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Search: a349324 -id:a349324
Displaying 1-8 of 8 results found. page 1
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A271576 Integers whose square is of the form x^2 + y^4, with x,y > 0 (see A111925). +10
16
5, 15, 20, 34, 39, 41, 45, 60, 65, 80, 85, 111, 125, 135, 136, 145, 150, 156, 164, 175, 180, 194, 219, 240, 245, 255, 260, 265, 306, 313, 320, 325, 340, 351, 353, 369, 371, 375, 405, 410, 444, 445, 455, 500, 505, 514, 540, 544, 580, 585, 600, 605, 609, 624, 629, 656, 671, 674, 689 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
If z is in this sequence, z*k^2 is, for k > 0. Also note that since there are no fourth powers in A111925, there are no squares in this sequence. - Altug Alkan, Apr 10 2016
From Karl-Heinz Hofmann, Oct 22 2021: (Start)
Terms must have at least one prime factor of the form p == 1 (mod 4), a Pythagorean prime (A002144).
If the terms additionally have prime factors of the form p == 3 (mod 4), which are in A002145, then they must appear in the prime divisor sets of x and y too. (End)
From Jon E. Schoenfield, Nov 15 2021: (Start)
Apparently, the vast majority of the terms of this sequence can be expressed as x^2 + y^4 with x,y > 0 in only one way (A345645 lists those terms), but some can be so expressed in exactly two, three, four, five, six, or more ways. Among the first 3976926961 terms of this sequence,
3948648229 are in A345645 (exactly 1 way),
25415062 are in A345700 (exactly 2 ways),
2697713 are in A345968 (exactly 3 ways),
161543 are in A346110 (exactly 4 ways),
3989 are in A348655 (exactly 5 ways),
424 are in A349324 (exactly 6 ways),
and just 1 -- a(3976926961) = 2474052064291275 = A346115(7) -- is a number whose square can be written as x^2 + y^4 with x,y > 0 in exactly 7 ways. The ratios of successive counts above, i.e., 3948648229/25415062, 25415062/2697713, 2697713/161543, 161543/3989, 3989/424, and 424/1, are approximately 155.4, 9.4, 16.7, 40.5, 9.4, and 424.0. What is it that (over the interval [1, 2474052064291275], at least) makes numbers whose squares can be written as x^2 + y^4 with x,y > 0 in more than 6 ways so much rarer than those that can be written thus in exactly 6 ways? (End)
LINKS
Chai Wah Wu, Table of n, a(n) for n = 1..10000
EXAMPLE
5^2 = 25 = 9 + 16 = 3^2 + 2^4, so 5 is a term.
MATHEMATICA
Select[Range@ 200, Resolve[Exists[{x, y}, Reduce[#^2 == x^2 + y^4, {x, y}, Integers], And[x > 0, y > 0]]] &] (* Michael De Vlieger, Apr 10 2016 *)
PROG
(PARI) isok(n) = n = n^2; for (k=1, n-1, if (issquare(k) && (p=ispower(n-k, 4)), return (1)))
(PARI) is(n)=my(n2=n^2); for(b=sqrtnint(2*n-2, 4)+1, sqrtint(n-1), if(issquare(n2-b^4), return(1))); 0 \\ Charles R Greathouse IV, Nov 16 2021
CROSSREFS
Cf. A111925, A271577, A345645 (in exactly 1 way).
Cf. A345700 (in exactly 2 ways), A345968 (in exactly 3 ways).
Cf. A346110 (in exactly 4 ways), A349324 (in exactly 6 ways), A346115 (the least solutions).
Cf. A002144 (p == 1 (mod 4)), A002145 (p == 3 (mod 4)).
KEYWORD
nonn
AUTHOR
Michel Marcus, Apr 10 2016
STATUS
approved
A346110 Numbers whose square can be represented in exactly four ways as the sum of a positive square and a positive fourth power. +10
10
469625, 1878500, 2224625, 4226625, 7514000, 8898500, 11740625, 15289625, 16906500, 20021625, 23011625, 25716665, 30056000, 35594000, 38039625, 46962500, 54316275, 55615625, 56824625, 61158500, 67626000, 79366625, 80086500, 92046500, 92481870 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
Terms are numbers z such that there are exactly four solutions to z^2 = x^2 + y^4, where x, y and z belong to the set of positive integers.
