Zhining Yang - cp's OEIS frontend

A387023 Integers w such that the Diophantine equation x^2 + y^3 + z^4 = w^5 where GCD(x,y,z)=1 has exactly 5 positive integer solutions.

9, 45, 70, 80, 120, 124, 125, 128, 133, 143, 170, 175, 180, 195, 201, 220, 224, 236, 252, 264, 275, 278, 296, 308, 311, 312, 330, 332, 336, 337, 352, 354, 355, 360, 362, 366, 374, 375, 380, 386, 390, 394, 399, 404, 411, 416, 418, 428, 430, 444, 461, 466, 477, 484, 488, 500

Offset: 1

Zhining Yang, Aug 13 2025

			444 is in the sequence because 444^5 = x^2 + y^3 + z^4 where GCD (x, y, z) = 1 has exactly 5 positive integer solutions: {676786, 25603, 343}, {342332, 25775, 345}, {4123199, 5503, 544}, {2451712, 21919, 919}, {3889117, 679, 1208}.

Zhining Yang, Table of n, a(n) for n = 1..328

Cf. A386373, A386377, A386521.

Do[w5=w^5;s={};c=0;
Do[yy=w5-z^4;Do[xx=yy-y^3;x=Sqrt@xx;
If[IntegerQ@x,If[GCD[x,y,z]==1,c++;AppendTo[s,{x,y,z}]]],{y,Floor[yy^(1/3)]}],{z,Floor[w5^(1/4)]}];
If[c==5,Print[w,s]],{w,100}]

A386521 Integers w such that the Diophantine equation x^2 + y^3 + z^4 = w^5 with GCD(x,y,z)=1 has no positive integer solutions.

Original entry on oeis.org

1, 3, 4, 5, 6, 10, 13, 22, 27, 34, 36, 42, 43, 47, 62, 72, 76, 87, 95, 102, 111, 183, 251, 279, 315, 322, 327, 344, 483, 490, 528, 615, 708, 762, 1170, 1302, 2295, 2526, 3282, 3382, 6012

Offset: 1

David A. Corneth and Zhining Yang, Jul 24 2025

a(42) > 6500. - Giovanni Resta, Aug 12 2025

			9 is not a term because 9^5 = x^2 + y^3 + z^4 where GCD(x,y,z)=1 has 5 positive integer solutions: {220,22,1}, {64,38,3}, {241,7,5}, {9,38,8}, {118,29,12}.

Cf. A386373, A386377.

f[w_]:=(c=0;zz=w^5;Do[yy=zz-z^4;Do[xx=yy-y^3;x=Sqrt@xx;
If[IntegerQ@x,If[GCD[x,y,z]==1,c++]],{y,Floor[yy^(1/3)]}],{z,Floor[zz^(1/4)]}];c);Select[Range@50,f@#==0&]

a(41) from Giovanni Resta, Aug 12 2025

A386377 a(n) is the number of solutions to the equation x^2 + y^3 + z^4 = w^5 where GCD(x, y, z)=1.

Original entry on oeis.org

0, 1, 0, 0, 0, 0, 2, 2, 5, 0, 1, 1, 0, 1, 1, 1, 3, 2, 1, 2, 2, 0, 2, 2, 4, 1, 0, 2, 2, 1, 2, 1, 13, 0, 2, 0, 1, 3, 1, 1, 4, 0, 0, 7, 5, 3, 0, 2, 10, 1, 1, 2, 7, 2, 1, 1, 8, 1, 2, 1, 7, 0, 4, 3, 8, 4, 4, 1, 1, 5, 1, 0, 11, 1, 2, 0, 3, 1, 3, 5, 12, 7, 2, 2, 2, 2, 0, 1, 14, 2, 2, 1

Offset: 1

David A. Corneth and Zhining Yang, Jul 20 2025

nonn

			a(9) = 5 because x^2 + y^3 + z^4 = 9^5 where GCD(x,y,z)=1 has 5 positive integer solutions :{220,22,1},{64,38,3},{241,7,5},{9,38,8},{118,29,12}.

Zhining Yang, Table of n, a(n) for n = 1..4500 (terms 1..856 from David A. Corneth)
David A. Corneth, Tuples (x, y, z, w) that are solutions to the equation.
David A. Corneth, PARI program

Cf. A386373, A386521.

f[w_]:=(c=0;zz=w^5;Do[yy=zz-z^4;Do[xx=yy-y^3;x=Sqrt@xx;
If[IntegerQ@x,If[GCD[x,y,z]==1,c++]],{y,Floor[yy^(1/3)]}],{z,Floor[zz^(1/4)]}];c);Array[f@#&, 30]

A386988 a(n) is the smallest integer w such that the equation x^2 + y^4 + z^6 = w^8 where GCD(x,y,z)=1 has exactly n positive integer solutions.

