A006339 -id:A006339 - cp's OEIS frontend

A290500 Hypotenuses for which there exist exactly 9 distinct integer triangles.

1953125, 3906250, 5859375, 7812500, 11718750, 13671875, 15625000, 17578125, 21484375, 23437500, 27343750, 31250000, 35156250, 37109375, 41015625, 42968750, 44921875, 46875000, 52734375, 54687500, 60546875, 62500000, 64453125, 70312500, 74218750, 82031250

Offset: 1

Hamdi Sahloul, Aug 04 2017

nonn

Numbers whose square is decomposable in 9 different ways into the sum of two nonzero squares: these are those with only one prime divisor of the form 4k+1 with multiplicity nine.

			a(1) = 1953125 = 5^9, a(5) = 11718750 = 2*3*5^9, a(101) = 375000000 = 2^6*3*5^9.

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 1000 terms from Hamdi Sahloul)

Cf. A002144, A006339, A046080, A046109, A083025.

Cf. A004144 (0), A084645 (1), A084646 (2), A084647 (3), A084648 (4), A084649 (5), A097219 (6), A097101 (7), A290499 (8), A097225 (10), A290501 (11), A097226 (12), A097102 (13), A290502 (14), A290503 (15), A097238 (16), A097239 (17), A290504 (18), A290505 (19), A097103 (22), A097244 (31), A097245 (37), A097282 (40), A097626 (67).

r[a_]:={b, c}/.{ToRules[Reduce[0Vincenzo Librandi, Mar 01 2016 *)

Terms are obtained by the product A004144(k)*A002144(p)^9 for k, p > 0 ordered by increasing values.

A290501 Hypotenuses for which there exist exactly 11 distinct integer triangles.

Original entry on oeis.org

48828125, 97656250, 146484375, 195312500, 292968750, 341796875, 390625000, 439453125, 537109375, 585937500, 683593750, 781250000, 878906250, 927734375, 1025390625, 1074218750, 1123046875, 1171875000, 1318359375, 1367187500, 1513671875, 1562500000, 1611328125

Offset: 1

Hamdi Sahloul, Aug 04 2017

nonn

Numbers whose square is decomposable in 11 different ways into the sum of two nonzero squares: these are those with only one prime divisor of the form 4k+1 with multiplicity eleven.

			a(1) = 48828125 = 5^11, a(5) = 292968750 = 2*3*5^11, a(101) = 9375000000 = 2^6*3*5^11.

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 1000 terms from Hamdi Sahloul)

Cf. A002144, A006339, A046080, A046109, A083025.

Cf. A004144 (0), A084645 (1), A084646 (2), A084647 (3), A084648 (4), A084649 (5), A097219 (6), A097101 (7), A290499 (8), A290500 (9), A097225 (10), A097226 (12), A097102 (13), A290502 (14), A290503 (15), A097238 (16), A097239 (17), A290504 (18), A290505 (19), A097103 (22), A097244 (31), A097245 (37), A097282 (40), A097626 (67).

r[a_]:={b, c}/.{ToRules[Reduce[0Vincenzo Librandi, Mar 01 2016 *)

Terms are obtained by the product A004144(k)*A002144(p)^11 for k, p > 0 ordered by increasing values.

A290502 Hypotenuses for which there exist exactly 14 distinct integer triangles.

Original entry on oeis.org

6103515625, 12207031250, 18310546875, 24414062500, 36621093750, 42724609375, 48828125000, 54931640625, 67138671875, 73242187500, 85449218750, 97656250000, 109863281250, 115966796875, 128173828125, 134277343750, 140380859375, 146484375000, 164794921875

Offset: 1

Hamdi Sahloul, Aug 04 2017

nonn

Numbers whose square is decomposable in 14 different ways into the sum of two nonzero squares: these are those with only one prime divisor of the form 4k+1 with multiplicity fourteen.

			a(1) = 6103515625 = 5^14, a(5) = 36621093750 = 2*3*5^14, a(101) = 1171875000000 = 2^6*3*5^14.

