11 2015

A272405 Numbers n such that sum of the divisors of n is not of the form x^2 + y^2 + z^2 where x, y, z are integers.

Original entry on oeis.org

4, 8, 12, 16, 18, 24, 25, 32, 38, 48, 59, 64, 75, 91, 96, 99, 114, 125, 128, 130, 135, 158, 166, 169, 177, 192, 196, 203, 205, 209, 221, 239, 242, 251, 256, 268, 273, 283, 290, 315, 324, 347, 358, 365, 367, 375, 378, 379, 384, 387, 390, 392, 403, 422, 423, 427, 443, 445, 460, 474, 476, 493

Offset: 1

Altug Alkan, Apr 29 2016

Numbers n such that sum of the positive divisors of n is the sum of 4 but no fewer nonzero squares.
Prime terms of this sequence are 59, 239, 251, 283, 347, 367, 379, 443, 571, ...
A006532 is a subsequence of complement of this sequence.
Pollack (2011) proved that the complementary sequence has asymptotic density 7/8. Therefore the asymptotic density of this sequence is 1/8. - Amiram Eldar, Apr 09 2020

			1 is not a term since sigma(1) = 1 = 0^2 + 0^2 + 1^2 is the sum of 3 squares.
4 is a term since sigma(4) = 7 is not the sum of 3 squares.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Paul Pollack, Values of the Euler and Carmichael functions which are sums of three squares, Integers, Vol. 11 (2011), pp. 145-161.

Cf. A000203, A004215, A006532.

Select[Range@ 500, ! SquaresR[3, DivisorSigma[1, #]] > 0 &] (* Michael De Vlieger, Apr 29 2016 *)

isA004215(n) = {n\4^valuation(n, 4)%8==7}
lista(nn) = for(n=1, nn, if(isA004215(sigma(n)), print1(n, ", ")));

from itertools import count, islice
from sympy import divisor_sigma
def A272405_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n:not (m:=(~(s:=int(divisor_sigma(n)))&s-1).bit_length())&1 and (s>>m)&7==7,count(max(startvalue,1)))
A272405_list = list(islice(A272405_gen(),30)) # Chai Wah Wu, Jul 09 2022

{n: A000203(n) in A004215}. - R. J. Mathar, May 02 2016

A297970 Numbers that are not the sum of 3 squares and a nonnegative 7th power.

Original entry on oeis.org

112, 240, 368, 496, 624, 752, 880, 1008, 1136, 1264, 1392, 1520, 1648, 1776, 1904, 2032, 2160

Offset: 1

XU Pingya, Jan 10 2018

The last term in this sequence is 2160. The reasons are as follows (let b, c, d, i, j, k, m, r, s, t, w, x, y and z be nonnegative integers).
For the Diophantine equation x^2 + y^2 + z^2 + w^7 = m:
(1) If m is not of the form 4^c * (8b + 7), then it follows from Legendre's three-square theorem that the equation has a solution with w = 0.
(2) 8b + 7 - 1^7 = 8b + 6. Then m = 8b + 7, the equation has a solution with w = 1.
(3) 4 * (8b + 7) - 1^7 = (8 * (4b + 3) + 3) = 8d + 3. Then m = 4 * (8b + 7), the equation has a solution with w = 1.
(4) For b >= 17, 16 * (8b + 7) - 3^7 = 8 * (16 * (b - 17) + 12) + 5 = 8i + 5. Then m = 16 * (8b + 7) and b >= 17, the equation has a solution with w = 3.
(5) 4^3 * (8b + 7) - 2^7 = 4^3 * (8b + 5). Then m = 4^3 * (8b + 7), the equation has a solution with w = 2. And 4^3 * (8b + 7) - 3^7 = 8 * (4^3 * (b - 4) + 38) + 5 = 8j + 5. Then m = 4^3 * (8b + 7) and b >= 4, the equation has a solution with w = 3.
(6) 4^4 * (8b + 7) - 2^7 = 4^3 * (8 * (4b + 3) + 3) = 4^3 * (8k + 3). 4^4 * (8b + 7) - 3^7 = 8 * (256b - 217) + 3 = 8r + 3. Then m = 4^4 * (8b + 7), the equation has a solution with w = 2 and when b > 0, the equation has a solution with w = 3.
(7) When c >= 5, 4^c * (8b + 7) - 2^7 = 4^3 * (8 * (b * 4^(c - 3) + 14 * 4^(c - 5) + 5) = 4^3 * (8s + 5). 4^c * (8b + 7) - 3^7 = 8 * (b * 4^(c - 3) + 14 * 4^(c - 3) - 273) + 3 = 8t + 3. Then n = 4^c * (8b + 7), the equation has solutions with w = 2 and 3.
In short, except for the 17 numbers in the sequence, every nonnegative integer can be represented as the sum of 3 squares and a nonnegative 7th power.

Wikipedia, Legendre's three-square theorem

Finite subsequence of A004215 and A296185.