Terms cannot be a square (see the comment from Altug Alkan in A111925).
Terms must have at least one prime factor of the form p == 1 (mod 4), a Pythagorean prime (A002144).
If the terms additionally have prime factors of the form p == 3 (mod 4), which are in A002145, then they must appear in the prime divisor sets of x and y too.
The special prime factor 2 has the same behavior. Means: If the term is even, x and y must be even too.
Apparently, all terms are divisible by 65. The divided terms are in A346594. Are there exceptions for n > 25? - Hugo Pfoertner, Jul 14 2021, Jul 29 2021
Yes, there are exceptions: a(44,46,53,95,97) are not divisible by 65 (5*13) but they have in common: They are divisible by 145 (5*29). - Karl-Heinz Hofmann, Aug 28 2021
LINKS
Jon E. Schoenfield, Table of n, a(n) for n = 1..10000 (first 97 terms from Karl-Heinz Hofmann)
Karl-Heinz Hofmann, All valid {z,x1,y1,x2,y2,x3,y3,x4,y4} sets up to 10^9.
Karl-Heinz Hofmann, A 3D Animation of the solutions up to 10^9.
Karl-Heinz Hofmann, Python code (not only for 4 Solutions).
EXAMPLE
29679^2 = 29640^2 + 39^4, so 29679 is not a term (only one solution).
60^2 + 5^4 = 63^2 + 4^4 = 65^2, so 65 is not a term (only two solutions).
572^2 + 39^4 = 1500^2 + 25^4 = 1575^2 + 20^4 = 1625^2, so 1625 is not a term (only three solutions).
165308^2 + 663^4 = 349575^2 + 560^4 = 433500^2 + 425^4 = 455175^2 + 340^4 = 469625^2, so 469625 is a term (four solutions).
PROG
(Python) see link above
CROSSREFS
Cf. A000290, A000583.
Cf. A271576 (all solutions), A345645 (one solution), A345700 (two solutions), A345968 (three solutions), A348655 (five solutions), A349324 (six solutions), A346115 (the least solutions).
Cf. A002144 (p == 1 (mod 4)), A002145 (p == 3 (mod 4)).
KEYWORD
nonn
AUTHOR
Karl-Heinz Hofmann, Jul 05 2021
STATUS
approved
A345645 Numbers whose square can be represented in exactly one way as the sum of a square and a biquadrate (fourth power). +10
9
5, 15, 20, 34, 39, 41, 45, 60, 80, 85, 111, 125, 135, 136, 150, 156, 164, 175, 180, 194, 219, 240, 245, 255, 265, 306, 313, 320, 325, 340, 351, 353, 369, 371, 375, 405, 410, 444, 445, 455, 500, 505, 514, 540, 544, 600, 605, 609, 624, 629, 656, 671, 674, 689 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
Numbers z such that there is exactly one solution to z^2 = x^2 + y^4.
From Karl-Heinz Hofmann, Oct 21 2021: (Start)
No term can be a square (see the comment from Altug Alkan in A111925).
Terms must have at least one prime factor of the form p == 1 (mod 4), a Pythagorean prime (A002144).
Additionally, if the terms have prime factors of the form p == 3 (mod 4), which are in A002145, then they must appear in the prime divisor sets of x and y too.
The special prime factor 2 has the same behavior, i.e., if the term is even, x and y must be even too. (End)
LINKS
Karl-Heinz Hofmann, Table of n, a(n) for n = 1..10000
EXAMPLE
3^2 + 2^4 = 9 + 16 = 25 = 5^2, so 5 is a term.
60^2 + 5^4 = 63^2 + 4^4 = 65^2, so 65 is not a term.
MATHEMATICA
Select[Range@100, Length@Solve[x^2+y^4==#^2&&x>0&&y>0, {x, y}, Integers]==1&] (* Giorgos Kalogeropoulos, Jun 25 2021 *)
PROG
(Python)
terms = []
for i in range(1, 700):
occur = 0
ii = i*i
for j in range(1, i):
k = int((ii - j*j) ** 0.25)
if k*k*k*k + j*j == ii:
occur += 1
if occur == 1:
terms.append(i)
print(terms)
(PARI) inlist(list, v) = for (i=1, #list, if (list[i]==v, return(1)));
isok(m) = {my(list = List()); for (k=1, sqrtnint(m^2, 4), if (issquare(j=m^2-k^4) && !inlist(vecsort([k^4, j^2])), listput(list, vecsort([k^4, j^2]))); ); #list == 1; } \\ Michel Marcus, Jun 26 2021
CROSSREFS
Cf. A000290, A000583, A180241, A271576 (all solutions).