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 1000 terms from Hamdi Sahloul)

Cf. A002144, A006339, A046080, A046109, A083025.

Cf. A004144 (0), A084645 (1), A084646 (2), A084647 (3), A084648 (4), A084649 (5), A097219 (6), A097101 (7), A290499 (8), A290500 (9), A097225 (10), A290501 (11), A097226 (12), A097102 (13), A290503 (15), A097238 (16), A097239 (17), A290504 (18), A290505 (19), A097103 (22), A097244 (31), A097245 (37), A097282 (40), A097626 (67).

r[a_]:={b, c}/.{ToRules[Reduce[0Vincenzo Librandi, Mar 01 2016 *)

Terms are obtained by the product A004144(k)*A002144(p)^14 for k, p > 0 ordered by increasing values.

A290503 Hypotenuses for which there exist exactly 15 distinct integer triangles.

Original entry on oeis.org

30517578125, 61035156250, 91552734375, 122070312500, 183105468750, 213623046875, 244140625000, 274658203125, 335693359375, 366210937500, 427246093750, 488281250000, 549316406250, 579833984375, 640869140625, 671386718750, 701904296875, 732421875000

Offset: 1

Hamdi Sahloul, Aug 04 2017

nonn

Numbers whose square is decomposable in 15 different ways into the sum of two nonzero squares: these are those with only one prime divisor of the form 4k+1 with multiplicity fifteen.

			a(1) = 30517578125 = 5^15, a(5) = 183105468750 = 2*3*5^15, a(101) = 5859375000000 = 2^6*3*5^15.

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 1000 terms from Hamdi Sahloul)

Cf. A002144, A006339, A046080, A046109, A083025.

Cf. A004144 (0), A084645 (1), A084646 (2), A084647 (3), A084648 (4), A084649 (5), A097219 (6), A097101 (7), A290499 (8), A290500 (9), A097225 (10), A290501 (11), A097226 (12), A097102 (13), A290502 (14), A097238 (16), A097239 (17), A290504 (18), A290505 (19), A097103 (22), A097244 (31), A097245 (37), A097282 (40), A097626 (67).

r[a_]:={b, c}/.{ToRules[Reduce[0Vincenzo Librandi, Mar 01 2016 *)

Terms are obtained by the product A004144(k)*A002144(p)^15 for k, p > 0 ordered by increasing values.

A290504 Hypotenuses for which there exist exactly 18 distinct integer triangles.

Original entry on oeis.org

3814697265625, 7629394531250, 11444091796875, 15258789062500, 22888183593750, 26702880859375, 30517578125000, 34332275390625, 41961669921875, 45776367187500, 53405761718750, 61035156250000, 68664550781250, 72479248046875, 80108642578125, 83923339843750

Offset: 1

Hamdi Sahloul, Aug 04 2017

nonn

Numbers whose square is decomposable in 18 different ways into the sum of two nonzero squares: these are those with only one prime divisor of the form 4k+1 with multiplicity eighteen.

			a(1) = 3814697265625 = 5^18, a(5) = 22888183593750 = 2*3*5^18, a(101) = 732421875000000 = 2^6*3*5^18.

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 1000 terms from Hamdi Sahloul)

Cf. A002144, A006339, A046080, A046109, A083025.

Cf. A004144 (0), A084645 (1), A084646 (2), A084647 (3), A084648 (4), A084649 (5), A097219 (6), A097101 (7), A290499 (8), A290500 (9), A097225 (10), A290501 (11), A097226 (12), A097102 (13), A290502 (14), A290503 (15), A097238 (16), A097239 (17), A290505 (19), A097103 (22), A097244 (31), A097245 (37), A097282 (40), A097626 (67).

r[a_]:={b, c}/.{ToRules[Reduce[0Vincenzo Librandi, Mar 01 2016 *)

Terms are obtained by the product A004144(k)*A002144(p)^18 for k, p > 0 ordered by increasing values.

A290505 Hypotenuses for which there exist exactly 19 distinct integer triangles.