Original entry on oeis.org

25, 9, 17, 53, 3

Offset: 1

Zhining Yang, Aug 12 2025

a(6)>1024.

			a(3) = 17 because 17^8 = 36840^2 + 273^4 + 20^6 = 82367^2 + 24^4 + 24^6 = 48^2 + 287^4  + 24^6 and for no integer smaller than 17 we have 3 solutions.

Fang, How to find non-trivial positive integer solutions to the Diophantine equation a^2 + b^4 + c^6 = d^8, where gcd(a,b,c) = 1?.

Cf. A386373.

f[w_]:=(v={};c=0;w8=w^8;
Do[yy=w8-z^6;Do[xx=yy-y^4;x=Sqrt@xx;
If[IntegerQ@x,If[GCD[x,y,z]==1,AppendTo[v,{x,y,z}];c++]],{y,Floor[yy^(1/4)]}],{z,Floor[w8^(1/6)]}];{c,w,v});
s=Table[{},5];
For[k=1,k<=60,k++,r=f[k][[1]];If[s[[r]]=={},s[[r]]=f[k];Print[s[[r]]]]]

A386373 a(n) is the smallest integer w such that the equation x^2 + y^3 + z^4 = w^5 where GCD(x,y,z)=1 has exactly n positive integer solutions.

Original entry on oeis.org

2, 7, 17, 25, 9, 100, 44, 57, 117, 49, 73, 81, 33, 89, 177, 193, 305, 161, 257, 273, 425, 289, 697, 441, 313, 689, 369, 593, 809, 233, 761, 1865, 2001, 857, 1121, 649, 1353, 865, 521, 1257, 577, 681, 2081, 1409, 1169, 1753, 1801, 1201, 1745, 2833, 3853, 3649, 3353, 1305, 793

Offset: 1

Zhining Yang, Jul 19 2025

nonn

From David A. Corneth, Jul 20 2025: (Start)
a(41) = 577. If a(41) is 1 (mod 8) then that values is exact.
For 10 <= n <= 30 we have a(n) == 1 (mod 8).
Heuristically this is no coincidence. There are 8^3 = 512 tuples (x, y, z) mod 8. The frequencies of k (mod 8) for x^2 + y^3 + z^4 for k = 0 through 7 are 64, 128, 96, 32, 64, 64, 32, 32 respectively. So 1 (mod 8) has the single largest value at 128 such tuples.
Extending this to other moduli like 56 we get the largest frequencies (7168) come from 9, 17, 25 and 33 (mod 56).
The second largest frequency is 6272 which occurs at 49 (mod 56). For n = 3, 4 and 10 <= n <= 20, 22, 30 we have a(n) == 9, 17, 25, 33 or 49 (mod 56). (End)

			a(4) = 25 because 25^5 = 1852^2 + 185^3 + 8^4 = 2711^2 + 134^3 + 10^4 = 2472^2 + 150^3 + 23^4 = 2973^2 + 15^3 + 31^4 and no integer less than 25 has 4 solutions.

Zhining Yang, Table of n, a(n) for n = 1..70

Cf. A386377, A386521.

f[w_]:=(v={};c=0;nn=w^5;
Do[yy=nn-z^4;Do[xx=yy-y^3;x=Sqrt@xx;
If[IntegerQ@x,If[GCD[x,y,z]==1,AppendTo[v,{x,y,z,d}];c++]],{y,Floor[yy^(1/3)]}],{z,Floor[nn^(1/4)]}];{c,w,v});
s=Table[{},20];
For[k=1,k<=100,k++,r=f[k][[1]];If[s[[r]]=={},s[[r]]=g[k];Print[s[[r]]]]]

a(21)-a(31) from David A. Corneth, Jul 20 2025

a(32)-a(58) from Zhining Yang, Jul 31 2025

A385976 Least integer k such that there are exactly n primitive Heron triangles having the same area and perimeter k.

