A000408 -id:A000408 - cp's OEIS frontend

A267215 Integers k such that k! is the sum of 3 squares.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 50, 51, 52, 53, 55, 56, 57, 58, 59, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 79, 80

Offset: 1

Altug Alkan, Jan 12 2016

nonn

Motivation for this sequence is the equation n! = x^2 + y^2 + z^2 where x, y and z are integers.

The asymptotic density of this sequence is 7/8 (Deshouillers and Luca, 2010). - Amiram Eldar, Jan 11 2021

			0 is a term because 0! = 1 = 0^2 + 0^2 + 1^2.
2 is a term because 2! = 2 = 0^2 + 1^2 + 1^2.
3 is a term because 3! = 6 = 1^2 + 1^2 + 2^2.
6 is a term because 6! = 720 = 0^2 + 12^2 + 24^2.

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. A000408, A004215, A084953.

Complement of A084953.

Select[Range[0, 18], SquaresR[3, #!] > 0 &] (* Michael De Vlieger, Jan 13 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, 1e2, if(!isA004215(n!), print1(n, ", ")));

A267312 Integers n such that n^3 is the sum of 3 nonzero squares.

Original entry on oeis.org

3, 5, 6, 9, 10, 11, 12, 13, 14, 17, 18, 19, 20, 21, 22, 24, 25, 26, 27, 29, 30, 33, 34, 35, 36, 37, 38, 40, 41, 42, 43, 44, 45, 46, 48, 49, 50, 51, 52, 53, 54, 56, 57, 58, 59, 61, 62, 65, 66, 67, 68, 69, 70, 72, 73, 74, 75, 76, 77, 78, 80, 81, 82, 83, 84, 85, 86, 88, 89, 90, 91, 93, 94, 96, 97, 98, 99, 100

Offset: 1

Altug Alkan, Jan 13 2016

nonn

Motivation for this sequence is the equation n^3 = x^2 + y^2 + z^2 where x, y and z are nonzero integers.

Corresponding cubes are 27, 125, 216, 729, 1000, 1331, 1728, 2197, 2744, 4913, 5832, 6859, 8000, 9261, 10648, 13824, 15625, 17576, 19683, 24389, 27000, ...

			3 is a term because 3^3 = 27 = 1^2 + 1^2 + 5^2.
5 is a term because 5^3 = 125 = 5^2 + 6^2 + 8^2.
6 is a term because 6^3 = 216 = 2^2 + 4^2 + 14^2.
9 is a term because 9^3 = 729 = 2^2 + 10^2 + 25^2.

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A000408.

Select[Range@ 100, Length[PowersRepresentations[#^3, 3, 2] /. {x_, , } /; x == 0 -> Nothing] != 0 &] (* Michael De Vlieger, Jan 13 2016 *)

is(n) = { my(a, b) ; a=1; while(a^2+1

A267686 Positive integers n such that n^4 = a^3 + b^3 = x^2 + y^2 + z^2 where x, y, z, a and b are positive integers, is soluble.

Original entry on oeis.org

9, 28, 35, 54, 65, 72, 91, 126, 133, 134, 152, 182, 183, 189, 201, 217, 219, 224, 243, 250, 273, 278, 280, 309, 341, 344, 351, 370, 399, 407, 422, 432, 453, 468, 497, 513, 520, 539, 559, 576, 579, 637, 651, 658, 686, 728, 730, 737, 756, 793, 854, 855

Offset: 1

Altug Alkan, Jan 19 2016

nonn

Inspired by intersection of A000408, A000583 and A003325.
Corresponding fourth powers are 6561, 614656, 1500625, 8503056, 17850625, 26873856, 68574961, 252047376, 312900721, 322417936, 533794816, 1097199376, 1121513121, 1275989841, 1632240801, 2217373921, 2300257521, 2517630976, 3486784401, ...
2 is the first number that its 4th power, 2^4, is the sum of 2 positive cubes and is not the sum of 3 nonzero squares. 16 is the second number for this case. So 2 and 16 are not in this sequence.

			9 is a term because 9^4 = 9^3 + 18^3 = 1^2 + 28^2 + 76^2.
28 is a term because 28^4 = 28^3 + 84^3 = 64^2 + 144^2 + 768^2.
35 is a term because 35^4 = 70^3 + 105^3 = 1^2 + 600^2 + 1068^2.
54 is a term because 54^4 = 162^3 + 162^3 = 12^2 + 264^2 + 2904^2.
399 is a term because 399^4 = 665^3 + 2926^3 = 17^2 + 11236^2 + 158804^2.

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A000408, A000583, A003325, A267702.

isA000408(n) = {my(a, b); a=1; while(a^2+1A003325(n)=#select(v->min(v[1], v[2])>0, thue(T, n))>0
for(n=3, 1e3, if(isA000408(n^4) && isA003325(n^4), print1(n, ", ")));

Added missing term a(32), Chai Wah Wu, Jan 31 2016

A272100 Integers n that are the sum of three nonzero squares while n*(n+1) is not.