Cf. A345700 (2 solutions), A345968 (3 solutions), A346110 (4 solutions), A348655 (5 solutions), A349324 (6 solutions), A346115 (the least solutions).
Cf. A002144 (p == 1 (mod 4)), A002145 (p == 3 (mod 4)).
KEYWORD
nonn
AUTHOR
Mohammad Tejabwala, Jun 21 2021
STATUS
approved
A345700 Numbers whose square can be represented in exactly two ways as the sum of a positive square and a positive fourth power. +10
9
65, 145, 260, 580, 585, 1025, 1040, 1105, 1305, 2320, 2340, 2465, 3185, 3625, 4100, 4160, 4335, 4420, 4810, 5220, 5265, 6625, 7105, 7585, 7865, 8405, 9225, 9280, 9360, 9860, 9945, 10985, 11745, 12740, 14500, 16400, 16465, 16640, 17340, 17545, 17680, 19240 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
Terms are numbers z such that there are exactly two solutions to z^2 = x^2 + y^4, where x, y and z belong to the set of positive integers.
Terms cannot be a square (see the comment from Altug Alkan in A111925).
Terms must have at least one prime factor of the form p == 1 (mod 4), a Pythagorean prime (A002144).
If the terms additionally have prime factors of the form p == 3 (mod 4), which are in A002145, then they must appear in the prime divisor sets of x and y too.
The lower limit of the ratio x/y is sqrt(2).
LINKS
Jon E. Schoenfield, Table of n, a(n) for n = 1..10000
Karl-Heinz Hofmann, All valid {z,x1,y1,x2,y2} sets up to 10^7
Karl-Heinz Hofmann, Python code for multiple solutions
EXAMPLE
29679^2 = 29640^2 + 39^4, so 29679 is not a term (only one solution).
17680^2 = 15088^2 + 96^4 = 17472^2 + 52^4, so 17680 is a term.
36900^2 = 36000^2 + 90^4 = 36828^2 + 48^4, so 36900 is a term.
18785^2 = 18207^2 + 68^4 = 17340^2 + 85^4 = 13983^2 + 112^4, so 18785 is not a term (three solutions).
PROG
(Python) # see link above.
(PARI) inlist(list, v) = for (i=1, #list, if (list[i]==v, return(1)));
isok(m) = {my(list = List()); for (k=1, sqrtnint(m^2, 4), if (issquare(j=m^2-k^4) && !inlist(vecsort([k^4, j^2])), listput(list, vecsort([k^4, j^2]))); ); #list == 2; } \\ Michel Marcus, Jun 26 2021
(PARI) is(n)=my(n2=n^2, s); for(y=sqrtnint(2*n-2, 4)+1, sqrtint(n-1), if(issquare(n2-y^4) && s++>2, return(0))); s==2; \\ Charles R Greathouse IV, Jul 02 2021
CROSSREFS
Cf. A111925, A271576, A345645 (in exactly 1 way), A345968 (in exactly 3 ways), A346110 (in exactly 4 ways), A348655 (in exactly 5 ways), A349324 (in exactly 6 ways), A346115 (the least solutions).
Cf. A002144 (p == 1 (mod 4)), A002145 (p == 3 (mod 4)).
KEYWORD
nonn
AUTHOR
Karl-Heinz Hofmann, Jun 24 2021
STATUS
approved
A345968 Numbers whose square can be represented in exactly three ways as the sum of a positive square and a positive fourth power. +10
9
1625, 6500, 14625, 18785, 24505, 26000, 40625, 58500, 75140, 79625, 88985, 98020, 104000, 120250, 131625, 162500, 169065, 196625, 220545, 234000, 274625, 296225, 300560, 318500, 355940, 365625, 392080, 416000, 481000, 526500, 547230, 586625, 611585, 612625 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
Terms are numbers z such that there are exactly 3 solutions to z^2 = x^2 + y^4, where x, y and z belong to the set of positive integers.
No term can be a square (see the comment from Altug Alkan in A111925).
Terms must have at least one prime factor of the form p == 1 (mod 4), a Pythagorean prime (A002144).