Cf. A004771, A022552, A022557, A022561, A022566, A111151.

t1={};
Do[Do[If[x^2+y^2+z^2+w^7==n, AppendTo[t1,n]&&Break[]], {x,0,n^(1/2)}, {y,x,(n-x^2)^(1/2)}, {z,y,(n-x^2-y^2)^(1/2)}, {w,0,(n-x^2-y^2-z^2)^(1/7)}], {n,0,3000}];
t2={};
Do[If[FreeQ[t1,k]==True, AppendTo[t2,k]], {k,0,3000}];
t2

a(n) = 128n - 16 = 16 * A004771(n - 1), 1 <= n <= 17.

A063951 Every number is the sum of 4 squares; these are the odd numbers n such that the first square can be taken to be any positive square < n.

Original entry on oeis.org

1, 3, 5, 7, 9, 13, 15, 17, 21, 25, 33, 41, 45, 49, 57, 65, 73, 81, 89, 97, 105, 129, 145, 153, 169, 177, 185, 201, 209, 217, 225, 257, 273, 297, 305, 313, 329, 345, 353, 385, 425, 433, 441, 481, 513, 561, 585, 609, 689, 697, 713, 817, 825, 945

Offset: 1

N. J. A. Sloane, Sep 04 2001

Odd numbers n such that for all k with 1 <= k < sqrt(n), n - k^2 is not in A004215. - Robert Israel, Jan 24 2018

The only numbers for which allowing k = 0 would make a difference are 7 and 15: These two are not in A063954.

J. H. Conway, personal communication, Aug 27, 2001.

Cf. A004215, A063949, A063950, A063952, A063953, A063954.

isA004215:= proc(n)
  local t;
  t:= padic:-ordp(n,2);
  t::even and (n/2^t) mod 8 = 7
end proc:
filter:= proc(n) andmap(not(isA004215), [seq(n - k^2, k=1..floor(sqrt(n-1)))]) end proc:
select(filter, [seq(i,i=1..1000,2)]); # Robert Israel, Jan 24 2018

ok[n_] := Range[ Floor[ Sqrt[n] ]] == DeleteCases[ Union[ Flatten[ PowersRepresentations[n, 4, 2]]], 0, 1, 1]; A063951 = Select[ Range[1, 999, 2], ok] (* Jean-François Alcover, Sep 12 2012 *)

is_A063951(n)=bittest(n,0)&&!forstep(k=sqrtint(n-1),1,-1,isA004215(n-k^2)&&return) \\ M. F. Hasler, Jan 26 2018

A063951=select(is_A063951,[1..945]) \\ M. F. Hasler, Jan 26 2018

This A063951 = A063954 U { 7, 15 }. - M. F. Hasler, Jan 27 2018

A084953 Numbers k such that k! is the sum of 4 but no fewer nonzero squares.

Original entry on oeis.org

10, 12, 24, 25, 48, 49, 54, 60, 78, 91, 96, 97, 107, 114, 120, 121, 142, 151, 167, 170, 172, 180, 192, 193, 212, 222, 226, 238, 240, 241, 246, 252, 270, 279, 301, 307, 309, 318, 327, 333, 344, 345, 357, 360, 361, 367, 375, 379, 384, 385, 403, 405, 421, 424, 425

Offset: 1

Hugo Pfoertner, Jun 15 2003

nonn

The asymptotic density of this sequence is 1/8 (Deshouillers and Luca, 2010). - Amiram Eldar, Jan 11 2021

			a(1) = 10 because 10! cannot be written as the sum of fewer than 4 squares.

Hugo Pfoertner, Table of n, a(n) for n = 1..5000
Dario Alpern, Sum of squares web application. See also code on GitHub.
Rob Burns, Factorials and Legendre's three-square theorem, arXiv:2101.01567 [math.NT], 2021.
Jean-Marc Deshouillers and Florian Luca, How often is n! a sum of three squares?, in: The legacy of Alladi Ramakrishnan in the mathematical sciences, Springer, New York, 2010, pp. 243-251.

Cf. A000142, A004215, A084966.

Complement of A267215.

C
```
/* See Alpern link. */
```

Select[Range[500], Mod[#!/4^IntegerExponent[#!, 4], 8] == 7 &] (* Amiram Eldar, Jan 11 2021 *)

isA004215(n)= n\4^valuation(n, 4)%8==7;
isok(n) = isA004215(n!); \\ Michel Marcus, Dec 30 2020

from math import factorial
from itertools import count, islice
def A084953_gen(startvalue=1): # generator of terms >= startvalue
        return filter(lambda n:(factorial(n)>>((n-n.bit_count())&-2))&7==7,count(max(startvalue,1)))
A084953_list = list(islice(A084953_gen(),30)) # Chai Wah Wu, Jul 09 2022

Equivalently, k! is of the form (4^i)*(8*j+7), i >= 0, j >= 0.

a(n) ~ 8n. - Charles R Greathouse IV, Jan 06 2025

Edited and extended by Robert G. Wilson v, Jun 17 2003

Added missing term 357 by Rob Burns, Dec 30 2020

A180347 The number of n-digit numbers requiring 4 nonzero squares in their representation as sum of squares.