Original entry on oeis.org

12, 70, 98, 2002, 2842, 20026, 91698, 721786

Offset: 1

Zhining Yang, Jul 13 2025

			a(3) = 98 because there exists 3 primitive Heron triangles: {{29, 29, 40}, {25, 34, 39}, {24, 37, 37}} with same area 420 and same perimeter 98.
a(6) = 20026  because there exists 6 primitive Heron triangles: {{2108,8493,9425}, {2173,8398,9455}, {2261,8277,9488},{2418,8075,9533},{4123,6205,9698},{4588,5729,9709}} with same area 8410920 and same perimeter 20026.
a(8) = 721786  because there exists 8 primitive Heron triangles: {{188105,189428,344253}, {179133,198458,344195},{124338,256523,340925}, {116093,266029,339664}, {91448,299013,331325}, {88253,304980,328553}, {85981,310493,325312}, {85618,311627,324541}} with same area 13338605280 and same perimeter 721786.

BBS Math Blog, Primitive Heron triangles with equal perimeter and area (in Chinese).

Cf. A385819.

sol=Association[];
For[n=6,n<=3000,n+=2,For[z=Ceiling[n/3],zz>=y>=x&&GCD[x,y,z]==1,p=(x+y+z)/2;A=Sqrt[p (p-x) (p-y) (p-z)];
If[IntegerQ[A],d=ToString@n<>"->"<>ToString@A;t={x,y,z};
If[KeyExistsQ[sol,d]==False,sol[d]={d}];
If[KeyExistsQ[sol,d],AppendTo[sol[d],t]]]]]]];
Do[Print[k,SelectFirst[sol,Length@#==k+1&]],{k,5}]

from math import gcd
from collections import Counter
from itertools import count
from sympy.ntheory.primetest import is_square
def A385976(n):
    for k in count(6,2):
        c = Counter()
        for x in range(1,k//3+1):
            for y in range(x,k//2+1):
                s, m = k>>1, x+y
                if  (m<<1)>k>=m+y and gcd(x,y,k-m)==1 and is_square(s*(a:=(s-x)*(s-y)*(m-s))):
                    c[a]+=1
                    if c[a]>n:
                        break
            else:
                continue
            break
        else:
            if any(c[d]==n for d in c):
                return k # Chai Wah Wu, Jul 25 2025

a(7)-a(8) from Xianwen Wang, Jul 13 2025

a(3) corrected by Xianwen Wang, Aug 19 2025

A385819 Numbers k such that there are least five primitive Heron triangles having the same area and perimeter k.

Original entry on oeis.org

2842, 3542, 5642, 5750, 6314, 7238, 7546, 9790, 15470, 15778, 17710, 20026, 21658, 21970, 22610, 26962

Offset: 1

Zhining Yang, Jul 09 2025

			3542 is a term because there exists 5 primitive Heron triangles: {{421,1518,1603}, {511,1375,1656}, {583,1288,1671},{759,1096,1687}, {851,1001,1690}} with same perimeter 3542 and same area 318780.
20026 is a term because there exists 6 primitive Heron triangles: {{2108,8493,9425}, {2173,8398,9455}, {2261,8277,9488}, {2418,8075,9533}, {4123,6205,9698}, {4588,5729,9709}} with same perimeter 20026 and same area 8410920.

BBS Math Blog, Primitive Heron triangles with equal perimeter and area (in Chinese).

Cf. A051516, A070138, A188158

sol = Association[];
For[n = 2, n <= 6000, n += 2,
For[z = Ceiling[n/3], z < Floor[n/2], z++,
For[x = 1, x < Floor[n/3], x++, y = n - x - z;
   If[x + y > z > y > x && GCD[x, y, z] == 1, p = (x + y + z)/2;
    A = Sqrt[p (p - x) (p - y) (p - z)];
    If[IntegerQ[A], d = ToString@n <> "->" <> ToString@A; t = {x, y, z};
     If[KeyExistsQ[sol, d], AppendTo[sol[d], t], sol[d] = {t}]]]]]];
Select[sol, Length@# > 4 &]

A385709 Least prime p such that the decimal expansion of p^2 contains exactly n distinct primes as substrings.

Original entry on oeis.org

11, 5, 23, 61, 73, 239, 487, 523, 569, 3461, 1319, 3373, 8923, 4937, 12619, 11489, 15569, 32189, 105173, 135319, 46619, 56473, 177127, 234161, 295861, 471923, 664319, 2366387, 3183613, 1092389, 3513877, 7702319, 4632077, 10666177, 13977923, 20825939, 35821939

Offset: 1

Zhining Yang, Jul 07 2025

			a(9) = 569 because 569^2 = 323761, which contains 9 distinct primes as substring:{2,3,7,23,37,61,761,3761,23761}, and no prime less than 569 has 9 solutions.