Original entry on oeis.org

203125, 265625, 406250, 453125, 531250, 578125, 609375, 640625, 796875, 812500, 828125, 906250, 953125, 1062500, 1140625, 1156250, 1218750, 1281250, 1359375, 1390625, 1421875, 1515625, 1578125, 1593750, 1625000, 1656250, 1703125, 1734375, 1765625, 1812500

Offset: 1

Hamdi Sahloul, Aug 04 2017

nonn

Numbers whose square is decomposable in 19 different ways into the sum of two nonzero squares: these are those with exactly two distinct prime divisors of the form 4k+1 with one, and six respective multiplicities, or with only one prime divisor of this form with multiplicity nineteen.

			a(1) = 203125 = 5^6*13, a(5) = 531250 = 2*5^6*17, a(281) = 12796875 = 3^2*5^6*7*13.

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 1000 terms from Hamdi Sahloul)

Cf. A002144, A006339, A046080, A046109, A083025.

Cf. A004144 (0), A084645 (1), A084646 (2), A084647 (3), A084648 (4), A084649 (5), A097219 (6), A097101 (7), A290499 (8), A290500 (9), A097225 (10), A290501 (11), A097226 (12), A097102 (13), A290502 (14), A290503 (15), A097238 (16), A097239 (17), A290504 (18), A097103 (22), A097244 (31), A097245 (37), A097282 (40), A097626 (67).

r[a_]:={b, c}/.{ToRules[Reduce[0Vincenzo Librandi, Mar 01 2016 *)

Terms are obtained by the products A004144(k)*A002144(p1)*A002144(p2)^6, or A004144(k)*A002144(p1)^19 for k, p1, p2 > 0 ordered by increasing values.

A054994 Numbers of the form q1^b1 * q2^b2 * q3^b3 * q4^b4 * q5^b5 * ... where q1=5, q2=13, q3=17, q4=29, q5=37, ... (A002144) and b1 >= b2 >= b3 >= b4 >= b5 >= ....

Original entry on oeis.org

1, 5, 25, 65, 125, 325, 625, 1105, 1625, 3125, 4225, 5525, 8125, 15625, 21125, 27625, 32045, 40625, 71825, 78125, 105625, 138125, 160225, 203125, 274625, 359125, 390625, 528125, 690625, 801125, 1015625, 1185665, 1221025, 1373125, 1795625

Offset: 1

Bernard Altschuler (Altschuler_B(AT)bls.gov), May 30 2000

This sequence is related to Pythagorean triples regarding the number of hypotenuses which are in a particular number of total Pythagorean triples and a particular number of primitive Pythagorean triples.
Least integer "mod 4 prime signature" values that are the hypotenuse of at least one primitive Pythagorean triple. - Ray Chandler, Aug 26 2004
See A097751 for definition of "mod 4 prime signature"; terms of A097752 with all prime factors of form 4*k+1.
Sequence A006339 (Least hypotenuse of n distinct Pythagorean triangles) is a subset of this sequence. - Ruediger Jehn, Jan 13 2022

			1=5^0, 5=5^1, 25=5^2, 65=5*13, 125=5^3, 325=5^2*13, 625=5^4, etc.

Ray Chandler, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Pythagorean Triple

Subsequence of A097752.

Cf. A002144, A006278, A006339, A071383, A097751, A097752, A097753, A097754, A097755, A097756.

maxTerm = 10^15;(* this limit gives ~ 500 terms *) maxNumberOfExponents = 9;(* this limit has to be increased until the number of reaped terms no longer changes *) bmax = Ceiling[ Log[ maxTerm]/Log[q]]; q = Reap[For[k = 0 ; cnt = 0, cnt <= maxNumberOfExponents, k++, If[PrimeQ[4*k + 1], Sow[4*k + 1]; cnt++]]][[2, 1]]; Clear[b]; b[maxNumberOfExponents + 1] = 0; iter = Sequence @@ Table[{b[k], b[k + 1], bmax[[k]]}, {k, maxNumberOfExponents, 1, -1}]; Reap[ Do[an = Product[q[[k]]^b[k], {k, 1, maxNumberOfExponents}]; If[an <= maxTerm, Print[an]; Sow[an]], Evaluate[iter]]][[2, 1]] // Flatten // Union (* Jean-François Alcover, Jan 18 2013 *)