Original entry on oeis.org

1, 14, 150, 1500, 14999, 150000, 1499999, 15000000, 149999998, 1500000001, 14999999999, 149999999999, 1500000000001, 14999999999999, 149999999999999, 1500000000000001, 14999999999999998, 149999999999999999, 1500000000000000003, 14999999999999999996

Offset: 1

Martin Renner, Jan 18 2011

A049415(n) + A180426(n) + A180429(n) + a(n) = A052268(n).

Eric W. Weisstein: MathWorld : Lagrange's Four-Square Theorem.
Eric W. Weisstein: MathWorld: Sum of Squares Function.

Cf. A004215, A167615.

Cf. A049415, A052268, A180426, A180429.

a(n) = A167615(n)-A167615(n-1).

A267321 Perfect powers that are not of the form x^2 + y^2 + z^2 where x, y and z are integers.

Original entry on oeis.org

343, 3375, 12167, 16807, 21952, 29791, 59319, 103823, 166375, 216000, 250047, 357911, 493039, 658503, 759375, 778688, 823543, 857375, 1092727, 1367631, 1404928, 1685159, 1906624, 2048383, 2460375, 2924207, 3442951, 3796416, 4019679, 4657463, 5359375, 6128487

Offset: 1

Altug Alkan, Jan 13 2016

nonn

Perfect powers that are the sum of 4 but no fewer nonzero squares. See first comment in A004215.
Intersection of A001597 and A004215.
A134738 is a subsequence.
Motivation for this sequence is the equation m^k = x^2 + y^2 + z^2 where x, y, z are integers and m > 0, k >= 2.
Corresponding exponents are 3, 3, 3, 5, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 5, 3, 7, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 5, 3, 3, 3, 3, 3, ...
Numbers of the form (4^i*(8*j+7))^(2*k+3) where i,j,k>=0. - Robert Israel, Jan 14 2016

			16807 is a term because 16807 = 7^5 and there is no integer values of x, y and z for the equation 7^5 = x^2 + y^2 + z^2.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A001597, A004215, A134738.

N:= 10^10; # to get all terms <= N
sort(convert({seq(seq(seq((4^i*(8*j+7))^(2*k+3),
    k=0..floor(1/2*(log[4^i*(8*j+7)](N)-3))),
     j = 0 .. floor((N^(1/3)*4^(-i)-7)/8)),
i=0..floor(log[4](N^(1/3)/7)))},list)); # Robert Israel, Jan 14 2016

isA004215(n) = { my(fouri, j) ; fouri=1 ; while( n >=7*fouri, if( n % fouri ==0, j= n/fouri -7 ; if( j % 8 ==0, return(1) ) ; ) ; fouri *= 4 ; ) ; return(0) ; } { for(n=1, 400, if(isA004215(n), print1(n, ", ") ; ) ; ) ; }
for(n=0, 1e7, if(isA004215(n) && ispower(n), print1(n, ", ")));

A273324 Integers n such that n^2 + 3 is the sum of 4 but no fewer nonzero squares.

Original entry on oeis.org

2, 5, 6, 10, 11, 14, 18, 21, 22, 26, 27, 30, 34, 37, 38, 42, 43, 46, 50, 53, 54, 58, 59, 62, 66, 69, 70, 74, 75, 78, 82, 85, 86, 90, 91, 94, 98, 101, 102, 106, 107, 110, 114, 117, 118, 122, 123, 126, 130, 133, 134, 138, 139, 142, 146, 149, 150, 154, 155, 158, 162, 165, 166, 170

Offset: 1

Altug Alkan, May 20 2016

If n^2 + k is a term of A004215, then the minimum positive value of k is 3, obviously.

See also the first differences (A278536) of this sequence.

			2 is in the sequence because 2^2 + 3 = 7 is a term of A004215.

Antti Karttunen, Table of n, a(n) for n = 1..10001

Cf. A000196, A004215, A117950, A278491, A278536.

isA004215(n) = {n\4^valuation(n, 4)%8==7}
lista(nn) = for(n=1, nn, if(isA004215(n^2+3), print1(n, ", ")));

a(n) = A000196(1+A278491(n)). - Antti Karttunen, Nov 26 2016

cp's OEIS Frontend

A167615 Total number of positive integers below 10^n with 4 positive squares in their representation as sum of squares.

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A261904 Largest x such that n can be written as n = x^2 + y^2 + z^2 with x >= y >= z >= 0, or -1 if no such x exists.

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A261915 Smallest x such that n can be written as n = x^2 + y^2 + z^2 with x >= y >= z >= 0, or -1 if no such x exists.

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A272405 Numbers n such that sum of the divisors of n is not of the form x^2 + y^2 + z^2 where x, y, z are integers.

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A297970 Numbers that are not the sum of 3 squares and a nonnegative 7th power.

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A063951 Every number is the sum of 4 squares; these are the odd numbers n such that the first square can be taken to be any positive square < n.

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A084953 Numbers k such that k! is the sum of 4 but no fewer nonzero squares.

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