Original entry on oeis.org

12, 19, 44, 51, 76, 83, 108, 115, 140, 147, 172, 179, 204, 211, 236, 243, 268, 275, 300, 307, 332, 339, 364, 371, 396, 403, 428, 435, 460, 467, 492, 499, 524, 531, 556, 563, 588, 595, 620, 627, 652, 659, 684, 691, 716, 723, 748, 755, 780, 787, 812, 819, 844, 851, 876, 883, 908, 915

Offset: 1

Altug Alkan, Apr 20 2016

Values of a^2 + b^2 + c^2 such that (a^2 + b^2 + c^2)^2 + a^2 + b^2 + c^2 is not of the form x^2 + y^2 + z^2 where a, b, c, x, y, z are nonzero integers.

First differences of this sequence are 7, 25, 7, 25, 7, 25, 7, 25, 7, 25, ...

			12 is a term because 12 = 2^2 + 2^2 + 2^2 = A000408(5) and 12*13 = A002378(12) = 156 is not in A000408.

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A000408, A002378.

Select[Range[10^3], Length[PowersRepresentations[#, 3, 2] /. {0, } -> Nothing] > 0 && Length[PowersRepresentations[# (# + 1), 3, 2] /. {0, } -> Nothing] == 0 &] (* Michael De Vlieger, Apr 20 2016, Version 10.2 *)
LinearRecurrence[{1,1,-1},{12,19,44},60] (* Harvey P. Dale, Mar 13 2017 *)

isA000408(n) = my(a, b) ; a=1 ; while(a^2+1A000408(n) && !isA000408(n*(n+1)), print1(n, ", ")));

Vec(x*(12+7*x+13*x^2)/((1-x)^2*(1+x)) + O(x^50)) \\ Colin Barker, Apr 30 2016

From Colin Barker, Apr 30 2016: (Start)
a(n) = (32*n-17-9*(-1)^n)/2.
a(n) = 16*n-13 for n even.
a(n) = 16*n-4 for n odd.
a(n) = a(n-1)+a(n-2)-a(n-3) for n>3.
G.f.: x*(12+7*x+13*x^2) / ((1-x)^2*(1+x)).
(End)

A302359 Numbers that are the sum of 3 squares > 1.

Original entry on oeis.org

12, 17, 22, 24, 27, 29, 33, 34, 36, 38, 41, 43, 44, 45, 48, 49, 50, 54, 56, 57, 59, 61, 62, 65, 66, 67, 68, 69, 70, 72, 74, 75, 76, 77, 78, 81, 82, 83, 84, 86, 88, 89, 90, 93, 94, 96, 97, 98, 99, 101, 102, 104, 105, 106, 107, 108, 109, 110, 113, 114, 115, 116, 117, 118, 120, 121, 122, 123, 125, 126, 129

Offset: 1

Ilya Gutkovskiy, Apr 06 2018

nonn

			33 is in the sequence because 33 = 2^2 + 2^2 + 5^2.

Michael S. Branicky, Table of n, a(n) for n = 1..10000
Index entries for sequences related to sums of squares

Cf. A000378, A000408, A085989, A214514, A281155.

max = 130; f[x_] := Sum[x^(k^2), {k, 2, 20}]^3; Exponent[#, x] & /@ List @@ Normal[Series[f[x], {x, 0, max}]]
With[{nn=15},Select[Union[Total/@Tuples[Range[2,nn]^2,3]],#<=nn^2+8&]] (* Harvey P. Dale, Jul 05 2021 *)

from itertools import count, takewhile, combinations_with_replacement as mc
def aupto(N):
    sqrs = list(takewhile(lambda x: x<=N, (i**2 for i in count(2))))
    sum3 = set(sum(c) for c in mc(sqrs, 3) if sum(c) <= N)
    return sorted(sum3)
print(aupto(129)) # Michael S. Branicky, Dec 17 2021

A065940 Integers of the form bc/a + ac/b + ab/c, where a,b,c are positive integers.

Original entry on oeis.org

3, 6, 9, 11, 12, 14, 15, 17, 18, 19, 21, 22, 24, 26, 27, 28, 29, 30, 33, 34, 35, 36, 38, 39, 41, 42, 43, 44, 45, 46, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 72, 73, 74, 75, 76, 77, 78, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91

Offset: 0

Floor van Lamoen, Dec 08 2001

nonn

F.M. van Lamoen et al., POLYA010, Integers of the form bc/a + ca/b + ab/c.

Positive multiples of A000408.

A144020 Numbers of the form 1+i^2+j^2+k^2 with 1 <= i <= j <= k.