list(lim)=
{
  my(u=[1], v=List(), w=v, pr, t=1);
  forprime(p=5,,
    if(p%4>1, next);
    t*=p;
    if(t>lim, break);
    listput(w,t)
  );
  for(i=1,#w,
    pr=1;
    for(e=1,logint(lim\=1,w[i]),
      pr*=w[i];
      for(j=1,#u,
        t=pr*u[j];
        if(t>lim, break);
        listput(v,t)
      )
    );
    if(w[i]^2Charles R Greathouse IV, Dec 11 2016

def generate_A054994():
    """generate arbitrarily many elements of the sequence.
    TO_DO is a list of pairs (radius, exponents) where
    "exponents" is a weakly decreasing sequence, and
    radius == prod(prime_4k_plus_1(i)**j for i,j in enumerate(exponents))
    An example entry is (5525, (2, 1, 1)) because 5525 = 5**2 * 13 * 17.
    """
    TO_DO = {(1,())}
    while True:
        radius, exponents = min(TO_DO)
        yield radius #, exponents
        TO_DO.remove((radius, exponents))
        TO_DO.update(successors(radius,exponents))
def successors(radius,exponents):
    # try to increase each exponent by 1 if possible
    for i,e in enumerate(exponents):
        if i==0 or exponents[i-1]>e:
            # can add 1 in position i without violating monotonicity
            yield (radius*prime_4k_plus_1(i), exponents[:i]+(e+1,)+exponents[i+1:])
    if exponents==() or exponents[-1]>0: # add new exponent 1 at the end:
        yield (radius*prime_4k_plus_1(len(exponents)), exponents+(1,))
from sympy import isprime
primes_congruent_1_mod_4 = [5] # will be filled with 5,13,17,29,37,...
def prime_4k_plus_1(i): # the i-th prime that is congruent to 1 mod 4
    while i>=len(primes_congruent_1_mod_4): # generate primes on demand
        n = primes_congruent_1_mod_4[-1]+4
        while not isprime(n): n += 4
        primes_congruent_1_mod_4.append(n)
    return primes_congruent_1_mod_4[i]
for n,radius in enumerate(generate_A054994()):
    if n==34:
        print(radius)
        break # print the first 35 elements
    print(radius, end=", ")
# Günter Rote, Sep 12 2023

Sum_{n>=1} 1/a(n) = Product_{n>=1} 1/(1 - 1/A006278(n)) = 1.2707219403... - Amiram Eldar, Oct 20 2020

More terms from Henry Bottomley, Mar 14 2001

A328151 a(n) is the smallest nonnegative integer k where exactly n ordered pairs of positive integers (x, y) exist such that x^2 + y^2 = k.

Original entry on oeis.org

0, 2, 5, 50, 65, 1250, 325, 31250, 1105, 8450, 8125, 19531250, 5525, 488281250, 105625, 211250, 27625, 305175781250, 71825, 7629394531250, 138125, 5281250, 126953125, 4768371582031250, 160225, 35701250, 1221025, 2442050, 3453125

Offset: 0

Felix Fröhlich, Oct 05 2019

a(n) is the smallest nonnegative i such that A063725(i) = n.

If a(n) exists, then a(n) is of the form 2*m^2 if and only if n is odd. - Chai Wah Wu, Jun 28 2024

			For n = 3: The sums of the two members of each of the pairs (1, 49), (25, 25) and (49, 1) is 50 and 50 is the smallest nonnegative integer where exactly 3 such pairs exist, so a(3) = 50.

Chai Wah Wu, terms a(n) for n odd and a(n) <= 3433037534088061250.