Additionally, if the terms have prime factors of the form p == 3 (mod 4), which are in A002145, then they must appear in the prime divisor sets of x and y too.
The special prime factor 2 has the same behavior, i.e., if the term is even, x and y must be even too.
LINKS
Jon E. Schoenfield, Table of n, a(n) for n = 1..10000
Karl-Heinz Hofmann, All valid {z,x1,y1,x2,y2,x3,y3} sets up to 10^8
EXAMPLE
29640^2 + 39^4 = 29679^2; 29679 is not a term (only 1 solution).
60^2 + 5^4 = 63^2 + 4^4 = 65^2; 65 is not a term (only 2 solutions).
572^2 + 39^4 = 1500^2 + 25^4 = 1575^2 + 20^4 = 1625^2; 1625 is a term (3 solutions).
165308^2 + 663^4 = 349575^2 + 560^4 = 433500^2 + 425^4 = 455175^2 + 340^4 = 469625^2; 469625 is not a term (4 solutions).
CROSSREFS
Cf. A271576 (1 and more solutions), A345645 (1 solution), A345700 (2 solutions), A346110 (4 solutions), A348655 (5 solutions), A349324 (6 solutions), A346115 (the least solutions).
Cf. A002144 (p == 1 (mod 4)), A002145 (p == 3 (mod 4)).
KEYWORD
nonn
AUTHOR
Karl-Heinz Hofmann, Jun 30 2021
STATUS
approved
A346115 Least number k such that k^2 can be expressed in exactly n ways as x^2 + y^4 with {x, y} >= 1. +10
9
5, 65, 1625, 469625, 642916625, 15697403475, 2474052064291275 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
a(5) <= 642916625. - Hugo Pfoertner, Jul 07 2021
From Karl-Heinz Hofmann, Sep 02 2021: (Start)
a(6) <= 15697403475.
2 * 10^10 < a(7) <= 2474052064291275.
These two conjectured values arise from the "green group". Up to term a(5) the least solutions are in the "blue group". Follow the links below to get more information about the different colored groups.
Terms cannot be a square (see the comment from Altug Alkan in A111925).
Terms must have at least one prime factor of the form p == 1 (mod 4), a Pythagorean prime (A002144).
If the terms additionally have prime factors of the form p == 3 (mod 4), which are in A002145, then they must appear in the prime divisor sets of x and y too. The special prime factor 2 has the same behavior, i.e., if the term is even, x and y must be even too. (End)
LINKS
Table of n, a(n) for n=1..7.
Karl-Heinz Hofmann, All valid {z,x1,y1,x2,y2,x3,y3,x4,y4} sets up to 10^9
Karl-Heinz Hofmann, A 3D Animation of the 4 solutions and the least 5 solution up to 10^9
EXAMPLE
a(1)=A345645(1); a(2)=A345700(1); a(3)=A345968(1); a(4)=A346110(1).
CROSSREFS
Cf. A000290, A000583, A111925.
Cf. A271576 (1 or more solutions), A345645 (1 solution), A345700 (2 solutions), A345968 (3 solutions), A346110 (4 solutions), A348655 (5 solutions), A349324 (6 solutions).
Cf. A002144 (p == 1 (mod 4)), A002145 (p == 3 (mod 4)).
KEYWORD
nonn,hard,more
AUTHOR
Karl-Heinz Hofmann, Jul 05 2021
EXTENSIONS
a(5) confirmed by Martin Ehrenstein, Jul 09 2021
a(6) confirmed by Karl-Heinz Hofmann, Oct 15 2021
a(7) confirmed by Jon E. Schoenfield, Nov 15 2021
STATUS
approved
A348655 Numbers whose square can be represented in exactly five ways as the sum of a positive square and a positive fourth power. +10
9
642916625, 2571666500, 4418701625, 5786249625, 10286666000, 16072915625, 17674806500, 20931496625, 23144998500, 31502914625, 39768314625, 41146664000, 52076246625, 57801168750, 64291662500, 70699226000, 77792911625, 83725986500, 92579994000, 108652909625 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
Numbers z such that there are exactly 5 solutions to z^2 = x^2 + y^4.
Terms cannot be a square (see the comment from Altug Alkan in A111925).
Terms must have at least one prime factor of the form p == 1 (mod 4), a Pythagorean prime (A002144).