Zhining Yang, Table of n, a(n) for n = 1..66

Cf. A018834, A046837.

b = Table[{}, 9]; Do[d = IntegerDigits[p^2];
 t = Union@Select[FromDigits /@ Flatten[Table[Partition[d, k, 1], {k, Length@d}], 1], PrimeQ]; c = Length@t;
 If[b[[c]] == {}, b[[c]] = {p, p^2, t, c}], {p, Prime@Range@120}]; b // Grid

A385724 The least integer of n consecutive numbers where each has its sum of prime factors, with multiplicity, being a prime.

Original entry on oeis.org

17, 2, 5, 10, 1547, 8837, 1293224, 52445796, 3267037, 896531141, 183208285259

Offset: 1

Zhining Yang, Jul 08 2025

According to Primepuzzles, puzzle 445:
a(10) = 896531141 given by Carlos Rivera.
a(11) = 183208285259 given by Jens Kruse Andersen.

			a(4) = 10 because 10=2*5, 11=11, 12 = 2^2*3, 13 = 13 and sum of prime factors of the four consecutive numbers are all primes : 2 + 5 = 7, 11, 2 + 2 + 3 = 7, 13.

Fang, Can we find that the sum of prime factors of each of twelve consecutive positive integers is a prime number?
G. L. Honaker, Jr., Prime Curio 183208285259.
Carlos Rivera, Puzzle 445. Consecutive integers with prime SPF, The Prime Puzzles & Problems Connection.

Cf. A100118, A337310.

f[n_] := PrimeQ@Total[Times @@@ FactorInteger[n]]; a = Table[0, 6];
c = 0; Do[If[c == 0, s = n]; If[f[n] == False, If[a[[c]] == 0, a[[c]] = s]; c = 0, c++], {n, 2, 10000}]; a

A385409 a(n) is the smallest positive integer k such that the Diophantine equation x^3 + y^3 + z^3 + w^3 = k^2, where 0 < x < y < z < w has exactly n integer solutions.

Original entry on oeis.org

10, 42, 39, 153, 126, 276, 273, 312, 315, 476, 588, 336, 546, 777, 1053, 756, 1216, 1386, 1560, 1134, 1323, 1488, 1365, 1368, 1344, 1596, 2366, 2496, 2988, 1680, 2548, 1736, 2184, 3003, 3720, 2520, 3185, 3552, 2268, 3564, 4095, 3213, 4578, 4392, 5208, 4004, 4599, 5733

Offset: 1

Zhining Yang, Jun 27 2025

nonn

Conjecture: a(n) exists for all n.

			a(4)=153, because 153^2 = 5^3 + 15^3 + 21^3 + 22^3 = 2^3 + 7^3 + 15^3 + 27^3 = 6^3 + 8^3 + 9^3 + 28^3 = 1^3 + 5^3 + 11^3 + 28^3 and no integer less than 153 has 4 solutions.

Zhining Yang, Table of n, a(n) for n = 1..1341

Cf. A024975, A025419, A377444, A384430, A385354.

s = Table[{k, Length@Select[PowersRepresentations[k^2, 4, 3],
      0 < #[[1]] < #[[2]] < #[[3]] < #[[4]] &]}, {k, 500}];
a = Table[SelectFirst[s, #[[2]] == k &], {k, 10}][[All, 1]]

cp's OEIS Frontend

User: Zhining Yang

A387023 Integers w such that the Diophantine equation x^2 + y^3 + z^4 = w^5 where GCD(x,y,z)=1 has exactly 5 positive integer solutions.

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A386521 Integers w such that the Diophantine equation x^2 + y^3 + z^4 = w^5 with GCD(x,y,z)=1 has no positive integer solutions.

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A386377 a(n) is the number of solutions to the equation x^2 + y^3 + z^4 = w^5 where GCD(x, y, z)=1.

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A386988 a(n) is the smallest integer w such that the equation x^2 + y^4 + z^6 = w^8 where GCD(x,y,z)=1 has exactly n positive integer solutions.

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A386373 a(n) is the smallest integer w such that the equation x^2 + y^3 + z^4 = w^5 where GCD(x,y,z)=1 has exactly n positive integer solutions.

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A385976 Least integer k such that there are exactly n primitive Heron triangles having the same area and perimeter k.

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A385819 Numbers k such that there are least five primitive Heron triangles having the same area and perimeter k.

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A385709 Least prime p such that the decimal expansion of p^2 contains exactly n distinct primes as substrings.

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