Cf. A063725, A093195.

a063725(n) = if(n==0, return(0)); my(f=factor(n)); prod(i=1, #f~, if(f[i, 1]%4==1, f[i, 2]+1, f[i, 2]%2==0 || f[i, 1]==2)) - issquare(n) \\ after Charles R Greathouse IV in A063725
a(n) = for(x=0, oo, if(a063725(x)==n, return(x)))

# uses Python code from A063725
from itertools import count
def A328151(n): return next(m for m in ((k**2<<1) if n&1 else k for k in count(0)) if A063725(m)==n) # Chai Wah Wu, Jun 28 2024

Conjecture: a(2k) = A093195(k) for k >= 1, a(2k+1) = 2*A006339(k)^2 for k >= 0. - Jon E. Schoenfield, Jan 23 2022

a(13)-a(22) from Bert Dobbelaere, Oct 20 2019

a(23)-a(28) from Chai Wah Wu, Jun 28 2024

A046112 a(n) is smallest integral radius of circle centered at (0,0) having 8n-4 lattice points on its circumference; a(n)/2 is smallest half-integral radius circle centered at (1/2,0) having 4n-2 lattice points; a(n)/3 is smallest third-integral radius circle centered at (1/3,0) having 2n-1 lattice points.

Original entry on oeis.org

1, 5, 25, 125, 65, 3125, 15625, 325, 390625, 1953125, 1625, 48828125, 4225, 1105, 6103515625, 30517578125, 40625, 21125, 3814697265625, 203125, 95367431640625, 476837158203125, 5525, 11920928955078125, 274625

Offset: 1

Eric W. Weisstein

nonn

Ray Chandler, Table of n, a(n) for n = 1..1439 (a(1440) exceeds 1000 digits).
Eric Weisstein's World of Mathematics, Circle Lattice Points
Eric Weisstein's World of Mathematics, Pythagorean Triple

Except for offset, same as A006339.

Cf. A046109, A046110, A046111, A000328.

A097756 Table read by rows of A054994 ordered by A046080.

Original entry on oeis.org

1, 5, 25, 125, 65, 625, 3125, 15625, 325, 78125, 390625, 1953125, 1625, 9765625, 48828125, 4225, 244140625, 1105, 8125, 1220703125, 6103515625, 30517578125, 40625, 152587890625, 21125, 762939453125, 3814697265625, 203125

Offset: 1

Ray Chandler, Aug 26 2004

Row n of table is A054994(k) such that A046080(A054994(k)) = n; number of terms in row n is A001055(2n+1).

Column 1 of table gives A006339 (or A046112).

			Table begins:
0: 1,
1: 5,
2: 25,
3: 125,
4: 65,625,
5: 3125,
6: 15625,
7: 325,78125,
8: 390625,
9: 1953125,
10: 1625,9765625,
11: 48828125,
12: 4225,244140625,
13: 1105,8125,1220703125,
14: 6103515625,
15: 30517578125,
16: 40625,152587890625,
17: 21125,762939453125,
18: 3814697265625,
19: 203125,19073486328125,
20: 95367431640625,
...

Ray Chandler, Table of n, a(n) for n = 1..4522, 1431 rows flattened.
Eric Weisstein's World of Mathematics, Pythagorean Triple

Cf. A097751, A097752, A097753, A097754, A097755, A054994.

cp's OEIS Frontend

A290500 Hypotenuses for which there exist exactly 9 distinct integer triangles.

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A290501 Hypotenuses for which there exist exactly 11 distinct integer triangles.

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A290502 Hypotenuses for which there exist exactly 14 distinct integer triangles.

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A290503 Hypotenuses for which there exist exactly 15 distinct integer triangles.

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A290504 Hypotenuses for which there exist exactly 18 distinct integer triangles.

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A290505 Hypotenuses for which there exist exactly 19 distinct integer triangles.

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A054994 Numbers of the form q1^b1 * q2^b2 * q3^b3 * q4^b4 * q5^b5 * ... where q1=5, q2=13, q3=17, q4=29, q5=37, ... (A002144) and b1 >= b2 >= b3 >= b4 >= b5 >= ....

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