If the terms additionally have prime factors of the form p == 3 (mod 4), which are in A002145, then they must appear in the prime divisor sets of x and y too.
Some other terms of the sequence: 20931496625, 23144998500, 31502914625, 41146664000, 52076246625, 64291662500, 77792911625, 83725986500, 92579994000, 108652909625, 126011658500, 144656240625, 164586656000. - Chai Wah Wu, Oct 29 2021
LINKS
Jon E. Schoenfield, Table of n, a(n) for n = 1..3989 (all terms < A346115(7) = 2474052064291275)
EXAMPLE
5786249625^2 = 5785404300^2 + 9945^4
5786249625^2 = 5608211175^2 + 37740^4
5786249625^2 = 5341153500^2 + 47175^4
5786249625^2 = 4307113575^2 + 62160^4
5786249625^2 = 2036759868^2 + 73593^4
CROSSREFS
Cf. A111925, A271576, A345645 (in exactly 1 way), A345700 (in exactly 2 ways), A345968 (in exactly 3 ways), A346110 (in exactly 4 ways), A349324 (in exactly 6 ways), A346115 (the least solutions).
Cf. A002144 (p == 1 (mod 4)), A002145 (p == 3 (mod 4)).
KEYWORD
nonn
AUTHOR
Karl-Heinz Hofmann, Oct 27 2021
EXTENSIONS
a(8) and beyond from Jon E. Schoenfield, Nov 14 2021
STATUS
approved
A349664 a(n) is the number of solutions for n^4 = z^2 - x^2 with {z,x} >= 1. +10
2
0, 1, 2, 3, 2, 7, 2, 5, 4, 7, 2, 17, 2, 7, 12, 7, 2, 13, 2, 17, 12, 7, 2, 27, 4, 7, 6, 17, 2, 37, 2, 9, 12, 7, 12, 31, 2, 7, 12, 27, 2, 37, 2, 17, 22, 7, 2, 37, 4, 13, 12, 17, 2, 19, 12, 27, 12, 7, 2, 87, 2, 7, 22, 11, 12, 37, 2, 17, 12, 37, 2, 49, 2, 7, 22 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,3
COMMENTS
If n is an odd prime^i, the number of solutions is 2*i.
If n = 2^i, the number of solutions is 2*i-1.
These two facts are not generally valid in reverse for terms > 6.
If a(n) = 2, n is an odd prime. This is generally valid in reverse.
For more information about these facts see the link.
The calculation of the terms is done with an algorithm of Jon E. Schoenfield, which is described in A349324.
Conditions to be satisfied for a valid, countable solution:
- z cannot be a square.
- z must have at least one prime factor of the form p == 1 (mod 4), a Pythagorean prime (A002144).
- If z has prime factors of the form p == 3 (mod 4), which are in A002145, then they must appear in the prime divisor sets of x and n too.
- If z is even, x and n must be even too.
- The lower limit of the ratio x/n is sqrt(2).
- high limits of z and x:
| n is odd | n is even
---------+------------------+------------------
z limit | (n^4 + 1)/2 | (n^4 + 4)/4
x limit | (n^4 + 1)/2 - 1 | (n^4 + 4)/4 - 2
LINKS
Karl-Heinz Hofmann, Table of n, a(n) for n = 1..10000
Karl-Heinz Hofmann, What the terms can tell about n.
EXAMPLE
a(6) = 7 (solutions): 6^4 = 1296 = 325^2 - 323^2 = 164^2 - 160^2 = 111^2 - 105^2 = 85^2 - 77^2 = 60^2 - 48^2 = 45^2 - 27^2 = 39^2 - 15^2.
MATHEMATICA
a[n_] := Length[Solve[n^4 == z^2 - x^2 && x >= 1 && z >= 1, {x, z}, Integers]]; Array[a, 75] (* Amiram Eldar, Dec 14 2021 *)
PROG
(PARI) a(n) = numdiv(if(n%2, n^4, n^4/4))\2; \\ Jinyuan Wang, Dec 19 2021
CROSSREFS
Cf. A000290, A000583, A271576, A349663, A002144, A002145, A346115.
Cf. A345645, A345700, A345968, A346110, A348655, A349324.
KEYWORD
nonn
AUTHOR
Karl-Heinz Hofmann, Dec 13 2021
STATUS
approved
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