Original entry on oeis.org

4, 7, 10, 12, 13, 15, 18, 19, 20, 22, 23, 25, 27, 28, 30, 31, 34, 35, 36, 37, 39, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 54, 55, 57, 58, 60, 62, 63, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 82, 83, 84, 85, 87, 89, 90, 91, 92, 94, 95, 97, 98, 99, 100, 102, 103, 105, 106

Offset: 1

N. J. A. Sloane, Dec 02 2008, Dec 21 2008

nonn

No number == -1 mod 8 is a sum of three squares, so multiples of 8 are missing from this sequence.

It appears that numbers == 1 mod 16 are also missing from the sequence.

a(n) = 1+A000408(n). [From R. J. Mathar, Dec 03 2008]

A242130 Sum of squares of three numbers x, y, z such that 3*floor(sum of squares/3) = x+y+z.

Original entry on oeis.org

3, 12, 14, 18, 27, 29, 33, 35, 41, 48, 50, 54, 56, 62, 66, 72, 74, 75, 77, 81, 83, 89, 93, 99, 101, 107, 108, 110, 114, 116, 122, 126, 132, 134, 140, 146, 147, 149, 153, 155, 161, 165, 171, 173, 179, 185, 189, 192, 194, 198, 200

Offset: 1

Carmine Suriano, May 05 2014

nonn

Three times the quadratic average of x,y,z in most cases is greater than their sum.

			a(7)=33 for 33=1^2+4^2+4^2; floor(sqrt(33/3))=3; 3*3=9=1+4+4.

Cf. A000408.

A267431 Indices of Catalan numbers that are not of the form x^2 + y^2 + z^2 where x, y and z are integers.

Original entry on oeis.org

10, 24, 37, 43, 46, 48, 49, 51, 69, 87, 96, 97, 102, 103, 109, 114, 117, 120, 133, 157, 170, 175, 187, 190, 192, 193, 198, 207, 226, 240, 241, 243, 261, 285, 300, 308, 332, 344, 351, 356, 360, 375, 384, 385, 390, 404, 411, 414, 415, 420, 424, 445, 450, 459, 462, 477, 480, 481

Offset: 1

Altug Alkan, Jan 15 2016

nonn

See first comment in A004215.
Corresponding Catalan numbers are 16796, 1289904147324, 45950804324621742364, 150853479205085351660700, ...
It is obvious that minimum value of a(n) - a(n-1) is 1. Is there a maximum value of a(n) - a(n-1)?

			10 is a term because the 10th Catalan number is 16796 and there are no integer values of x, y and z for the equation 16796 = x^2 + y^2 + z^2.

Cf. A000108, A004215, A000408.

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, ", ") ; ) ; ) ; }
c(n) = binomial(2*n,n)/(n+1);
for(n=0, 1e3, if(isA004215(c(n)), print1(n, ", ")));

A272119 Values of a^2 + b^2 such that the equation (a^2 + b^2)^2 = x^2 + y^2 + z^2 is soluble where a, b, x, y, z are nonzero integers.

Original entry on oeis.org

13, 17, 18, 25, 26, 29, 34, 37, 41, 45, 50, 52, 53, 58, 61, 65, 68, 72, 73, 74, 82, 85, 89, 90, 97, 98, 100, 101, 104, 106, 109, 113, 116, 117, 122, 125, 130, 136, 137, 145, 146, 148, 149, 153, 157, 162, 164, 169, 170, 173, 178, 180, 181, 185, 193, 194, 197, 200, 202, 205, 208, 212, 218

Offset: 1

Altug Alkan, Apr 21 2016

52 is the first term that is not a member of A046711.

			13 is a term because 13 = 2^2 + 3^2 and 13^2 = 3^2 + 4^2 + 12^2.

Cf. A000404, A000408, A046711.

isA000404(n) = {for( i=1, #n=factor(n)~%4, n[1, i]==3 && n[2, i]%2 && return); n && ( vecmin(n[1, ])==1 || (n[1, 1]==2 && n[2, 1]%2))}
isA000408(n) = {my(a, b) ; a=1 ; while(a^2+1A000404(n) && isA000408(n^2), print1(n, ", ")));

cp's OEIS Frontend

A267215 Integers k such that k! is the sum of 3 squares.

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A267312 Integers n such that n^3 is the sum of 3 nonzero squares.

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A267686 Positive integers n such that n^4 = a^3 + b^3 = x^2 + y^2 + z^2 where x, y, z, a and b are positive integers, is soluble.

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A272100 Integers n that are the sum of three nonzero squares while n*(n+1) is not.

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A302359 Numbers that are the sum of 3 squares > 1.

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A065940 Integers of the form bc/a + ac/b + ab/c, where a,b,c are positive integers.

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A144020 Numbers of the form 1+i^2+j^2+k^2 with 1 <= i <= j <= k.

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A242130 Sum of squares of three numbers x, y, z such that 3*floor(sum of squares/3) = x+y+z.

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A267431 Indices of Catalan numbers that are not of the form x^2 + y^2 + z^2 where x, y and z are integers.

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